Replacement in Logic

J Philos Logic (2013) 42:49–89DOI 10.1007/s10992-011-9212-4

Replacement in Logic

Lloyd Humberstone

Received: 8 December 2010 / Accepted: 17 August 2011 / Published online: 7 September 2011© Springer Science+Business Media B.V. 2011

Abstract We study a range of issues connected with the idea of replacing oneformula by another in a fixed (linguistic) context. The replacement core ofa consequence relation � is the relation holding between a set of formulas{A1, . . . , Am, . . .} and a formula B when for every context C(·), we haveC(A1), . . . , C(Am), . . . � C(B). Section 1 looks at some differences betweenwhich inferences are lost on passing to the replacement cores of the classicaland intuitionistic consequence relations. For example, we find that while theinference from A and B to A ∧ B, sanctioned by both these initial conse-quence relations, is retained on passage to the replacement core in the classicalcase, it is lost in the intuitionistic case. Further discussion of these two (andsome other) logics occupies Sections 3 and 4. Section 2 looks at the m = 1 case,describing A as replaceable by B according to � when B is a consequence ofA by the replacement core of �, and inquiring as to which choices of � renderthis induced replaceability relation symmetric. Section 5 investigates furtherconceptual refinements— such as a contrast between horizontal and verticalreplaceability—suggested by some work of R. B. Angell and R. Harrop (anda comment on the latter by T. J. Smiley) in the 1950s and 1960s. Appendix 1examines a related aspect of term-for-term replacement in connection withidentity in predicate logic. Appendix 2 is a repository for proofs which wouldotherwise clutter up Section 3.

Keywords Consequence relations · Classical logic · Intuitionistic logic ·Modal logic · Identity · Contexts · Replacement

L. Humberstone (B)Department of Philosophy, Monash University, Clayton, VIC 3800, Australiae-mail: [email protected]

50 L. Humberstone

1 Introduction

The present study of various replacement-related phenomena in logic—mainly concerning formula-by-formula replacement in propositional logic, butconsidering term-by-term replacement in predicate logic (with identity) inAppendix 1—began with the simple observation of a difference, not widelyremarked on, between modal and non-modal classical propositional logic.For the sake of recalling this observation we treat logics as sets of formulas,passing later to a treatment of logics as consequence relations or—in passing—generalized (i.e., ‘multiple-conclusion’) consequence relations. Modal logicsbased on a language supplementing some functionally complete set of booleanprimitives by the additional 1-ary connective �,1 and stipulated to containall truth-functional tautologies and be closed under uniform substitution andModus Ponens are sometimes (e.g. [44]) said to have the aggregation propertywhen for all formulas A, B, they contain the formula

(�A ∧ �B) → � (A ∧ B) .

For example, the class of regular modal logics comprises those with thisproperty which are also monotone in the sense that �A → �B is in the logicwhenever A → B is. More generally, we will want to speak of aggregative‘contexts’. For this purpose we take a formula we denote by C(p), and call acontext, which may—but need not—contain the propositional variable p, andmay also contain occurrences of further propositional variables. We denoteby C(A) the result of substituting A uniformly for p in C(p); we often writeC(p) as C(·) to emphasize the idea of splicing the formula A into the context.We now define a context C(p) to be aggregative according to a logic if for anyformulas A, B, the logic proves (i.e. contains) the formula

(C(A) ∧ C(B)) → C (A ∧ B) .

The context C(p) = �p is the special case used in defining the aggregationproperty above.

The definition given in the previous paragraph is of a 1-ary context. Forthe n-ary case, we take C(p1, . . . , pn)—in which, again, variables other thanthose displayed are allowed to occur, not all those displayed are requiredto occur, and the result of splicing formulas A1, . . . , An into the context isthe substitution instance C(A1, . . . , An). We mostly need the n = 1 case in

1What is meant here by “functionally complete set of boolean primitives” is a set: of connectivesfor which the set of truth-functions associated with them on all boolean valuations is functionallycomplete. By a boolean valuation is meant an assignment of truth-values T, F, to formulas whichrespects the usual truth-table conditions: assigning T to ¬A iff it assigns F to A, etc. We use ∧, ∨,→, ¬, for the connectives conjunction, disjunction, implication and negation, resp.

Replacement in Logic 51

what follows, and use the term context with the default understanding of 1-arycontext.2 The informal talk of distinguished variables can be made precise invarious ways, such as by identifying the context as the n-tuple 〈C, p1, . . . , pn〉,as is done for the n = 1 case in Williamson [54] (or [55]). To enjoy theconvenience of identifying contexts with formulas rather than with such tuples,while still leaving no doubt as to which variables are to be substituted for,let us take it that the propositional variables come in two disjoint countablesequences, p1, p2, . . . , pn, . . . and q1, q2, . . . , qn, . . ., reserving the pi as gap-markers for contexts and the qi as the propositional variables not serving inthat capacity, with the additional conventions that p1 is abbreviated to p, andq1, q2, to q, r. The separating out of gap-marking variables (pi) is purely fornotational convenience; for all purposes other than deciphering metalinguisticnotation they have the same significance as the other propositional variables(qi). In particular, when a consequence relation is described as substitution-invariant (and only such consequence relations will be under considerationhere), this imposes the requirement that uniform substitution of arbitraryformulas for variables applies no less to the qi and to the pi.

Returning to the 1-ary case, the reason that we do not actually require p tooccur in C(p), that is) is that in various inductive arguments—as in the proofof Lemma 2.2—we need to consider the cases in which C(p) is disjunctive,for example, rewriting it accordingly as C0(p) ∨ C1(p) and would not wantto handle separately a plethora of further subcases arising as to whether poccurs in this or that disjunct or in both. Of course, when p does not occurin C(p), C(A) is just C(p) = C; we call C(p) an improper context in thiscase—otherwise C(p) is a proper context. This means that, considered as afunction from formulas to formulas, C(·) is a constant function in these cases,so we encounter here one disadvantage of the present policy as compared withWilliamson: it conflates the distinction between a constant function and thevalue constantly assumed by that function.

Slightly more generally, whenever p is the only variable occurring in C(p),the context amounts to a primitive or derived 1-ary connective of the language,and it is a familiar fact that whereas if � enjoys the monotone property abovein a modal logic, so does the derived connective ♦, with ♦A = ¬�¬A, it isnot the case that aggregativity is similarly inherited by ♦ from �. Indeed if weare informally reading ♦ as expressive of some kind of possibility, we typicallydo not want ♦ to be aggregative according to any modal logic plausible forsuch an interpretation (with � as the correlative notion of necessity)—and♦ is not aggregative according to S5 or therefore according to any weakerlogic. Thus these logics provide non-aggregative contexts, and it is here that theobservation alluded to arises: in classical non-modal propositional logic every

2We shall avoid using the schematic letter “C” for anything other than such contexts, with in givingthe axiom schemes for BCI in Section 2 (and for the implication connective in some Polish notationquoted in note 4).

52 L. Humberstone

context is aggregative.3 The aggregativity of all such contexts was perhaps firstexplicitly noted by Łukasiewicz, who proves it in the course of an axiomaticdevelopment of his Ł-modal logic, as Theorem 60 (on p. 383 of [30]);4 the proofgiven below is, by contrast, semantic. Unfortunately not only boolean but alsomodal contexts emerge as aggregative in the Ł-modal system, including thecase of C(p) = ♦p we have just touched on—a feature many have objected toin the literature (e.g., references [14] and [15] from the bibliography of Smiley[49]; see Font and Hájek [12] for more recent discussion and references).

Let us rephrase this point about the aggregativity of all contexts inclassical (non-modal) propositional logic in terms of the consequence re-lation concerned in the latter case, �CL.5 For any context C(p) and anyformulas A, B:

C(A), C(B) �CL C (A ∧ B) .

The justification for this claim is that for any boolean valuation v—anassignment of truth-values T, F, to formulas respecting the usual truth-tableconditions (v(A ∧ B) = T iff v(A) = v(B) = T, and so on) with v(C(A)) =v(C(B)) = T we must also have C(A ∧ B) = T, since C(p) induces a functionf (x, y1, . . . , yn), where v(p) = x and v(qi) = yi for 1 ≤ i ≤ n, with—for sim-plicity, but without loss of generality—q1, . . . , qn the propositional variables

3This difference underlies the difference expressed at p. 578f. of [20] in terms of the failure ofnormal modal logics in general, by contrast with classical non-modal propositional logic, to beclosed under what is there called ‘internal’ uniform substitution. Classical (non-modal) predicatelogic also presents us, thanks to “∃”, with non-aggregative contexts and corresponding failuresof internal uniform substitution, once the notion of a context is redefined suitably to cover theapparatus of variable-binding.4Only after the present paper was substantially written did the author realise that the conclusion,arrived at on semantic grounds, that all contexts in classical propositional logic were aggregative,was already to be found as one of the buried treasures in the appendix of [30]. The formulaproved at this point is, as Łukasiewicz writes it in Polish notation: CδpCδqCδKpq, where δ is a‘variable functor’ (in the object language). Similarly, Łukasiewicz’s version of Proposition 1.1(ii)below appears in Theorem 30 of [30], where the formula concerned is written as CδCpqCδpδq.5We assume familiarity with the notion of a consequence relation here, and note that all theconsequence relations to be considered will be what are sometimes called standard, meaning: bothfinitary and substitution-invariant (or ‘structural’). To complete the contrast in the text betweenthe modal and non-modal cases, we invoke the (local) consequence relation �S4 associated withS4—taken here and elsewhere merely as one of normal modal logics which serve to illustratethe point at hand—and put the above observation as: the context C(p) = ♦p is not aggregativeaccording to �S4. (Take the term local here to mean that � �S4 A when A can be obtainedby repeated applications of Modus Ponens to formulas beginning with those in � or among thetheorems of S4, as opposed to ‘global’, which would allow applications of Necessitation alongsidethose of Modus Ponens. The terms have a semantic etymology from preservation of truth at anarbitrary point, in the former case, and at all points in the latter, in Kripke models on suitableframes.) We use the customary abbreviations deployed in connection with consequence relations,writing, e.g., “A1, A2 � B” for “{A1, A2} � B” and “� B” for “∅ � B”.


other than p occurring in C(p), for which v(C(A)) = f (v(A), y1, . . . , yn). Forbrevity we write this as f (v(A), yi). Now suppose f (v(A), yi) = f (v(B), yi) =T, with a view to showing that f (v(A ∧ B), yi) = T. If v(A) = v(B) = x thenv(A ∧ B) = x—by the booleanness condition as it pertains to ∧ (specifically,the idempotence of the associated truth-function)—so since f (v(A), yi) =T, f (v(A ∧ B), yi) = T. On the other hand, if v(A) �= v(B) then at leastone of v(A), v(B)—v(A), say —is F, and also v(A ∧ B) = F. So fromthe fact that f (v(A), yi) = T, we know that f (F, yi) = T, and thus thatf (v(A ∧ B), yi) = T.

We can formulate the observation about aggregativity more concisely andas well as more suggestively with the aid of the following notation, inspired bythe conventional distinction between the relation of C(p) to C(A), the latterarising from the former by (uniform) substitution of A for p, and the relationbetween C(A) and C(B), the latter arising from the former by (not necessarilyuniform) replacement of A by B. (“Not necessarily uniform”, because thecontext C(p) may already contain occurrences of A as subformulas and theseare not replaced in the transition from C(A) to C(B). We return to thisuniformity issue in Section 2.6) Given a consequence relation � on a languagewith L as its set of formulas, let the replacement core—more explicitly onemight say the “omni-contextual replacement core”—�rep, of �, be definedthus, for all B ∈ L, � ⊆ L:

� �rep B ⇔ for every context C(p) ∈ L: {C(A) | A ∈ �} � C(B).

We call this the replacement ‘core’ of � to reflect the fact that we alwayshave �rep ⊆ �, in view of replacement in the (‘null’) context C(p) = p.7 Let usexpress in this new notation what was shown above, as part (i) of the following;a similar argument establishes part (ii) (or: see the ‘quick check’ described inSection 3, Proposition 3.1):

6The same terminological distinction between substitution and replacement is also standard inequational logic—as in Henkin [16, p. 599], or Burris [8, p. 175], for example—with individualvariables in place of propositional variables and arbitrary terms for formulas; we return to thecase of replacement licensed by “=” in Appendix 1 below, though in the setting of general first-order logic rather than specifically equational logic. The terminology is not always used this way,however: Bonnay and Westerståhl (in [7] and elsewhere) use replacement for the uniform case—though not quite in the sense of substitution, since only atomic expressions are substituted forother atomic expressions of the same syntactic type. In later sections we shall also consider certainkinds of uniform replacement, still not counting as substitutions because the expression replacedis not required to be a variable. Finally, let us note an alternative to the present use of thereplacement: when the author was learning logic as an undergraduate, the class was admonishednever to confuse closure under uniform substitution with the substitutivity of equivalents.7Compare the notion of the substitution core of a logic, from van Benthem [53], comprising theformulas (or more generally sequents) all of whose substitution instances belong to the logic, whichgives a sublogic, and indeed a proper sublogic when the original logic is not substitution-invariant.

