Pertinent Reasoning

8/9/2019 Pertinent Reasoning

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Pertinent Reasoning

Katarina Britz

Meraka Institute and UNISAPretoria, South Africa

[email protected]

Johannes Heidema

University of South AfricaPretoria, South Africa

[email protected]

Ivan Varzinczak

Meraka Institute, CSIRPretoria, South Africa

[email protected]

Abstract

In this paper we venture beyond one of the fundamentalassumptions in the non-monotonic reasoning commu-nity, namely that non-monotonic entailment is supra-classical. We investigate reasoning which uses an infra-classical entailment relation that we call pertinent en-tailment. The notion of pertinence proposed here is

induced by a binary accessibility relation on worldsestablishing a link (representing some form of perti-nence) between premiss and consequence. We showthat this notion can be captured elegantly using a sim-ple modal logic without nested modalities. One roadto infra-classicality has been studied extensively, that ofsubstructural logics, which weaken the generating en-gine of axioms and inference rules for producing en-tailment pairs (X, Y). Here we follow an alternativestrategy: we first demand that X entails Y classically,and then, with supplementary information provided byan accessibility relation, more, trimming down the setof entailment pairs to infra-classicality. It turns out thatpertinent entailment restricts well-known paradoxesavoided by relevance/relevant logic in an interestingway. We present its properties, showing that it pos-sesses other non-classical properties, like strong non-explosiveness and non-monotonicity, and we discusswhich inference rules traditionally considered in the lit-erature it satisfies.

Introduction

Classical logic is, in a sense, the logic of complete igno-rance. In a classical entailment X |= Y no informationwhatsoever beyond that encapsulated locally in X and Y plays any role at all. Extra information may be employedto construct altered entailment relations, which sometimesallow more pairs (X, Y) into the relation, going supra-

classical, or fewer, going infra-classical, or just going non-classical.

If rather specific, the extra information is usually ex-pressed as syntactic rules or is of a semantic nature and typ-ically involves an (often binary) relation on W, the set ofworlds. More generally and vaguely the extra may be adesire to adapt classical entailment |= in order to obtain anentailment relation which more closely resembles common-sense human reasoning as precipitated in natural language.

Pertinent reasoning is quite specific. It is based on infra-classical entailment relations which employ the informationpresent in a binary accessibility relation R on Win so far asthis is embodied in standard modal operators 2 and 3 withtheir R- and R (converse ofR)-semantics. The informationin R is considered to be pertinentto the sense in which X en-tails Y. This will be motivated in more detail and illustratedby means of examples.

In this work we consider infra-classical pertinence rela-tions. One road to infra-classicality is well known, that ofsubstructural logics (Restall 2006), which weaken the gen-erating engine of axioms and inference rules for producingentailment pairs (X, Y). In pertinent reasoning we follow, ina sense, the opposite strategy: we first demand that X |= Y,but then (invoking R) more, trimming down the set of entail-ment pairs to infra-classicality.

The present text is structured as follows: after some logi-cal preliminaries, we motivate and define infra-classical per-tinent entailment. Following that, we investigate the non-classical properties satisfied by our entailment relation. Wethen derive a set of inference rules for pertinent reasoning.

After a discussion of and comparison with related work, weconclude with an overview and future directions of research.

Logical Background

We work in a propositional language L over a set of propo-sitional atoms P, together with the two distinguished atoms (verum) and (falsum), and with the standard model-theoretic semantics. Atoms will be denoted by p , q , . . . Weuse X , Y , . . . to denote classical (Boolean) formulas. Theyare recursively defined in the usual way, with connectives ,, , and .

We denote by W the set of all worlds (alias propositionalvaluations or interpretations) w : P {0, 1}, with 0 de-

noting falsity and 1 truth. Satisfaction ofX by w is denotedby w X. With Mod(X) we denote the set of all modelsofX (propositional valuations satisfying X).

Classical logical consequence (semantic entailment) andlogical equivalence are denoted by |= and respectively.Given sentences X and Y, the meta-statement X |= Ymeans Mod(X) Mod(Y). X Y is an abbreviation(in the meta-language) ofX |= Y andY |= X.