54 L. Humberstone

Proposition 1.1

(i) For any formulas A, B, we have: A, B �repCL A ∧ B.

(ii) For any formulas A, B, we have: A, A → B �repCL B.

We regard the provision of such succinct formulations as the main meritof the �rep notation, though an informal reading (suggested by a referee)of such things as “A, B �rep

CL A ∧ B” along the lines of: “the inference fromA, B to A ∧ B is safe within an arbitrary context, by the lights of classicalpropositional logic” may be heuristically helpful. There are certainly manyinteresting relationships of replacement and contexts which this notation doesnot cover. For example, we may want record the fact that for certain formu-las A1, . . . , Am, B1, . . . , Bn, D and a given consequence relation � we haveA1, . . . , Am, C(B1), . . . , C(Bn) � C(D) for all contexts C(p), which the ‘re-placement core’ idea cannot convey because the Ai are not context-embedded.In Section 5 below, for example, we shall be recalling (R. B. Angell’s ‘ruleof excision’) that with m = n = 1 and A1 = q, B1 = q → r and D = r, thisrelationship obtains for � = �CL.

A statement dual to Proposition 1.1(i) can be made if we adopt a similarapproach to the generalized consequence relation of classical logic, i.e., thatrelation �CL between pairs of sets of formulas such that for all �, �: � �CL �

iff there is no boolean valuation v with v(A) = T for all A ∈ � and v(B) = Ffor all B ∈ �. Then with a definition of the mapping (·)rep as above but for thegeneralized case, we could similarly observe that for arbitrary formulas A, B:

A ∨ B �repCL A, B.

We touch on the case of generalized consequence relations in Section 3, butmostly continue to concentrate on consequence relations themselves.

The case of Proposition 1.1(ii) brings out a point worthy of comment, giventhe customary association for �CL (and �IL) between commas on the left of“�” and applications of ∧. This association is broken when it comes to thereplacement cores of such consequence relations, because in the replacementcore we consider each formula on the left separately embedded into variouscontexts C(·). For example, whereas, as Proposition 1.1(ii) tells us, q, q →r �rep

CL r, it is clear—consider embedding under negation—that, by contrast,q ∧ (q → r) �

repCL r. (Dualizing this example, we see that commas on the right in

the generalized consequence relation case cannot be traded in for disjunctions:q �rep

CL r, q ∧ ¬r, while q �repCL r ∨ (q ∧ ¬r).8)

We saw with the example of ♦ that the analogue of Proposition 1.1(i)typically fails in modal logic (as does part (ii), e.g., for �S4 again, taking

8This is not surprising in view of the fact that although A ∨ B �repCL A, B, we do not in general have

A �repCL A ∨ B or B �rep

CL A ∨ B, just as in the consequence relation—or generalized consequencerelation—case, we lose the inference from conjunctions to conjuncts on passage to the replacementcore.


C(p) = ♦p); given the translations of intuitionistic logic into S4, this suggeststhat it might also fail for the consequence relation �IL of intuitionistic logic.

Proposition 1.2 Both (i) and (ii) of Proposition 1.1 fail for �repIL .

Proof The failure of (i) is addressed under Example 3.11 of [22], in which C(p)

is the formula (p → q) → r; the discussion there explains why, as we may putit in the present notation, we do not in general have:

(A → q) → r, (B → q) → r �IL ((A ∧ B) → q) → r.

For (ii) the same context works to give a counterexample. Suppose that for anarbitrarily selected pair of formulas A, B, we had

((A → B) → q) → r, (A → q) → r �IL (B → q) → r.

Without loss of generality, we can assume that q and r do not occur in A, B.(If they do, re-letter A, B, to A′, B′, to satisfy this condition, and re-letterthem back again at the end of the proof which follows.) Then we can substitute((A → B) → q) ∨ (A → q) for r, rendering the left-hand formulas IL-provableand delivering

�IL (B → q) → (((A → B) → q) ∨ (A → q)) .

Finally, substituting B for q and detaching, we would get

�IL ((A → B) → B) ∨ (A → B)),

which is well known not to be the case for arbitrary A, B, the schema hereindicated being a variant on Dummett’s LC axiom. ��

Two points deserve mention concerning this proof. First: as to whether thechoice of C(p) used for both halves of the above proof can be simplified—e.g., by reducing the total number of variables of which there are occurrencesin C(p)—the author is unsure; certainly, à propos of the failure of (i), wecannot replace the occurrence of r with a second occurrence of q, or witha second occurrence of p, in the current C(p), as the resulting contexts areintuitionistically aggregative. (This suggests that one might usefully consideran alternative notion of the replacement core of �, defined as above butwith C(p) restricted to formulas in which no variables other than p occur.(This means we can regard C(p) as a derived 1-ary connective.) Though thismakes no difference for � = �CL—see Section 4—it may well yield a propersubrelation of the current �rep

IL .9) Secondly, the description of the schema atthe end of the proof as a variant on the LC schema is correct as long as we

9Another line of inquiry that may be worth pursuing was suggested by Greg Restall: obtaining areplacement core which looks more natural as a relation of logical consequence by quantifyingover monotone contexts rather than arbitrary contexts. (Actually Restall said “positive” contexts,so I am reformulating the suggestion a little here.) Related considerations are aired in Zeman [56].

56 L. Humberstone

are working with all the customary intuitionistic primitives, though in the ∧-free fragment (primitives: →, ∨, ¬) the given schema is strictly weaker thanthe LC schema—by an adaptation in Humberstone [21, p. 250], of an argumentfrom Prior [38]. (A semantic analysis of the weaker-than-LC logic involvedhere is provided in Section 6 of [25].) We return to �IL and the examples ofProposition 1.2 in Section 4, closing this section with another example in asimilar vein and a simple general observation to the effect that the operation(·)rep on consequence relations is monotone (w.r.t. inclusion as the ordering).

Example 1.3 By the same reasoning as was given for Proposition 1.1—andmore specifically part (i) thereof, since, as with the ∧ case there, the truth-function associated with ∨ is idempotent—we have (for all A, B):

A, B �repCL A ∨ B,

which changes ∧ to ∨ in Proposition 1.1(i). As in the case of Proposition1.2, this becomes false if �rep

CL is changed to �repIL , though in the present case

the context used there to show this does not suffice. A suitable context isprovided by the proof of Proposition 4.1 below, however (which also providesan alternative proof for Proposition 1.2).

Proposition 1.4 The condition that for consequence relation �0 and �1:

�0 ⊆ �1 ⇒ �rep0 ⊆ �rep

1

is satisf ied whenever the language of �1 coincides with that of �0.

Proof Suppose that �0 ⊆ �1. If the languages coincide, we can infer from thisthat �rep

0 ⊆ �rep1 because the “for all contexts C(p)” in the definition of �rep

i(i = 0, 1) ranges over the same set of contexts. ��

Note the necessity of the condition in Proposition 1.4 that the languagesof �0 and �1 coincide. If they do not, then certainly the language of �0 isincluded in that of �1 by our supposition (since for all formulas A of the formerlanguage A �0 A, and so A �1 A, placing A in the latter language), but theremay be additional contexts to serve as counterexamples, as indeed we saw inthe passage from �CL to, e.g., �S4, in which we lost the aggregation inference onpassage from the first replacement core to the second. We return to the caseof modal logic in Section 4, though there are also some modal asides in theSection 2, concluding the present discussion with some general observations,under three headings:

(1) If � is substitution-invariant, then so is �rep (for any �).(2) Although the definition of �rep quantifies over all 1-ary contexts pro-

vided by the language of �, for specific choices of � there may bea finite set C of such contexts for which A1, . . . , An �rep B when-ever C(A1), . . . , C(An) � C(B) for each C(·) ∈ C. in the case of theconjunction–disjunction fragment of classical or intuitionistic logic, for


example, one can make such a simplification with |C| = 1 (see Proposi-tion 2.5); in the case of (functionally complete) classical logic �CL, wecan provide such a C with |C| = 2 (see Proposition 3.1). While from atheoretical point of view, this may seen as trivializing the relation �rep

CL ,we continue to use it, as in the present section, to give simple examplesand draw instructive contrasts. The author does not know the extent towhich such finite choices of C are available for the (replacement coresof the) intuitionistic—or more generally intermediate—and modal andother consequence relations.

(3) The mapping (·)rep is not injective, and we include the following exampleto illustrate this, for which the following definition is required. Givena class V of valuations—functions from the set of formulas to the set{T, F}—we say that the consequence relation � is determined by V tomean that for all sets of formulas � ∪ {A} we have � � A iff for all v ∈ V,if v(B) = T for all B ∈ �, then v(A) = T.

Example 1.5 Understanding by �CL the consequence relation determined bythe class of boolean valuations for some functionally complete language—call this V—and by �Suszko the consequence relation determined by V ∪ {vF},where vF is the (non-boolean) valuation assigning F to every formula of thatlanguage. Thus:

� �Suszko B if and only if � �CL B and � �= ∅.

Thus �Suszko ⊆ �CL; so �repSuszko ⊆ �rep

CL by Proposition 1.4. For the converseinclusion, suppose, to derive a contradiction, that

(1) � �repCL B but (2) � �

repSuszko B.

By (2), there is some context C(·) for which C(�) �Suszko C(B), where C(�) ={C(A) | A ∈ �}. But from (1) we have C(�) �CL C(B), so since �CL and �Susko

agree on the consequences of all non-empty sets, C(�) = ∅ and therefore � =∅, so by (1) again, we have �rep

CL B. But this is impossible: consider the contextC′(p) = ¬p, or again C′′(p) = p ∧ q. Thus �CL and �Suszko, although they aredistinct consequence relations, have the same replacement core. (For historicalinformation, with references, as to why the turnstile here has been labelled withSuszko’s name, see Indrzejczak [26].)

The last part of the reasoning in Example 1.5 raises the question of whetherany consequence relations � satisfy, as we have just noted �CL does not: �rep Bfor some formula B. For example, if we consider the restriction of �CL tothe one 1-place constant true connective, �1, as we may write it,10 and noother (primitive) connectives, one has, for this choice of �, � � B iff B ∈ �

10That is: � is the consequence relation on the language whose sole primitive connective is 1-ary �1, determined by the class of all valuations v for this language satisfying the condition thatv(�1 A) = T for all formulas A.

58 L. Humberstone

or B = �1 A for some formula A, and one might accordingly expect thathere we have �rep �1q, for example, on the grounds that any context C(p)

in which �1q is to be embedded will itself be of the form �1(�1(. . . �1(p) . . .)),given the highly impoverished logical vocabulary available, and will thereforea consequence of the empty set by �, meaning that �rep �1q. However, let usrecall the availability of improper contexts, such as C(p) = r: then C(�1q) isthe result of substituting �1q for all occurrences—of which there are none—of p in C(p), which means that C(�1q) is the formula r. Since � r, �

rep �1qafter all. This rather unexpected conclusion could be avoided by defining(·)rep differently—as with the suggestions mentioned in note 9—namely byquantifying not over all contexts but only over all proper contexts.

2 Replaceability as a Binary Relation

Formulas A and B in the language of a consequence relation � are equivalentaccording to � when A � B and B � A. Following a usage of Smiley (as in[48, 49]), we say that A and B are synonymous according to � when A andB are equivalent according to the replacement core �rep of �. A consequencerelation � is congruential when any two formulas equivalent according to �are synonymous according to �.11 Synonymy as here defined is a matter ofthe interreplaceability of the formulas concerned in arbitrary contexts. Butlet us pause to remove the “inter”, here, and define the binary relation ofreplaceability for � as that relation holding between formulas (of the languageof �) A, B, just when A �rep B; we say in this case that A is replaceableby B according to �. It is immediate from the definition that this relation isreflexive and transitive, and our attention in this section will mainly be onwhether—or, more accurately, for which choices of �—it is also symmetric.The extent to which it is will explain why one hears so little of the unilaterallydefined replaceability relation by comparison with its more commonly invokedbilateral companion.

The restriction of �CL to any fragment containing ¬ evidently gives rise toa symmetric replaceability relation, since when A �rep

CL B we have, taking C(p)

first as p and then as ¬p: A �CL B and ¬A �CL ¬B, and thus B �CL A. Soby the congruentiality of �CL, B �rep

CL A. The same applies in the case of anyfragment containing →, a point we prefer to illustrate in the weaker settingof the non-congruential consequence relation �BCI, whose language is here

11The ‘congruentiality’ terminology is that of Segerberg [46], adapted from the usage ofMakinson—e.g., in [32]. The term “self-extensional” is used in much of the contemporaryliterature of abstract algebraic logic (AAL) for this property. (The term “congruential” is useddifferently, it should be noted since we shall be citing some of his publications below, by WolfgangRautenberg, for a stronger property—which is essentially congruentiality ‘with side formulas’(on the left); current AAL literature uses the term Fregean for this property. Note that weare following the convention here that take any terms for a property of contexts (relative toconsequence relations) is applied to consequence relations themselves when every context hasthe property according to the consequence relation in question.)


taken to have → as its sole connective, with � �BCI D if and only if D can beobtained by applications of Modus Ponens from formulas in � together withany formulas of the forms

(B → C) → ((A → B) → (A → C)) ,

(A → (B → C)) → (B → (A → C)) ,

A → A,

(which schemata are respectively known as B, C and I). Information aboutthe formula logic BCI and the consequence relation �BCI can be found inKabzinski [27], Hindley [18], Blok and Pigozzi [6], and Raftery and van Alten[40]. (Kowalski [28] is also very informative.)

Proposition 2.1 Replaceability is symmetric for �BCI.

Proof Suppose A �repBCI B. Then by taking C(p) as p → q we have A → q �BCI

B → q and thus by substitution-invariance A → A �BCI B → A, so since �BCI

A → A (by I), we have �BCI B → A. Similarly, by taking C(p) as q → p, weget q → A �BCI q → B, and thus again A → A �BCI A → B and so �BCI A →B. It is well known that from its being the case that �BCI A → B and �BCI B →A, the synonymy (according to this consequence relation) of A and B follows,so B �rep

BCI A. ��

Three remarks on this proof are in order. First: it was not necessary to takeC(p) as p → q and then as q → p and then make the substitutions as above:we could simply (though, arguably, less clearly) have taken C(p) as p → A andthen as A → p. Secondly, a further simplification that might suggest itself doesnot work. One might think that since, although A �BCI B and B �BCI A do notsuffice for the synonymy of A with B in this setting, �BCI A → B and �BCI

B → A do, we could make do with one of these for the ‘replaceability’ half ofinterreplaceability (= synonymy). The reader may enjoy the (easy) exercise offinding counterexamples to the claim that either �BCI A → B or �BCI B → Asuffices for B �rep

BCI A. Finally, we could make do with a weaker consequencerelation for Proposition 2.1: namely �BB′I, defined analogously except using B′in place of C, where B′ is the schema B with its two antecedents permuted.According to a famous result of Martin and Meyer in [34] (answering a long-open question of Belnap), A → B and B → A are only both provable usingModus Ponens from instances of the axiom schemes B, B′ and I, then A is thesame formula as B; thus in the present terminology, we have the following: therelation of replaceability for �BB′I is the relation of identity.12

As indicated, this last point was made in [34] not for �BB′I but for theformula logic (or Hilbert system) BB′I: no two formulas are synonymous, or

12Two years earlier, Mortensen [35] had shown that in a range of logics considered by Newton daCosta, any formula is synonymous only with itself. See also Porte [37].

60 L. Humberstone

interchangeable salva provabilitate in this logic. As we see from the end of thefollowing quotation from Smiley [48], a similar observation had been madeconsiderably earlier by Harrop for a logic devised by Ackermann.13 (See thereferences at the end of the quotation.) But we are citing this passage heremore precisely because of its connection with the first of the above threeremarks on the proof of Proposition 2.1, in support of a generalization of thepoint: we can always choose our contexts C(p), when considering the replace-ability of A by B, as formulas in which A does not occur as a subformula (seenote 6 above), in which case the replacement concerned is uniform—providedthat we are working with a substitution-invariant consequence relation tobegin with:

Interreplaceability is being taken here in the strictest sense, to cover notonly uniform replacement but also partial replacements, though in fact apair of formulae which are interreplaceable under uniform replacementwill automatically be interreplaceable under non-uniform replacement aswell, since the effect of any non-uniform replacement can be producedby substitution in a uniform one. E.g., A ⊃ A � A ⊃ B could follow bysubstitution from p ⊃ A � p ⊃ B, in which the substitution14 is uniform.If on the other hand synonymity is defined (as is common) solely withreference to replacement in theorems, so that A and B are said to besynonymous if, for every φ(A) and φ(B), � φ(A) if and only if � φ(B),the process of substitution just described cannot be carried out: from thefact that, for a given A and B, if � p ⊃ A then � p ⊃ B, it does notfollow that if � A ⊃ A then � A ⊃ B, any more than from the undoubtedfact that if � p then � q does it follow that if � A then � B. Thuswe may well have formulae which are ‘synonymous’ in the theorem-sense as far as uniform replacements are concerned, without being fullyinterreplaceable. This is what happens (despite the author’s intention)in the propositional calculus of Ackermann’s [1], of which Harrop,[15], has shown that no two formulae are fully interreplaceable even intheorems. Smiley [48], footnote 1.

Here Smiley’s notation has been preserved intact (though the citations are re-numbered to match our bibliography), and in particular φ(A), etc., appear inplace of what we have been calling C(A), etc., and the variable p is used in

13Rosser [43] had already drawn attention to a similar problem in connection with singular termsin Ackermann’s system. Some of Ackermann’s axioms have a decidedly artificial and ad hocappearance to them, a feature shared by da Costa’s, alluded to in note 12.14That is, the substitution of A for p. Note that p should not occur in A or B here. We alsohave a uniform replacement, of A by B in the example, within a single �-statement. We return tothis contrast of changes within such statements and changes on passage from one to another suchstatement, in the discussion of horizontal vs. vertical replacement in Section 5.


other than its role here to do duty for the gap in a context, to be filled byformulas substituted for it.15 In accordance with the finer-grained approachtaken here, the talk of interreplaceability in the first part of the passagewould be replaced by talk of replaceability. Thus emended, Smiley’s pointis that, defining the uniform replaceability relation for � to be that relationholding between A and B when C(A) � C(B) for all C(p) not having A as asubformula—or equivalently, in which C(A) has all occurrences of A replacedby B—then when � is substitution-invariant, the uniform replaceability rela-tion for � coincides with the replaceability relation for � (as defined above: i.e.,with no such restriction on the contexts C(p)). We return to some aspects ofHarrop’s and Smiley’s contributions in Section 5, and, specifically to Smiley’s,in Appendix 1.

The proof of Proposition 2.1 continues to apply as we pass to stronger conse-quence relations, including the →-including fragments of the congruential �IL

and �CL. For the latter case, as we have already seen, to get a counterexampleto the symmetry of replaceability, we need to avoid not only → but also ¬; wedo this in Corollary 2.6 below. In the former case, things are more interestingin that we do not have to avoid ¬. (The reasoning in the second paragraph ofthis section, expressed by “¬A �CL ¬B, and thus B �CL A”, would of coursefail for �IL.) Accordingly, let �IL− be the restriction of �IL to →-free formulas,or more explicitly to the language with primitive connectives ∧, ∨, and ¬.16

Lemma 2.2 For formulas A, B, in the language of �IL−: A is replaceable by Baccording to �IL− if and only if A �IL− B and ¬A �IL− ¬B.

Proof The ‘only if’ direction being evident, for the converse we assumethat A �IL− B and ¬A �IL− ¬B, and show by induction on the complexity( = number of connectives in the construction of) C(p) that we must haveC(A) �IL− C(B) and ¬C(A) �IL− ¬C(B). (The claim to be proved is simplythat C(A) �IL− C(B) for all C(p), and the second part here—¬C(A) �IL−¬C(B)—provides some needed induction loading.) The basis case is imme-diate (whether p occurs in C(p) or not), and the inductive cases for ¬ and∨ are straightforward, the former requiring the fact that the prefix ¬¬ ismonotone in intuitionistic logic; in fact now that we are explicitly addressingonly {∧, ∨, ¬}-formulas, we can write “�IL” rather than “�IL−” without risk

15And of course ⊃ appears for →. Note that in Smiley’s example, “A ⊃ A � A ⊃ B couldfollow by substitution from p ⊃ A � p ⊃ B,” the fact that one might expect A ⊃ A to beprovable outright has no bearing on the point, which could equally well have been made bysaying that, for instance, A ⊃ ¬A � A ⊃ ¬B follows, by the substitution-invariance of �, fromp ⊃ ¬A � p ⊃ ¬B.16This fragment has attracted considerable attention in the literature. See Section 3 of Rautenberg[41], Blok and Pigozzi [6], Section 5.2.5 ([6] adding � and ⊥ to the primitive logical vocabularyhere).

62 L. Humberstone

of confusion. We look at the case of ∧, i.e., the case in which C(p) isC0(p) ∧ C1(p). We must show (1) C0(A) ∧ C1(A) �IL C0(B) ∧ C1(B) and (2)¬(C0(A) ∧ C1(A)) �IL ¬(C0(B) ∧ C1(B)), with the inductive hypothesis beingthat Ci(A) �IL Ci(B) and ¬Ci(A) �IL ¬Ci(B), i = 0, 1. (1) is immediate fromthe first part of the inductive hypothesis. For (2), the second part of theinductive hypothesis gives:

¬C0(A) ∨ ¬C1(A) �IL ¬C0(B) ∨ ¬C1(B).

Since �IL ⊆ �CL, we can replace the “IL” here with “CL” and then, by someclassically (though not in general intuitionistically) available equivalences,rewrite this as:

¬ (C0(A) ∧ C1(A)) �CL ¬ (C0(B) ∧ C1(B)) ,

from which we conclude by Glivenko’s Theorem that we also have this for �IL,completing the proof. ��

Proposition 2.3 Replaceability for �IL− is not symmetric.

Proof q is replaceable by ¬¬q according to �IL−, by Lemma 2.2, since q �IL

¬¬q and ¬q �IL ¬¬¬q, whereas ¬¬q is not replaceable by q since ¬¬q �IL q.��

We turn to the case of dropping negation from the fragment just considered.Let �∧,∨ be the restriction to the language with connectives ∧, ∨ of �CL; recallthat this coincides with the corresponding fragment of intuitionistic logic. Italso coincides with its own replacement core,17 as we shall find (Proposition2.5) en route to showing that it provides a further example of a non-symmetricreplaceability relation. First, an ancillary observation concerning disjunctionsof contexts is called for.

Lemma 2.4 Suppose that for 1-ary contexts C0(p), C1(p), and formulasA1, . . . , Am and B1, . . . , Bm of the language of �CL conditions (0) and (1) aresatisf ied:

(0) C0(A1), . . . , C0(Am) �CL C0(B) (1) C1(A1), . . . , C1(Am) �CL C1(B).

Then we also have:

C0(A1) ∨ C1(A1), . . . , C0(Am) ∨ C1(Am) �CL C0(B) ∨ C1(B).

17This is a seldom considered strengthening of the notion of the congruentiality of a consequencerelation �. The latter amounts to saying that if each of A and B is a consequence of theother according � then they stand in this relation according to �rep, while the current notionunilateralizes this, amounting to: if B is a consequence of A according to �, then it is a consequenceof A according to �rep.


Proof Suppose, for a contradiction, that (0) and (1) are satisfied but theclaimed conclusion fails. Thus for some boolean valuation v, v(C0(Ai) ∨C1(Ai)) = T for i = 1, . . . , m while v(C0(B) ∨ C1(B)) = F. Thus v(C0(B)) = Fand v(C1(B)) = F. Since v(C0(B)) = F, in view of (0), v(C0(A j)) = F for atleast one j (1 ≤ j ≤ m), and since v(C1(B)) = F, by (1) we have v(C1(Ak)) = Ffor some k (1 ≤ k ≤ m). Note that we cannot have j = k since v(C0(Ai) ∨C1(Ai)) = T for each i. The story so far, with v’s assignments written beneaththe formulas concerned:

C0(A j) ∨ C1(A j) C0(Ak) ∨ C1(Ak) C0(B) ∨ C1(B)

F T T T T F F F F

Relative to v’s assignments to the values of variables other than p, which bycontrast with the discussion in Section 1 (where they were collected togetheras yi) we will not explicitly register here, C0(p) and C1(p) induce 1-ary truth-functions, say f (·) and g(·) respectively, so with x, y, z, as v(A j), v(Ak), v(B),respectively, what we have is:

f (x) = T, g(x) = F; f (y) = F, g(y) = T; f (z) = F, g(z) = F.