We now extend our propositional language with one


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modal operator 2 (Blackburn, van Benthem, and Wolter2006). We will denote complex formulas (possibly withmodal operators) by , , . . . They are recursively definedas follows:

::= X | 2X | | | | |

With Fwe denote the set of all complex formulas of our lan-guage. Note that no nesting of modal operators is allowed.Our only reason for this restriction is expositional simplic-

ity. Our work extends naturally to a fully (multi-) modalcontext. Here we will also use the modal operator 3, whichis the dual operator of2, defined by 3X def 2X.

Definition 1 A frame is a tuple F = W,R, with W the setofworlds and R W W the accessibility relation on W.

For simplicity of exposition our notion of frame does notfollow the standard notion from modal logics: here no twoworlds satisfy the same valuation. Nevertheless, all we shallsay in the sequel can be straightforwardly formulated forstandard frames.

Sometimes it will be useful to consider the identity rela-tion on W. It is defined as idW := {(w, w) | w W}.For purposes that will become clear in the sequel, in this pa-

per we consider only reflexive frames, i.e., we assume thatidW R.

Definition 2 Given a frame F = W,R,

w F

p (p is true at worldw of frame F) iffw p;

w F2X iffw

FX for every w such that(w, w) R;

w F

iffw F

, i.e., if it is not the case thatw F

;

w F

iffw F

andw F

; truth conditions for the other connectives are classical.

Given a frame F = W,R and formulas , , we say

that entails with respect to frame F (denoted |=F

)

if and only if for every w W, ifw F

, then w F

. If

|=

F

, we say that is (logically) valid (or a tautology)in frame F and we denote this as |=

F. Clearly, for X and

Y both without modal operators, X |=F

Y is equivalent toX |= Y (X entails Y classically).

Given modal operators 2 and 3, we can speak of theirconverse operators: 2 and 3, respectively. The follow-ing definition follows straightforwardly from Definition 2by applying the converse of the accessibility relation R, butsince we are going to refer constantly to these notions westate them here:

Definition 3 Given a frame F = W,R,

w F2Xiffw

FX for every w such that(w, w) R;

wF3

X iffwF

X for some w such that(w, w

) R.Finally, we have another useful definition:

Definition 4 LetF = W,R. For any U W:

R[U] := {w W | there is a w U s.t. (w, w) R}; R[U] := {w W | there is a w U s.t. (w, w) R}.

Hence, given a frame F = W,R and a formula X, it iseasy to see that e.g.

Mod(3X) = R[Mod(X)] and Mod(3X) = R[Mod(X)].

The Road to Pertinence

Classical semantic entailment X |= Y says that Mod(X) Mod(Y), i.e., that every X-world is a Y-world. This for-mal definition does of course not capture all of the intu-itive connotations of natural language phrases like if X,then Y, X entails Y, or from X, Y follows logically.Many of the properties of |= that may strike some peopleas odd result from the following fact: As long as every

X-world is a Y-world, X |= Y and hence (equivalently)Y X(Y X) hold, and the Y-worlds which are notinMod(X) are completely free and arbitrary, in the sense thatthey need have nothing whatsoever to do with X or any ofthe X-worlds. Any arbitrary (trivial) dilation ofMod(X)yields a Y such that X |= Y. One intuitive connotation ofentailment is that more, some additional relation of rele-vance or pertinence, should hold between X and Y.

Existing relevance/relevant logics (Anderson and Belnap1975; Anderson, Belnap, and Dunn 1992) share some of theaims that we have with the present paper, but (at least in ourview) they harbour certain less attractive features:

Remark 1 Most of the literature on relevance/relevant log-

ics confuse and conflate entailment with the conditional con-nective or material implication (), the first being a no-tion at the meta-level and the second at the object level.According to Anderson and Belnap, it is philosophicallyrespectable to confuse implication or entailment with theconditional, and indeed philosophically suspect to harp onthe danger of such a confusion (Anderson and Belnap1975, p. 473).

Remark 2 Relevance logics traditionally tend to start outfrom syntactic considerations to rule out some classical en-tailments as irrelevant and then afterwards contrive to con-structing a matching semantics not always completelyconvincingly, it must be said. Syntax is protean (shape-

shifting): infinitely many syntactically different sentencesrepresent the same proposition. Granted: there are normalforms. But our contention is that we should start from se-mantic notions and then find apt syntax to simulate the se-mantics.