Since f (x) �= f (y), we have x �= y. Thus either (i) z = x or (ii) z = y. But (i) isincompatible with our having f (x) = T and f (z) = F, while (ii) is incompatiblewith having g(y) = T and g(z) = F. This contradiction shows that (0) and (1)cannot after all be satisfied while the conclusion of the lemma fails. ��

The above proof would not be needed for the cases m = 0, 1, of Lemma2.4, for which the familiar properties of disjunction suffice, as in the proof ofLemma 2.2. There (the m = 1 case) we need only that D � D′ and E � E′imply D ∨ E � D′ ∨ E′ and not the further information that there are formulasA, B, with D and D′ being C0(A) and C0(B) for some boolean contextC0(p), and E and E′ being C1(A) and C1(B) for another such context. Wecan provide a simple illustration of this point using modal contexts. Take thesmallest normal bimodal logic, writing �0, �1, for the non-boolean primitivesinvolved.18 Then, where � is the corresponding (local) consequence relation,we have:

�0q, �0r � �0 (q ∧ r) and �1q, �1r � �1 (q ∧ r) ,

while

�0q ∨ �1q, �0r ∨ �1r � �0 (q ∧ r) ∨ �1 (q ∧ r) .

18A context C(p) is normal according to � when � � B implies {C(A) | A ∈ �} � C(B) for all�, B. A normal n-modal logic (“bimodal” for n = 2) is one for which each of the contextsC(p) = �0 p, . . . , �n−1 p, is normal. If we restrict the condition to the case of � non-empty thenthis coincides with the notion of a regular modal logic as defined in Section 1, when this isgeneralized to the case of more than one � operator.

64 L. Humberstone

Proposition 2.5 � �rep∧,∨ A if and only if � �∧,∨ A.

Proof The “if” direction is all that needs to be proved. So, taking advantageof the fact that �∧,∨ is finitary, we need to show that A1, . . . , Am �∧,∨ Bimplies C(A1), . . . , C(Am) � C(B), for every context C(p), which is done byinduction on the complexity of C(p), using Lemma 2.4 for the case of C(p) =C0(p) ∨ C1(p). ��

The situation here is somewhat reminiscent of the slogan “positive impliesmonotone” in first-order model theory (see [2] and references there, in par-ticular [31]), except that we have something stronger for our 1-ary sententialcontexts: positive implies normal, as the latter is defined in note 18. In the first-order case this strengthening, as is the intermediate strengthening to “regular”,is blocked by the presence of existential quantification—cf. note 3. In the caseof �∧,∨, indeed, the difference between regularity and normality evaporates,since this consequence relation is atheorematic, or ‘purely inferential’ (i.e. theset of consequences of ∅ is ∅).

Note, à propos of Proposition 2.5 that while the right-hand side means thesame as “� �CL A for all � ∪ {A} constructed at most using ∧ and ∨”, the left-hand side does not mean the same as “� �rep

CL A for all � ∪ {A} constructedat most using ∧ and ∨”, since the superscript here would mean we have toconsider contexts themselves constructed with the aid of other connectives,and in particular, so re-written, the if direction would fail in view of the factthat q �

repCL q ∨ r in view of negative contexts.

Corollary 2.6 Replaceability is not symmetric for �∧,∨.

Proof Given Proposition 2.5 it suffices to observe, for example, that q ∧ r �∧,∨q while q �∧,∨ q ∧ r. ��

Let us pause to take stock: according to Proposition 2.1 replaceability issymmetric for �BCI and for extensions of this consequence relation even onexpanded languages (including �IL, �CL and all intermediate consequencerelations), while according to Propositions 2.3 and 2.6 replaceability is not sym-metric for �∧,∨ or �IL−, respectively. These examples may suggest a certainhypothesis: namely that the (substitution-invariant) consequence relations forwhich replaceability is symmetric are precisely the equivalential consequencerelations, which is to say,19 those � for which there is a set of formulasE(p1, p2) in the language of � with the properties that (i) every formula in

19See Czelakowski [11], p. 185; p. 241 supplies historical information.


E(A, A)20 is a �-consequence of the empty set, (ii) A, E(A, B) � B for allformulas A, B, and finally:

(iii) E(A1, B1), . . . , E(An, Bn) � E(#(A1, . . . , An), #(B1, . . . , Bn)),

for all formulas A1, . . . , An, B1, . . . , Bn of the language of �, and primitiven-ary connectives # of that language. Note that although a set of formulasrather than an individual formula appears on the right of the “�” in (iii), weare not dealing with a generalized consequence relation (in which case wewould have written “�” in any case, to accord with our current conventions),but abbreviating the claim that each formula in this set is a �-consequence ofthe set of formulas on the left. A set of formulas E(p1, p2) satisfying theseconditions is called a set of equivalence formulas for �. It is not difficult tosee that (ii) and (iii) can together be replaced with the condition that for allcontexts C and all formulas A and B:

(ii)* E(A, B), C(A) � C(B),

a fact worth noting because of the resemblance of the two conditions (i) and(ii)* to the usual fundamental principles governing identity in predicate logic.In a natural deduction treatment (i) is an =-Introduction principle while (ii)*is an =-Elimination principle. The parallel is closest when the set E(p1, p2)

contains only one formula, and that formula is of the form p1 # p2 for aprimitive binary connective (perhaps written as “↔” or “≡”), that connectivethen being analogous to the predicate symbol “=”. Such analogies have beenextensively considered in the literature (e.g., Suszko [50]). In Appendix 1 tothis paper, an issue arising for ordinary “=” in predicate logic analogous tothat about to be raised for E will be aired.

Now, (ii)* looks as though it presents E(p1, p2) as a generalizedconnective—or when this set contains only one formula a definiens for aconnective proper—which expresses in the object language the relation ofreplaceability rather than interreplaceability, since it may be read informally assaying that subject to the condition (expressed by) E(A, B), A is replaceableby B. And indeed in Czelakowski [10] a further condition was included on E inthe definition of equivalentiality, to the effect that E(A, B) � E(B, A), whichgives every appearance of boosting the replaceability idea to interreplaceabil-ity. This appearance, however, is deceptive, and this condition follows from(i), (ii) and (iii), as Czelakowski [11], Corollary 3.1.4 points out. The easiestway to see this is in terms of the (i)–(ii)* formulation, as follows. Let E(p1, p2)

20As with the “C(p)” notation (and recall that p simply abbreviates p1), E(A, B) is the set offormulas resulting from formulas in E(p1, p2) on uniformly substituting A and B for p1 and p2respectively; unlike the C(p) case, however, here we insist that no propositional variables otherthan p1 and p2 occur in the formulas in E(p1, p2), each of which formulas is a 2-ary context.

66 L. Humberstone

consist, for some index set I, of the formulas Ei(p1, p2) for i ∈ I. Take Ci(p)

as Ei(p, A). Then by (ii)*, for each i ∈ I, E(A, B), Ci(A) � Ci(B), which isto say:

E(A, B), Ei(A, A) � Ei(B, A).

Since by (i), we have � Ei(A, A), we conclude that E(A, B) � Ei(B, A), andsince we have this for each i ∈ I, we have E(A, B) � E(B, A). Thus thecommutativity of the ‘generalized connective’ E follows from the identity-likeconditions (i) and (ii)* just as the symmetry of the identity relation (to whichwe return in Appendix 1) follows from the usual rules governing identity.21

Returning to the matter of when replaceability is symmetric, we have:

Proposition 2.7 If a consequence relation � is equivalential, then replaceabilityfor � is symmetric.

Proof Suppose � is equivalential, with equivalence formulas E(p1, p2), andthat A is replaceable by B according to �. We want to show that B isreplaceable by A. It suffices to observe that for any formulas F1 and F2, F1

is replaceable by F2 according to � if and only if � E(F1, F2). Thus from oursupposition we get � E(A, B), so by the above remarks, � E(B, A), so B isreplaceable by A according to �. ��

Thus with the present apparatus we can see Proposition 2.1 and the anal-ogous results for classical and intuitionistic logic as corollaries of Proposition2.7, given the fact that E(p1, p2) = {p1 → p2, p2 → p1} is a set of equivalenceformulas for the consequence relations involved. However, to the question ofwhether having being equivalential is not only sufficient, but also necessary, forhaving a symmetric replaceability relation, we must return a negative answerin view of another example already aired at the start of this section. Let �CL¬be the restriction of �CL to formulas in which the only connective to appearis ¬. Note that for this consequence relation, we have, for any formulas F1,F2 in its language: F1 �CL¬ F2 iff F2 �CL¬ F1; from this it follows that forany A, B, if for all C(p), we have C(A) �CL¬ C(B), then for all C(p), wehave C(B) �CL¬ C(A), i.e., replaceability is symmetric. Yet this consequencerelation is not equivalential, since, being atheorematic, there is no way ofchoosing E(p1, p2) so as to satisfy condition (i) above. (In fact for this samereason, �CL¬ is not even protoalgebraic, which requires only the existenceof E(p1, p2) satisfying (i) and (ii); see [11, Chapter 1].) A natural questionon which to end this section would accordingly be: Is there an informativecharacterization in independently defined terms of this property of having asymmetric replaceability relation?

21The same goes for a condition similarly analogous to the transitivity of identity (see againCorollary 3.1.4 in [11]), another condition cited as part of the definition of equivalentiality in [10],following the originators of the notion [39].


3 A Closer Look at the Classical Case

Although the definition of �rep involves a universal quantification over allcontexts, we can whittle down the number of contexts that need to be con-sidered in the case in which � is �CL. In the first place, although additionalvariables other than p can enter into the construction of C(p), we do notneed to consider candidate C(p)s for which this is the case (when treatingclassical logic). For suppose with C constructed from p and q1, . . . , qn we haveC(A1), . . . , C(Am) �CL C(B). We can re-express this by saying that there issome derived (n + 1)-ary connective # for which:

# (A1, q1, . . . , qn) , . . . # (Am, q1, . . . , qn) �CL # (B, q1, . . . , qn) .

Thus for some boolean valuation v, we have v(#(Ai, q1, . . . , qn)) = T (1 ≤ i ≤m), while v(#(B, q1, . . . , qn)) = F. To ‘freeze’ the effects of v on the positionsoccupied by the q1, . . . , qn, define formulas Q j (1 ≤ j ≤ n) as follows:

Q j ={

p → p if v(q j) = Tp ∧ ¬p if v(q j) = F

and consider C′(p) = #(p, Q1, . . . , Qn), in which the only variable appearing isp. Evidently C′(A1), . . . , C′(Am) �CL C′(B), so we need only consider the fourderived 1-ary connectives of the language �CL which for uniformity we shalltemporarily—just for this paragraph—call C1, C2, C3 and C4 where C1(p) = p,C2(p) = ¬p, C3(p) = p → p and C4(p ∧ ¬p), corresponding to the four one-place truth-functions (identity, negation, constant-true, constant-false): C(B)

follows classically from C(A1), . . . , C(Am) for all C(p) just in case C(B)

follows classically from C(A1), . . . , C(Am) for C(p) = C1(p), C2(p), C3(p),C4(p). We can remove C3 from this list because this context never providesa counterexample to the hypothesis that A1, . . . , Am �rep

CL B, since we cannothave v(C3(B)) = F. And we can also remove C4 from the list, since the onlyway for a boolean valuation to assign T to each of C4(A1), . . . , C4(Am) is tohave m = 0, in which case v serves as a countervaluation to the hypothesis that�repCL B, so we can use C1 and C2 already to tell us that �

repCL B, since no formula

and its negation can both be truth-functional tautologies. Let us summarizethese findings, dropping the “C1”, “C2” notation, and using ¬� to denote{¬C | C ∈ �}:

Proposition 3.1 � �repCL B if and only if � �CL B and ¬� �CL ¬B.

This provides the ‘quick check’ alluded to in Section 1, which allows us tosee with only twice the time it takes to do a truth table test (in the case offinite �), whether or not � �CL B. Still concentrating on the case of � finite,we have a simple variation on the second conjunct of the condition provided byProposition 3.1: B �CL

∨�, where

∨� is the disjunction of formulas in �. If

68 L. Humberstone

we were working with generalized consequence relations—an option mootedin the discussion after Proposition 1.1—then we would be able to dispense notonly with negation, but also with disjunction, observing that:

� �repCL � if and only if � �CL � and � �CL �,

or, more succinctly, denoting the converse (sometimes called the dual) of ageneralized consequence relation � by �−1:

�repCL = �CL ∩ �−1

CL .

However, we stick with the consequence relations framework here.22

Proposition 3.1 gives us a simple characterization of �repCL in terms of �CL, but

we may wish for a syntactical characterization of the former relation in its ownterms, and it is to this project that we turn our attention for the remainder ofthis section. In the interests of simplicity let us declare our boolean primitivesto be ∧ and ¬, and distinguish two classes of valuations, V∧:∧,¬ and V∧:∨,¬. The“∧ : ∧” and “∧ : ∨” parts of the subscripts here are meant to suggest valuationson which ∧ is treated as ∧ is treated on boolean valuations, and valuationson which ∧ is treated as ∨ is treated on boolean valuations, respectively.That is, V∧:∧,¬ comprises those valuations v such that for all formulas A,B, v(A ∧ B) = T iff v(A) = v(B) = T, and v(¬A) = T iff v(A) = F, whileV∧:∨,¬ comprises those v such that v(A ∧ B) = F iff v(A) = v(B) = F, andv(¬A) = T iff v(A) = F. Thus the valuations in V∧:∧,¬ are just the booleanvaluations for the present language, while those in V∧:∨,¬ treat ¬ in the booleanmanner but treat ∧ as though it was ∨. Note that V∧:∧,¬ ∩ V∧:∨,¬ = ∅, since forall formulas A, for each v ∈ V∧:∧,¬, v(A ∧ ¬A) = F , while for each v ∈ V∧:∨,¬,v(A ∧ ¬A) = T. This means that the following function (·)δ (“δ” for “dual”) iswell-defined over V∧:∧,¬ ∪ V∧:∨,¬:

vδ =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

the unique u in V∧:∨,¬ with u(pi) �= v(pi) if v ∈ V∧:∧,¬for each variable pi

the unique u in V∧:∧,¬ with u(pi) �= v(pi) if v ∈ V∧:∨,¬for each variable pi

By induction on the complexity of A, one verifies:

Lemma 3.2 For all formulas A, and all v ∈ V∧:∧,¬ ∪ V∧:∨,¬, v(A) �= vδ(A).

Since we have promised a syntactical characterization (“in its own terms”) ofthe replacement core of the classical consequence relation, the appearance ofall these semantic considerations may not seem immediately to the point. Butthe idea is to present such a syntactical characterization in the form of a proof

22The removal of C3 from the list of contexts needing consideration cannot proceed as forconsequence relations, since there we have ∅ on the right of “�”, but an argument like that usedto eliminate C4 in the consequence relation case can be used here instead.


system which is shown to do the job by showing its soundness and completenessw.r.t. this semantic apparatus. For this we need one further observation, and areminder about a familiar (and readily verified) fact, namely that that if �1 and�2 are consequence relations on the same language determined respectively byV1 and V2, then the consequence relation �1 ∩ �2 is determined by V1 ∪ V2.

Proposition 3.3 �repCL is determined by V∧:∧,¬ ∪ V∧:∨,¬.

Proof By Proposition 3.1 and the familiar fact just recalled, the result followsfrom the following two:

(1) � �CL B iff, if for all v ∈ V∧:∧,¬, v(A) = T for each A ∈ �, then v(B) = T.(2) ¬� �CL ¬B iff, if for all v ∈ V∧:∨,¬, v(A) = T for each A ∈ �, then

v(B) = T.

(1) reports that �CL is determined by the class of boolean valuations, under-standing this as restricted to the language with connectives ∧ and ¬, and weassumed this to be a familiar result in need of no further explanation. In termsof determination, (2) claims that the consequence relation �δ

CL, say (though weshall not make further use this notation), defined by:

� �δCL B iff ¬� �CL ¬B,

is determined by V∧:∨,¬.For the “only if” half of (2), suppose we have v ∈ V∨,¬, with v(A) = T for

each A ∈ � but v(B) = F, with a view to showing that ¬� �CL ¬B. Considerthe dual valuation vδ , which lies in V∧,¬, and so is a boolean valuation, andreverses all the verdicts of v, by Lemma 3.2. Thus vδ falsifies all formulas in� and verifies B, showing, as required, that ¬� �CL ¬B. The “if” half of (2) issimilarly established, this time taking vδ for a suitable v ∈ V∧,¬. ��

As a prelude to our syntactic characterization of �repCL , let us look at the

situation in the absence of negation. Suppose, that is, that our only primitiveconnective is ∧ and now consider the classes of valuations V∧:∧ and V∧:∨ whichare defined as for V∧:∧,¬ and V∧:∨,¬ but omitting the condition pertaining to¬. The consequence relation determined by V∧:∧ ∪ V∧:∨ gives the “commonlogic” of conjunction and disjunction, and studied under that description itwould be most naturally conducted with an object language that does notnotationally bias either of these connectives over the other, just using a specialbinary connective ∗ for which those (unconditional) �-statements we areinterested in are those which hold classically whether ∗ is interpreted as ∧ or∨, such as q ∗ r � r ∗ q. This is the notation used in Rautenberg [42], whichprovides a remarkably simple syntactic characterization of the consequence

70 L. Humberstone

relation concerned, namely as the least consequence relation on this languagesatisfying the following conditions23 for all formulas:

(R0) A ∗ (B ∗ C) � (A ∗ B) ∗ C; (R1) A, B ∗ C � A ∗ B;(R2) A ∗ B � B ∗ A; (R3) A � A ∗ A; (R4) A ∗ A � A.

Thus we could rewrite these conditions as sequent schemata, replacing the “�”by a sequent separator, thinking of sequents as having a finite set of formulasto its left and a single formula to its right, and of the defining conditionson consequence relations as sequent-to-sequent rules (the ‘Cut’ rule, etc.),and (R0)–(R4) as further such (in their case, 0-premiss) rules. The sequentsprovable in this system—not a sequent calculus, to be sure, but a sequent-basedproof system all the same—are then precisely those valid over V∗:∧ ∪ V∗:∨(i.e., truth-preserving from left to right, for each valuation in this set), whereV∗:∧ comprises the valuations treating ∗ as boolean valuations treat ∧ and V∗:∨those valuations treating ∗ as boolean valuations treat ∨. We are interestednot in the consequence relation determined by V∗:∧ ∪ V∗:∨, however, but inits extension, in the language expanded by ¬, determined by V∗:∧,¬ ∪ V∗:∨,¬.Essentially this consequence relation—though with a more generous choiceof primitive connectives—is the focus of attention in [33], where Malinowskiexplores its strengthenings (= extensions in the same language), finding thereto be eight proper consistent substitution-invariant extensions in all, but isnot so much interested in providing an explicit syntactic or proof-theoreticcharacterization of the consequence relation itself.24 In providing our suchcharacterization here, we continue with Rautenberg’s ∗ notation.

We need some conditions pertaining to ¬ and to its interaction with ∗. Fornegation by itself, we use:

(N1)A, ¬A � B; (N2) If �, A � B and �, ¬A � B, then � � B.

(N1) and (N2) are familiar principles governing classical negation. (One mightthink that (N2) needs to be restricted to the case of � �= ∅ on the grounds thatwe can never have � B, if � is to line up—re-notating ∗ as ∧—with �rep

CL . But the

23These conditions are labelled here as in the theorem on p. 533 of [42], with “R” added (for“Rautenberg”), except that our (R0) and (R1) are Rautenberg’s (1) and (0), respectively, and (R3)and (R4) are Rautenberg’s (4) and (3). These interchanges are made for notational convenience(as we are going to be dropping the schemas (R0) and (R4) presently, and can then refer to whatremains by “(R1)–(R3)”).24This Malinowski isolates as the intersection of the classical consequence relation and its dual, i.e.,as capturing the inferences which are both truth-preserving and falsity-preserving. (See the proofof Proposition 3.3 above.) To be quite accurate: Malinowski [33] couches his discussion in termsof the corresponding consequence operations. Herrmann and Rautenberg [17] give a syntacticdescription of the consequence relation on the language with connectives ∧, ∨, and ¬, for which aformula counts as a consequence of set of formulas when it is a classical consequence of them andalso a consequence of them when ∧ and ∨ are interchanged. While this project is evidently close toour present concerns—and we could for instance introduce a defined dual �, say, for the neutral∗, putting A � B for ¬(¬A ∗ ¬B)—the methods and motivations in [17] are rather different. (Seethe second last paragraph of this section.)


correct conclusion to draw is rather that we never have both of the “premisses”A � B and ¬A � B needed for such an application of (N2); the rule does notneed the envisaged restriction.) As to “mixed” conditions, we offer:

(M1) D ∗ ¬D, A � A ∗ B;(M2) ¬ (D ∗ ¬D) , A ∗ B � A;(M3) A ∗ B, ¬A � B.

Let �∗ be the smallest consequence relation on the present language whichsatisfies conditions (R1)–(R3), (N1)–(N2), (M1)–(M3). (The earlier conditions(R0) and (R4) are redundant in this setting; for example, (R4) can be derivedusing (M3) and (N2).)

Lemma 3.4 For all formulas A, B:

(i) A, B �∗ A ∗ B;(ii) �, A �∗ B implies �, ¬B �∗ ¬A;

(iii) A ∗ ¬A �∗ B ∗ ¬B;(iv) ¬ (A ∗ ¬A) �∗ ¬ (B ∗ ¬B).

Proof See Appendix 2. ��

Since those principles, (N1) and (N2), governing ¬ and making no referenceto ∗, are exactly sufficient for the classical logic of ¬, it is not strictly necessaryto include a proof for part (ii) of the above lemma. It is included for the benefitof anyone not familiar with using (N1) and (N2) in this way. (All classicalnegation principles formulated without explicit reference to ∗ can be similarlyobtained, such as A �∗ ¬¬A and ¬¬A �∗ A. We illustrate with the latterexample at the end of the proof of Proposition 3.8.)

Let us note also that (M1) could be replaced, in specifying �∗ as the leastconsequence relation � satisfying the conditions, by the condition emerging inLemma 3.4(iii) here: A ∗ ¬A � B ∗ ¬B. We can recover (M1) from this since(R1) and (R2) yield:

B ∗ ¬B, A �∗ A ∗ B,

from which we get (M1) by an appropriate instance (namely D ∗ ¬D �∗B ∗ ¬B) of the Lemma 3.4(iii) condition. As for (M2), this condition isredundant—see Proposition 3.4 below—though we have retained it for asimple presentation ((M1) and (M2) dealing with two subcases symmetrically)of the completeness proof which follows.

Theorem 3.5 The consequence relation �∗ is determined by V∗:∧,¬ ∪ V∗:∨,¬.

Proof Appendix 2. ��

72 L. Humberstone

Corollary 3.6 �repCL is �∗ with ∗ rewritten throughout as ∧.

Proof By Proposition 3.3 and Theorem 3.5. ��

This completes our attempt to provide a syntactic description, or proof-theoretic codification, of the replacement core of �CL. By way of incidentalspin-off, we also have the following, since the relation of ∗ to the two terms ofthe union V∗:∧,¬ ∪ V∗:∨,¬ is the same:

Corollary 3.7 �repCL is �∗ with ∗ rewritten throughout as ∨.

The examples of Proposition 1.1(i) and (ii) were: A, B �repCL A ∧ B, and

(rewriting the → in terms of disjunction) A, ¬A ∨ B �repCL B. Thus, illustrating

the point of this corollary—though this is already evident from Proposition3.1—we can interchange the conjunctions and disjunctions, giving variantexamples:

A, B �repCL A ∨ B and A, ¬A ∧ B �rep

CL B.

Similarly, whereas after Proposition 1.1, we noted that A ∨ B �repCL A, B, we

could equally well have given the example with ∨ replaced by ∧. Since we havebeen working with consequence relations rather than generalized consequencerelations, ‘multiple succedents’ are encoded (thanks to (N1) and (N2)) bynegating all but one of them and moving the negated formulas to the left ofthe �. Both of these examples are encompassed on the present approach bymeans of the condition (M3), formulated using the neutral “∗” notation.

Left over from our earlier discussion is the matter of the redundancy,mentioned above, of the condition (M2):

Proposition 3.8 The condition (M2) can be derived from the remaining condi-tions used to def ine �∗.

Proof Here is a formal derivation. Note that Lemma 3.4 was proved withoutappeal to (M2), legitimating its appearance in line justifications here.

(1) B ∗ A, ¬A � A ∗ ¬A (R1)(2) A ∗ B � B ∗ A (R2)(3) A ∗ B, ¬A � A ∗ ¬A 1, 2 Cut B ∗ A(4) A ∗ B, ¬A � D ∗ ¬D 3, Lemma 3.4(iii)(5) ¬ (D ∗ ¬D) , A ∗ B � ¬¬A 8, Lemma 3.4(ii).

We have ended up with ¬¬A on the right rather than just A, as in (M2). Asmentioned earlier, all such pure “¬” manipulations are as for �CL, thanks to(N1) and (N2), but for the record: to derive ¬¬A � A from these conditions,first note that ¬¬A, A � A, simply because � is a consequence relation, while¬¬A, ¬A � A in virtue of (N1). From these two facts the desired conclusionfollows by (N2). ��


As remarked in note 24, a syntactic description of essentially this conse-quence relation (with a different choice of primitives) was provided alreadyin [17], which has more unconditional conditions on the consequence relationbut no conditional conditions, such as the present (N2). (Recall that suchconditional �-conditions appear as non-zero-premiss sequent-to-sequent rulesin an explicit proof system; and note that (N2) could be simplified to the—stillconditional—condition, easily seen to be equivalent given (N1)—that �, ¬A �A implies � � A.) What is significant about Herrmann and Rautenberg’selimination of such rules is that the resulting proof system has only finitelymany sequent-schemata (corresponding to unconditional �-statements) asstarting points for proofs.25

A point mentioned in Malinowski [33] and Herrmann and Rautenberg[17] concerning what is essentially the same consequence relation as �∗ (seenote 24) is worth repeating here. Because of a violation of the conditioncalled ‘uniformity’ in Łos and Suszko [29], or ‘cancellation’ in Shoesmith andSmiley [47, p. 272], �∗ is determined by no single matrix. This is most evidentfrom the condition mentioned at Lemma 3.4(iii), according to which r ∗ ¬rfollows from q ∗ ¬q: these formulas have no common propositional variables,so uniformity/cancellation would require that either every formula follow fromq ∗ ¬q or else that r ∗ ¬r should follow from every formula. But neither ofthese is the case, when “follows from” is interpreted in terms of �∗.