Remark 3 Sometimes philosophical, metaphysical ideasget admixed into the relevance endeavour ideas like di-aletheism (the thesis that some contradictions are true) orbelief in impossible worlds, like inconsistent models ofarithmetic (Priest 2002). These notions may bemire an al-ready complex issue.

Remark 4 Relevance logics traditionally pay scant atten-tion to contexts. What is relevant in one reasoning context

may not be so in another context. For instance, legal argu-ment differs from intuitionistic proof in mathematics.

How our approach deals with these four issues will be-come clear in the sequel.

All of this is not to say, of course, that relevance/relevantlogics are not appropriate candidates for pertinent reasoning.Here we follow an alternative (not antagonistic) approach.This is what we develop in the rest of the paper.


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A Pertinent Entailment Relation

The notion of entailment is an asymmetric, directed relation.In the forward (from premiss to consequence) direction itpreserves truth, or at least plausibility; in the backward di-rection it carries along falsity, or at least implausibility. Inthe forward direction, it usually loses information, while inthe backward direction it usually gains information (think ofhypothesis generation or abduction, for example).

In a direct proof of an entailment there is a step-by-steplogical movement from premiss to consequence; in an in-direct proof, such as reductio ad absurdum or by contrapo-sition, from the negation of the consequence to the negationof the premiss directed movement, to and from.

This intuitive notion of entailment as a species of accessrelation between sentences or propositions starting at thepremiss access to the consequence, or starting at the conse-quence access from the premiss this idea of entailment asaccess has a natural analogue in the accessibility relationbetween worlds in modal logic. We intend to anchor ourbrand of relevant entailment, called pertinent entailment,in some accessibility relation on worlds. To be specific: inthe entailment relation which we choose as the focus of thispaper, those totally unconstrained Y-worlds which are notX-worlds in a classical entailment X |= Y, should be dis-ciplined. They should be admitted only if they have somepertinence to the premiss X a pertinence that those Y-worlds which are X-worlds of course automatically have.

In our new infra-classical entailment ofY by X, the con-dition that we impose upon the (previously wild) Y X-worlds is that now each of them must be accessible fromsome X-world. This establishes the mutual pertinence ofXand Y to each other. Of course, this assumes that the spe-cific accessibility relation chosen for this purpose reflectsthe required type of pertinence. (See below for more on thedefinition ofR and examples.)

Given a propositional language, a frame F = W,R,with R a reflexive (and for some results in the sequel transi-tive) accessibility relation on the set of possible worlds W;and a modal operator 3, corresponding to the converse re-lation R ofR:

Definition 5 X |< Y ifand onlyifX |=F

Y andY |=F3X.

Intuitively, Definition 5 states that premiss X and conse-quence Y are mutually pertinent if and only if X entailsY (classically) and every Y-world in frame F is accessi-ble from some X-world importantly, the Y X-worlds(Figure 1). (The X-worlds are each accessible from itself.)

W

X

Y

Figure 1: Pertinent entailment of Y by X: Mod(X) Mod(Y) and any Y-world is accessible from some X-world.

In the symbol |


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W

X XY

X

X

R R

Figure 3: The rock ofX-worlds shields the island of Y-worlds from the R-flow of the X-stream.

Furthermore, givenR1 andR2, ifR1 R2, then |


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semantic meaning, are not treated differently and can be sub-stituted anytime and anywhere by each other in our seman-tic approach. All the tautologies together are just one un-differentiated element in the Lindenbaum-Tarski algebraof propositions, or, if you like, logical equivalence classesof sentences. Relevance/pertinence makes only sense rela-tive to some extra semantic information (whether reflectedon the object/syntactic level in a sentence or available only

on the semantic/meta-level), while undifferentiated W hasnone. Pertinent entailment needs to move out of the domainof triviality, of tautologies, of huh? we know nothing!

Classically, we have contraposition: X |= Y is equivalentto Y |= X. Not so for |


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So, assuming X |< Y, we have no guarantee that X X |

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Pertinent Reasoning