4 Intuitionistic and Modal Observations

While it would be interesting to see a similarly explicit description “in its ownterms” of the replacement core of �IL (along the lines of Theorem 3.5, thatis) we do not have one to report here, and will instead supply an observationrelevant to its provision, which presents a striking contrast with the classicalcase. Call a consequence relation � left-prime when � � B always implies A �B for some A ∈ �.26 The best known example is the restriction of �CL (whichcoincides with that of �IL) to the language whose sole connective is ∨. Thereplacement core of classical consequence, �rep

CL is clearly not left-prime, sincein view of Proposition 1.1, we have:

q, r �repCL q ∧ r and q → r, q �rep

CL r,

while in neither case can either of the left-hand formulas be dropped (since wedon’t even have the corresponding � statements correct without the “rep”).

25Of course if we were working with generalized consequence relations instead of consequencerelations, we could replace (N2) with the single condition � A,¬A. Further discussion ofHerrmann and Rautenberg [17] may be found in Humberstone [23].26This terminology remains convenient in discussing consequence relations even though there isno corresponding notion (of interest) to go under the name “right-prime” here. The latter hasa natural application for generalized consequence relations, where it receives the obvious (dual)definition.

74 L. Humberstone

Now, in Proposition 1.2 we saw that we lost the above examples on passingfrom the replacement core of classical logic to that of intuitionistic logic, andwhat we shall see next—with the help of a proof reminiscent of one used byGödel in [13]—is that the loss of such examples is a special case of a generalphenomenon. (The same applies to Example 1.3.)

Proposition 4.1 The consequence relation �repIL is left-prime.

Proof Suppose A1, . . . , Am �repIL B. Thus C(A1), . . . , C(Am) �IL C(B) for all

contexts C(p) including C(p) = (p ↔ A1) ∨ . . . ∨ (p ↔ Am). But for thischoice of C we have �IL C(Ai) for i = 1, . . . , m, so we conclude that C(B) isintuitionistically provable:

�IL (B ↔ A1) ∨ . . . ∨ (B ↔ Am).

Then, by the Disjunction Property for �IL, some B ↔ Ai is provable, for whichi we have Ai �IL B and B �IL Ai. By the congruentiality of �IL this suffices forthe conclusion that Ai �rep

IL B. ��

Note that the above proof establishes more than the left-primeness of�repIL . It actually shows that this consequence relation is what we might call

degenerate (though a more accurate term for general purposes would be “left-degenerate”—see note 26), as we may describe a consequence relation �for which whenever a formula is a consequence (according to �) of a set offormulas, it is equivalent (according to �) to some formula in the set. The left-prime ∨-fragment consequence relation mentioned earlier lacks this strongerproperty.

Despite the use of ↔ and the appeal to the Disjunction Property in the proofof Proposition 4.1, the whole of the above argument works, with suitable andfamiliar adaptations, for the pure implicational fragment of intuitionistic logic,as we illustrate here with the sample case of m = 3. For the moment keepingthe “↔” intact, we take C(p) as:

(p ↔ A1� → q) → ((p ↔ A2� → q) → ((p ↔ A2� → q)

→ ((p ↔ A3� → q) → q))),

where q is a propositional variable not occurring in the Ai or in B (the formulaB, that is, for which we are supposing that A1, A2, A3 �rep

IL B). As before, wehave �IL C(Ai) for i = 1, 2, 3, so this would imply �IL C(B), where C(B) is:

(B ↔ A1� → q) → ((B ↔ A2� → q) → ((B ↔ A2� → q)

→ ((B ↔ A3� → q) → q))),

which again, since q occurs only as displayed, would imply the provabilityof some B ↔ Ai, by an implicational version of the Disjunction Property.27

27As in Rautenberg [41, p. 132, top].


Finally, to get rid of the ↔, change each of the p ↔ Ai� → q antecedents inC(p) to (p → Ai) → ((Ai → p) → q).

One problem left open here is that of providing a complete syntacticaldescription of �rep

IL in its own terms, like that given for �repCL in Section 3—

though the fact that �repIL is degenerate in the sense explained above perhaps

renders this less interesting than the case of �repCL . (Note that because �IL is

congruential, this means that � �repIL A if and only if some formula in � is �IL-

equivalent to A.) We turn to (classically based) modal logic and recall thata modal logic S is said to enjoy the rule of disjunction if for all n, whenever�A1 ∨ . . . ∨ �An is provable in S, then some Ai is provable in S (1 ≤ i ≤ n).As in Section 1 (see note 5) we treat these logics via the associated localconsequence relations, here restricting attention to the case of normal modallogics, for which the rule of disjunction can be made to play the role played bythe Disjunction Property in the proof of of Proposition 4.1. Our formulationmakes explicit the element of degeneracy remarked on after that proof:

Proposition 4.2 For any normal modal logic S enjoying the rule of disjunction,the consequence relation �rep

S is degenerate (and therefore left-prime).

Proof Replicate the proof of Proposition 4.1 but using the context C(p) =�(p ↔ A1) ∨ . . . ∨ �(p ↔ Am) to conclude that

�S �(B ↔ A1) ∨ . . . ∨ �(B ↔ Am),

whence the rule of disjunction we conclude that some B ↔ Ai is S-provable,which implies Ai ��S B, showing that �S is degenerate (and so left-prime).

��

(In the final line of the proof, “Ai ��S B” abbreviates “Ai �S B and B �S

Ai”.)To conclude with a question: what about normal modal logics which do not

enjoy the rule of disjunction, such as S5? Are the replacement cores of theirassociated (local) consequence relations always degenerate?

5 Angell and Harrop

In [3] and [4], Angell describes a generalization of Modus Ponens (whichemerges as the special case of C(p) = p), called by him the rule of excision,which appears thus in our preferred notation for contexts:

A C(A → B)

C(B)

Angell considers this rule both as a rule of proof (for obtaining new theoremsfrom old, in the axiomatic approach to logic) and also as a rule of inference

76 L. Humberstone

(for deriving conclusions from hypotheses or assumptions).28 The ‘rule ofinference’ interpretation is closest to the case considered in Proposition 1.1(ii),amounting to the following condition on a consequence relation �, for allformulas A, B and contexts C(p) of the language of �:

A, C (A → B) � C(B).

This differs from our ‘replacement’ version in Proposition 1.1 in that wehad C(A) rather than A simpliciter on the left.29 The latter condition, asreported there, is satisfied by �CL, but not (we found in Proposition 1.2)by �IL, whereas Angell’s condition is satisfied there, as is most easily seenby considering the semantics for �IL in terms of Heyting algebras taken asmatrices when (just) their top elements are designated. Calling the top elementof an arbitrary such algebra 1, the hypothesis that the left-hand formulas aredesignated on a given homomorphism (matrix evaluation) from the algebra offormulas to the Heyting algebra in question is the hypothesis that h(A) = 1 andh(C(A → B)) = 1. Let b be h(B), c(·) be the 1-ary polynomial correspondingto the context C(·), and write “→” for the operation corresponding to theconnective of the same name, h(C(A → B)) = c(h(A) → b) = c(1 → b), andsince 1 → x ≈ x is an identity of Heyting algebras, 1 → b = b , so c(1 → b) =c(b), so h(C(A → B)) = c(b) = h(C(B)). Thus since h(C(A → B)) = 1, wehave h(C(B)) = 1, as required.

If one wants to show that A, C(A → B) �IL C(B) for all contexts C(p)

without invoking the fact this consequence relation has an algebraic semanticsin the shape of Heyting algebras, then the best way to proceed is to show by asingle induction on the complexity of C(p) that both A, C(A → B) �IL C(B)

and A, C(B) �IL C(A → B).Since Boolean algebras are Heyting algebras, the earlier argument works

to establish A, C(A → B) �CL C(B) (for all values of the metalinguistic vari-ables), while, going downward instead of upward, we also have the same resultfor �BCK—or again the (same) alternative syntactic argument suffices in thesecases too. (Oddly enough, neither Angell, à propos of �CL in [3, 4], nor Thomas[51] for �IL, provides any reason for taking the logics concerned to satisfythe rule, perhaps because they saw it as obvious. Indeed there is a simplerargument than either of those sketched here for the �CL case, as we shallmention below.) Not so for �BCI, however:30

28The rule of inference/rule of proof terminology is from Smiley [49], and this way of reading theunclear formulation in Angell [4] is that suggested by Church in [9]. (In Angell [3] apparently onlythe ‘rule of proof’ interpretation is at issue, though again the formulation is unclear, and indeeddoes not make sense as it stands.) For more information on Smiley’s distinction, see [24].29What about the reverse pattern, in which the minor rather than the major premiss is embedded inthe context C? With few exceptions (see [36]) consequence relations boasting a connective notatedby “→” will provide counterexamples to the general claim that C(A), A → B � C(B) on takingC(p) to be p → q, since we do not expect it to be the case that A → q, A → B � B → q.30This reflects the well known fact (see [27]) that BCI-algebras do not provide an algebraicsemantics—let alone an equivalent algebraic semantics, in the sense of [6]—for the logic afterwhich they were named.


Proposition 5.1 We do not in general have A, C (A → B) �BCI C(B).

Proof A counterexample suffices: Take A = q → q, B = r, C(p) = p → (A →B). Then we have �BCI A and �BCI C(A → B), in each case in virtue of schemaI; but C(B) is B → (A → B), i.e., r → ((q → q) → r), and this formula is notBCI-provable. ��

Indeed the example in the above shows that Angell’s rule does not workfor the BCI case even as a rule of proof, since the transition considered takesus from theorems as premisses to a non-theorem as conclusion. A similarobservation was made in Harrop [15] for a logic (in the “set of formulas” sense)of Ackermann’s (from [1]) which Harrop calls A. Here is Theorem 2 of [15],which needs to be quoted first to make intelligible a later passage we quote onthe theme of Angell’s rule. Harrop’s own notation is used (except that we use¬ for negation, and—in the next quotation—∧ for conjunction), with U , V, Was schematic letters. The word “completely” means “uniformly”:

Theorem 2 If U, V and W are w.f.f. such that W, U → V, V → U,¬U → ¬V and ¬V → ¬U are all provable (in A), then the result W* ofcompletely replacing U by V in W is also provable. [15, p. 498].

In the passage of principal interest, Harrop writes as follows, using “P” forclassical propositional logic conceived of as a formula logic or any axiomatiza-tion thereof:

If X is a provable formula, it can be shown that for any formula Z ,X ∧ Z and Z satisfy the conditions imposed on U , V in Theorem 2.Hence it can be deduced that if φ(X ∧ Z ) is provable in A, and X is alsoprovable, then so is φ(Z ), where φ(Z ) is the result of replacing X ∧ Zby Z throughout by φ(X ∧ Z ).

It can similarly be proved that if φ(X ∨ Z ) and X are provable in Athen so is φ(X), where φ(X) denotes the result of completely replacingX ∨ Z by X in φ(X ∨ Z ).

The last two results are related to methods which may be used in thecalculus P for obtaining some provable formulae from others. The nextresult shows that the corresponding classical result for implication doesnot hold for the calculus A.

Theorem 3 There are provable formulae X and φ(X → Z ) for whichφ(Z ) is not provable in A where φ(Z ) is the result of completely replacingX → Z by Z in φ(X → Z ). [15, p. 499].

Harrop’s Theorem 3 here is what we found in Proposition 5.1, or moreparticularly the corresponding ‘rule of proof’ version noted after its proofabove, with one exception: our C(·) notion, unlike Harrop’s φ(·) notation,does not require the replacements to be uniform (or ‘complete’). Whateverthe case may be with the strange system A, for the case of Harrop’s P, all of

78 L. Humberstone

the replacement results alluded to hold in a doubly strengthened form, firststronger in holding in the ‘rule of inference’ form and not just the ‘rule ofproof’ form stated by him, and secondly, in the non-uniform (or better: notnecessarily uniform) rather than the uniform versions he describes. That is, forall A, B, C, we have:

A, C (A ∧ B) �CL C(B);A, C (A ∨ B) �CL C(A);A, C (A → B) �CL C(B).

of which the third is the rule-of-inference form later used by Angell. Indeeda further strengthening, analogous to that evident from the algebraic andthe syntactic arguments above concerning the third case, is available, in thatin each case the �CL-statement interchanging the two context embeddingformulas across the “�CL” is also correct. To state the strengthened versionconcisely, we reach for ↔; the simple proof is left to the reader:

Proposition 5.2 For � as �CL, we have the following for all A, B, C:

(i) A � C (A ∧ B) ↔ C(B);(ii) A � C (A ∨ B) ↔ C(A);

(iii) A � C (A → B) ↔ C(B).

It is this simple argument in the case of (iii) (or the part involving the →-direction of the ↔-formula on the right, more precisely) that we had in mindbefore the statement of Proposition 5.1 as establishing the Angell’s rule (ofinference) for classical logic. The same Heyting-algebraic considerations asapply in case (iii) apply for (i) and (ii) also, using the facts that 1 ∧ b = b and1 ∨ b = 1 in the way that 1 → b = b was used for (essentially) case (iii) above,so Proposition 5.2 is also correct for � = �IL. Combining (i) and (iii) we get forthe classical and intuitionistic cases that A � C(A ∧ B) ↔ C(A → B), whilefor the classical case we can rewrite (iii) as A � (¬A ∨ B) ↔ C(B), and takingas a special case of (ii), ¬A � C(¬A ∨ B) ↔ C(¬A), so by the law of excludedmiddle we get the following principle (writing the “→” back in):

�CL (C (A → B) ↔ C (¬A)) ∨ (C (A → B) ↔ C(B)) ,

in which, as with the original �rep statements, all of the varying formulas appearembedded in the context C(·). Of course this can be readily checked usingf (v(A), yi)-style calculations, as with the version in which disjunction appearsin place of implication:

�CL (C (A ∨ B) ↔ C(A)) ∨ (C (A ∨ B) ↔ C(B)) .

Indeed this last claim is correct for any idempotent binary connective # in placeof the inner (i.e. C-embedded) occurrences of ∨.

Although consequence relations are not in the forefront of Harrop’s dis-cussion, we can make some useful conceptual distinctions by continuing to


discuss them—distinctions which will lead us back to the point picked up bySmiley in the passage (from [48]) quoted in Section 2. Let us say that A isvertically replaceable by B according to a consequence relation � (assumedhere for expository convenience to be finitary) when for all 1-ary contextsC1(p), . . . , Cn+1(p), whenever the relationship indicated above the line ob-tains, so does that below:

C1(A), . . . , Cn(A) � Cn+1(A)

C1(B), . . . , Cn(B) � Cn+1(B)

For clarity the replaceability relation of Section 2 will be referred to by contrastas horizontal replaceability.31 In introducing the terminology of one formula’sbeing synonymous with another according to a consequence relation, Smileyobserved what in this terminology would be expressed by saying that formulasare horizontally interreplaceable according to any given � if and only if theyare vertically interreplaceable according to �; either of these would do, asSmiley [48, p. 426f.] notes, as a definition of synonymy (or synonymity, ashe calls it). This is recorded in Proposition 5.3. We can refine the point, bynoting that for A and B to be horizontally interreplaceable, we do not needthe hypothesis that A and B are vertically interreplaceable: it suffices that Ashould be vertically replaceable by B. We put this a bit differently in 5.3(ii), tomark the contrast with the horizontal replaceability relation, which we saw inSection 2 sometimes was and sometimes wasn’t symmetric:

Proposition 5.3

(i) Formulas A and B in the language of any consequence relation � arehorizontally interreplaceable according to � if and only if A and B arevertically interreplaceable according to �.

(ii) Vertical replaceability according to any consequence relation � issymmetric.

Proof (We prove (ii).) Suppose A is vertically replaceable by B according to�. Then for any context C(p)—assuming w.l.o.g. that p does not occur in A—since C(A) � C(A), we can choose C1(p) = C(A), C2(p) = C(p), and rewritethis as C1(A) � C2(A), from which, vertically replacing A with B, we concludethat C1(B) � C2(B), i.e., C(A) � C(B). (Note that C1(·), as a function fromformulas to formulas, is a constant function, returning always the value C(A).)A similar argument yields C(B) � C(A). Thus vertical replaceability implieshorizontal interreplaceability and hence vertical interreplaceability. ��

31This horizontal/vertical contrast—discussed further in [23] and [24]—is taken from Scott [45]; itsaptness will be evident from the above ‘rule’ notation for the vertical case: we make a verticalreplacement when we make a replacement from above the line to below the line, whereas ahorizontal replacement is made from the left to the right of the “�”.

80 L. Humberstone

The above proof takes conspicuous advantage of the fact that the definitionof vertical replaceability, like that of horizontal replaceability (and the re-placement core of a consequence relation generally) does not require uniformreplacement, in the sense that it allows occurrences of A to survive (those inC(p)) the transition from C(A) to C(B). Just as, in Section 2, we considereda notion of uniform horizontal replaceability by requiring that A not be asubformula of C(p), so we could here consider a relation of uniform verticalreplaceability by subjecting C1(p), . . . , Cn+1(p), in the above definition ofvertical replaceability, to this same restriction. Uniform substitution as appliedto sequents rather than just formulas32 is the special case of uniform verticalreplacement of a propositional variable by an arbitrary formula. As Smiley’sdiscussion, quoted in Section 2, points out, and as this example itself illustrates,we can by no means conclude that one formula is (generally) vertically replace-able by another from the hypothesis that it is uniformly vertically replaceableby the latter, relative to a given consequence relation, or indeed—the caseconsidered by Smiley—just relative to a set of formulas such as the theoremsof a Hilbert system. (These can be thought of as provable sequents—see thepreceding footnote—with � = ∅). Thus Smiley’s point in the quotation fromSection 2, though put in terms of synonymy “in the theorem-sense”, actuallyapplies more generally to vertical transitions; for more on this, see Appendix 1.Nor is the uniform replaceability relation symmetric, as again this exampleshows. Is uniform substitution indeed the sole source of essentially uniformvertical replaceability—of one’s formula’s being uniformly vertically replace-able by another without being vertically replaceable in the more general sense?The following example doesn’t fall under the letter of substitution invariancebut seems to fall under its spirit.

Example 5.4 Consider the consequence relation associated with bi-minimallogic, i.e., the smallest substitution-invariant consequence relation extendingthe consequence relation of positive logic (primitive connectives: ∧, ∨, →)with two additional nullary connectives ⊥1 and ⊥2. As in the familiar minimalcase (Johansson’s Minimalkalkül, that is), where we have only one of ⊥1, ⊥2,no special logical properties are conferred on these constants (in terms ofwhich we could proceed, if we wanted, to define two negations ¬i A = A → ⊥i,i = 1, 2). If a sequent (as in note 32) σ belongs to this consequence relation, �,then so does the result σ ′ of replacing all occurrences of ⊥1 by ⊥2, though this isnot so if not all occurrences are replaced—since for example we have ⊥1 � ⊥1

while ⊥2 � ⊥1. So we have uniform vertical replaceability without general

32Here we think of a sequent 〈�, A〉 as an element of some � under consideration; of course a moresuggestive notation is usually used for such sequents with a suitable sequent-separator for whichat some points in Section 3 we allowed “�” itself to do double duty, and one may well restrictattention to the case of � finite, or change tack and have � be a sequence or a multiset. But onthe simplest version first outlined here, the substitution-invariant consequence relations are justthose � for which whenever, for a sequent σ ∈ � we also have σ ′ ∈ �, for any substitution instanceσ ′ of σ .


vertical replaceability. (Adapting the context notation C(p) from formulas tosequents, to get σ(p), A is vertically replaceable in the general sense by Baccording to � when for all sequent-contexts σ(p), if σ(A) ∈ � then σ(B) ∈ �.)But the example is really too much like uniform substitution to present agenuine alternative. Although the ⊥i are not propositional variables, we knowfull well that they behave like variables (as far as � is concerned), since theresult of uniformly replacing either of them by any formula whatever, and notjust by the other, preserves membership in �.

Thus an example would be desirable of a case of vertical replaceabilitywhich holds uniformly though not in general, and is not just a variation onthe theme of uniform substitution. With this “all points bulletin”, we concludeour discussion of replacement in logic, though several issues have not beencovered. In particular we have not returned to the specific point picked up bySmiley in the passage quoted in Section 2 from Harrop [15], which pertainednot to the general issue of vertical uniform replacement but to vertical uniformreplacement in theorems.33 (One might think of this in terms of a consequencerelation � but considering only replacements made in the transition fromsequents 〈�, A〉 ∈ � to 〈�′, A〉 when � = ∅.) And although we have separatedreplaceability from interreplaceability and distinguished the uniform from thenon-uniform versions thereof, we have given no attention to what might becalled the relation of strict interchangeability, which, like everything else,comes in a vertical and a horizontal form. To illustrate with the vertical form:A and B would be strictly interchangeable according to � just in case for any 2-ary sequent context of two variables σ(p1, p2) in no formula in which either Aor B occurs as a subformula, we have σ(A, B) ∈ � if and only if σ(B, A) ∈ �.34

This kind of exchangeability is more commonly encountered for connectivesthan for formulas, as in the case of tense logic, where it surfaces in sayingof various systems that a formula (or a sequent) is provable if and only if itsmirror image is, where the mirror image is the result of interchanging pastand future tense operators. The latter operators are 1-place connectives. (Cf.Bonnay and Westerståhl [7], mentioned in note 6.) Of course in the case of a 0-place connective we also have a formula, and indeed the ⊥1, ⊥2, of Example 5.4

33Theorem 5 of [15] reports that no two formulas of the logic under investigation are unrestrictedlyinterreplaceable in theorems, salva provabilitate, even though uniform interreplaceability obtainsin such cases as described in Harrop’s Theorem 2, quoted above after Proposition 5.1—whichinclude the cases described by Harrop in the passage quoted from [15] immediately after that.Note that the conditions of Theorem 2—that the two formulas should provably imply each otherand that their negations should also provably imply each other—coincide with the conditionsnecessary and sufficient, and not formulated redundantly to this end, for interreplaceability inintuitionistic logic with strong negation, when “their negations” is interpreted as meaning: theirstrong negations. (See Gurevich [14].) But the interreplaceability secured in this latter case, unlikeHarrop’s, is not restricted to uniform interreplaceability.34Note that since p2 need not actually occur in σ(p1, p2), this means that any sequent—or in thespecial case of empty left-hand sided sequents, any formula—with A but not B as a subformulamust not change its provability status (membership in �) when A is uniformly replaced by B. Ofcourse if A and B both occur as subformulas, they must be swapped with each other.

82 L. Humberstone

are in this sense strictly interchangeable in that any σ(⊥1, ⊥2) is bi-minimallyprovable just in case σ(⊥2, ⊥1), understood as above, is provable—though thisdoes not illustrate the phenomenon in a particularly focussed way since on thesame understanding (for the reasons given in Example 5.4) this is just the kindof interchangeability exhibited by any two propositional variables.

Appendix 1: Identity

Bigelow and Pargetter [5, Sections 3.2 and 3.5] give an axiomatic treatment offirst-order predicate logic with identity, and some modal extensions thereof.Here our interest is in what they say about identity, which in the interestsof direct quotation we follow them in representing by “=” rather than “≈”.They lay down two axioms, (A19) and (A20) governing ‘=’. The first isan unobjectionable principle to the effect that everything is identical withitself (“∀x(x = x)”). The informal explanation of the second principle (anaxiom schema rather than an axiom, to be precise) is again unobjectionable:“The second requires that if something is true of something and not true ofsomething, then the thing it is true of cannot be identical with the thing it is nottrue of” (p. 141). This, Bigelow and Pargetter say, can be summarized by thefollowing axiom schema, in which rather idiosyncratically, the authors use “λ”and “σ” as schematic symbols for terms:(

α ∧ ¬α[σ/λ])

→ (σ �= λ) . (A20)

They explain:

In this schema, λ is any name or variable occurring in α (so that α assertssomething to be true of the referent of λ). Similarly, σ is any nameor variable and α[σ/λ] is the formula which results from replacing everyoccurrence of λ in α by σ . Hence α[σ/λ] asserts something to be true ofthe referent of σ . What it asserts to be true of the referent of σ is exactlywhat α asserts to be true of the referent of λ. Hence α ∧ ¬α[σ/λ] assertsthat what is true of the referent of σ is not true of the referent of λ. If thisis so, then the referents of λ and σ cannot be identical—hence the axiom.

There is no particular need for the negative formulation of (A20), whichcould equally well, given that resources have been supplied for classicalpropositional logic, be presented in the form:

(α ∧ σ = λ) toα[σ/λ], (A20′)

with α[σ/λ] understood as before. It is precisely this understanding of thenotation—namely as denoting “the formula which results from replacing everyoccurrence of λ in α by σ”—that is surprising, since one expects to see, not“every”, but “one or more”. This means that, contrary to expectation, whilewe do have as instances of the schema (A20) and (A20′):

(Raa ∧ ¬Rbb) → a �= b (Raa ∧ a = b) → Rbb ,


various further implications which the informal justification in the quotedpassage from [5] would succeed in justifying,35 and which in any case we wantsince we are doing classical first-order logic with identity and not somethingelse, are not forthcoming as instances of these schemata. For instance, we donot get

(Raa ∧ ¬Rab) → a �= b (Raa ∧ a = b) → Rab ,

and nor we do we get

(Raa ∧ ¬Rba) → a �= b (Raa ∧ a = b) → Rba,

in this way. Noting that in these cases Raa can be obtained from Rab orRba by a uniform replacement of b by a, however, one might think toliberalize or extend the logical apparatus by allowing such replacements—stillrequired to be uniform—in either direction. (For example one could add asymmetry axiom for identity or replace (A20′) with σ = λ → (α ↔ α[σ/λ].) Forthis reason it is instructive to consider something of a ‘worst case’ example forthe approach.36 Let S be a three-place predicate symbol and consider insteadthe negative formulation—failing to be an instance of (A20)—and the positiveformulation—failing to be an instance of (A20′), namely

(Saab ∧ ¬Sabb) → a �= b and (Saab ∧ a = b) → Sabb ,

respectively. Now, whether or not it was an oversight on the part of Bigelowand Pargetter to formulate the indiscernibility (or “=-Elimination”) principlefor identity in terms of uniform as opposed to arbitrary replacements, in thesense that they were not intending to depart from the standard practice here,37

it turns out not to have been a mistake in the sense of actually renderingunprovable some things that should be provable.38 As Timothy Williamsonobserved to the author in a discussion on these matters some years ago,even the ‘worst case’ presented by ternary S above is not so hard to dealwith after all, using a uniform replacement version of the indiscernibility

35Even if some parts of the quoted passage—for example “What it asserts to be true of the referentof σ is exactly what. . . ”—raise qualms, as noted in [19], it can be assumed that these can bereformulated away.36Thanks are due here to Thomas Bull, a graduate student attending the author’s seminar onlogical aspects of Bigelow and Pargetter [5] in the mid 1990s.37For the record, John Bigelow has confirmed (personal communication) that they had no suchintention.38There is the other kind of mistake, too, of course, of making provable some things whichshould not be: a failure of soundness w.r.t. the intended semantics. And this mistake Bigelowand Pargetter certainly do make, giving an axiomatization [5, p. 104] with the aid of a rule ofuniform substitution, according to which “[t]he result of uniformly replacing any atomic sentencein a theorem by any given sentence will also be a theorem.” This would allow us to pass from“∀x(Fx) → Fa” to “∀x(Fx) → ¬Fa” or to “∀x(Fx) → (Fa ∧ ¬Fa)”,. . . and in fact one sees easilythat every formula ends up being provable in the system. This can be fixed in the usual way byusing axiom-schemata without a rule of uniform substitution or by formulating that rule muchmore judiciously—and as a rule licensing substitutions for (non-logical) predicate symbols ratherthan atomic formulas.

84 L. Humberstone

(‘of identicals’) principle. Giving his argument in informal natural deductionmode here, we proceed by reductio. Suppose Saab and ¬Sabb , and (for acontradiction) a = b . From Saab, together with a = b , we infer—replacing alloccurrences of a with b : Sbbb; from ¬Sabb together with a = b , we infer—again replacing all occurrences of a by b—¬Sbbb. This contradiction, Sbbb ∧¬Sbbb , establishes that from Saab ∧ ¬Sabb , a �= b follows, the claim writtenin implicational form as the first of the two S-formulas inset above.

While this shows—somewhat surprisingly—is that the uniform replacementversion of indiscernibility is not weaker than the more familiar non-uniformversion against the backdrop of classical predicate logic, since the implicationalform just arrived at is equivalent to the ‘positive’ form given as the second ofthe two inset S formulas above, it does raise a question about whether similarmanoeuvres would be available if the underlying predicate logic (or, more tothe point, propositional logic) extended by principles for “=” were intuitionis-tic logic. As far as the author can see, the closest we get in this setting, by meansof this technique, is the inference from Saab and a = b to ¬¬Sabb .39 Be thatas it may, there is another technique which we can use, namely that providedby Smiley [48] in the passage quoted above in Section 2. Admittedly, thatdiscussion related to replacements of formulas by formulas in propositionallogic rather than terms by terms in predicate logic, so some adaptation is calledfor. Smiley’s idea can be put like this. If we want to make a horizontal non-uniform replacement of A by B, on the supposition that the correspondinguniform replacements are available, then we can mark the positions in thecontext of replacement where A is to remain by p2, say, and those where it isto be replaced using the propositional variable p1, giving what we may take tobe a 2-ary context C(p1, p2). Then to show that C(A, A) � C(B, A)—a non-uniform horizontal replacement—for substitution-invariant �, we begin withthe fact that C(A, p2) � C(B, p2), given by uniform horizontal replacement,and then exploit substitution-invariance to put A for all occurrences of p2.(Here we are assuming that p1 and p2 do not occur in A, B; otherwise changevariables.) The same moves apply if we are not considering a consequencerelation but just a set of formulas (closed under uniform substitution), withC(A, A) → C(B, A), and so on, in the set. But let us continue to conduct thediscussion in terms of consequence relations as we shift to the case of identity.

Since that case involves a conditional replaceability claim, it is more closelyanalogous to the situation with equivalentiality, described after Corollary 2.6,with its additional E(A, B) on the left of the “�”:

E (A, B) , C (A, p2) � C (B, p2) ,

into which we substitute A for B to get the non-uniform version. FollowingBigelow and Pargetter [5] in using α as a schematic letter for predicate-logical

39Alternatively—cf. (A20) vs. (A20′), no longer equivalent in this weaker logical setting—we getthe inference from Saab and ¬Sabb to a �= b .


formulas but using t as a place-holder for terms (about which we shall be moreprecise below), Smiley’s manoeuvre then becomes:

a = b , α(a, t) � α(b , t),

to which we apply a term-for-term substitution to replace all occurrences of tby a. In the case considered above, in which we wish to pass from a = b andSaab to Sabb by a uniform replacement principle for identity, we apply thatprinciple to obtain the following, in which t is not a:

a = b , Stab � Stbb ,

at which point we make the uniform vertical substitution of a for t to obtainthe desired result. In the propositional case the role played by t here wasplayed by a propositional variable—in the first-order case two options presentthemselves: t could taken either to be a free (individual) variable or to be anindividual constant.40 The former option is mentioned under Remark (iii) onp. 98 of Troelstra and Schwichtenberg [52] for a quantifier-free formulationof (primitive recursive) arithmetic; it amounts to generalizing the substitutionrule of equational logic to cover the presence of sentence connectives and non-equational predicates. The latter option would in this setting be inappropriatesince we have an individual constant (for zero) with special properties, but itcould perhaps be considered for purely logical applications, especially by thosewith a distaste for open formulas appearing in sequents or �-statements.

As to those with no such objections, one must of course distinguish twopolicies as to the interpretation of such variables as occur free on one or bothsides of the � (or sequent separator): the global interpretation which treatseach open formula as amounting to its own universal closure, and the localinterpretation which understands the whole � as tacitly prefaced by “for allvalues of the variables”. On the global interpretation, substitution becomes amatter of uniform horizontal replacement, while on the local interpretation,such replacements would be disastrous and substitution is a matter of uniformvertical replacement, exactly as in propositional logic with the conventionalconsequence relations in mind (�CL, �IL, etc.). The local interpretation in thecase of equational logic amounts to a consideration of quasi-identities as op-posed to conditional claims about equations holding as identities (to the effectthat if those on the left hold ‘identically’ in any algebra of the right similaritytype, those so does that on the right).41 Because of the naturalness of suchlocal interpretations, remarks like the following from Burris [8, p. 171] seemmisleading: “Replacement is essentially a nonuniform version of substitution.”For the most familiar consequence relations, replacement and substitutiondiffer not just in respect of uniformity but also, and no less significantly, over

40Evidently a term resulting from the application of a function symbol to a term—closed or open—would not be a suitable choice of t, since this would lack the required generality to allow thesubsequent substitution of a for t.41Does this mean that we need to distinguish two separate relations, one of equality and one ofidentity (say)? Certainly not!—pace Henkin [16, p. 608f.]

86 L. Humberstone

the fact that the former involves essentially horizontal and the latter essentiallyvertical transitions.

Appendix 2: Two Proofs for Section 3

Lemma 3.4 For all formulas A, B:

(i) A, B �∗ A ∗ B;(ii) �, A �∗ B implies �, ¬B �∗ ¬A;

(iii) A ∗ ¬A �∗ B ∗ ¬B;(iv) ¬(A ∗ ¬A) �∗ ¬(B ∗ ¬B).

Proof

(i) A, B ∗ B �∗ A ∗ B by (N1) and B �∗ B ∗ B by (R3), from which (i)follows (by a Cut on B ∗ B).

(ii) Suppose �, A �∗ B; from this and the fact that B, ¬B �∗ ¬A (by (N1))we get �, A, ¬B �∗ ¬A. But of course we have �, ¬A, ¬B �∗ ¬A. So(N2) yields (ii).

(iii) A ∗ ¬A, B �∗ B ∗ ¬B, by (M1), and A ∗ ¬A, ¬B �∗ B ∗ ¬B by (M1)and (R2). From these (iii) follows by (N2).

(iv) From (iii) by (ii) (with � = ∅) and re-lettering (A to B and vice versa).��

Theorem 3.5 The consequence relation �∗ is determined by V∗:∧,¬ ∪ V∗:∨,¬.

Proof Soundness is routine, so we check completeness. Denote by �∗ theconsequence relation here described syntactically, and suppose � �

∗ B. Let�+ be a superset of � for which �+

�∗ B while for all �′

� �+, �′ �∗ B. (ALindenbaum argument guarantees the existence of such a �+; we shall makeuse without special comment of the well-known fact that �+ is deductivelyclosed—i.e., for the present case that �+ �∗ D implies D ∈ �+.) In view of(N1) and (N2), for each formula A, exactly one of A, ¬A, is an element of �+.Thus the valuation v defined by:

v(A) = T if and only if A ∈ �+ (all formulas A),

obeys the boolean condition for negation, as well as verifying every formula in� but not B, so to show—what we require for completeness—that v ∈ V∗:∧,¬ ∪V∗:∨,¬ we need to show that v interprets ∗ as ∧ or else as ∨. That is, we needthat either (1) or (2), respectively, is satisfied:

(1) For all A, B, A ∗ B ∈ �+ if and only if A ∈ �+ and B ∈ �+;(2) For all A, B, A ∗ B ∈ �+ if and only if A ∈ �+ or B ∈ �+.

In view of the above point about ¬, picking a propositional variable q (anyformula could serve in this capacity), we have that either ¬(q ∗ ¬q) ∈ �+ orq ∗ ¬q ∈ �+. (As we know from Lemma 3.4(iii), (iv), this suffices for every


formula to behave w.r.t. �+ as q does in these two ‘marker’ compounds.) Wenow verify that in the former case, condition (1) is satisfied, while in the latter,condition (2) is satisfied, completing the proof. Accordingly, suppose first that¬(q ∗ ¬q) ∈ �+. The “if” direction of (1) does not even use this hypothesis,since it is given by Lemma 3.4. For the “only if” direction of (1), assumethat A ∗ B ∈ �+. Since ¬(q ∗ ¬q) ∈ �+, by (M2) we get A ∈ �+, and similarlyby (M2), with assistance from the commutativity condition (R2), which putsB ∗ A into �+, we have B ∈ �+. Thus condition (1) is satisfied. Now supposeinstead that q ∗ ¬q ∈ �+. We want to check that condition (2) is satisfied. The“only if” half does not require our supposition since if A ∗ B ∈ �+ and A /∈ �+,then B ∈ �+ in view of (M3). For the converse, if A ∈ �+ then we appeal to(M1) and our supposition that q ∗ ¬q ∈ �+ to conclude that A ∨ B ∈ �+, andsimilarly if B ∈ �+, with an added appeal to (R1), as in the case of establishing(1) on the supposition that ¬(q ∗ ¬q) ∈ �+. ��Acknowledgements For their comments on various portions of this material (see notes 9, 36,and the body of Appendix 1, respectively) I am grateful to Greg Restall, Thomas Bull and TimWilliamson; thanks also to Steve Gardner, as well as to two referees for this journal, for numerouscorrections.

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