An alternate formulation of Kripke's theory of truth

LAWRENCE DAVIS

AN ALTERNATE FORMULATION OF

KRIPKE’S THEORY OF TRUTH*

In his paper ‘Outline of a Theory of Truth’ (Jowd of Philosophy, Nov. 6, 1975) Saul Kripke gives the semantics for a language 2’which contains its own truth predicate. This semantics (hereafter, the ‘K semantics’) contains a mathematical defmition of the true and false wffs of 9, based on the properties of a hierarchy of interpretations of Z One impression which the reader may be left with upon reading the Kripke paper is that the K semantics is intrinsically a hierarchy-based system. In fact, although the semantics is presented in hierarchical terms and there is a good deal of discussion of language hierarchies in Kripke’s paper, the hierarchical apparatus is not essential to the assignment of truth values in the K semantics. I have devised a semantics which is equivalent to the K semantics which procedes from complex wffs to atomics, as most contemporary semantic systems do. As this version of the K semantics may make it more amenable to comparison with such systems, a sketch of it follows.

1. A DOWNWARD SEMANTICS FOR 2

The semantics to be given employs trees of a certain sort. Each node of the trees to be used contains as members a wff of q a truth-value, and an index to distinguish it from other nodes containing the same wff and truth- value. The trees we consider are constructed so that each node n with wff s and truth-value TV directly dominates a node or nodes with the wffs and truth-values required to guarantee TV for s. The only terminal nodes are atomic nodes; in order to be acceptable, each downward path must contain a terminal node for which the interpretation assigns its first member the truth-value which is its second member.

The following definitions make these ideas more precise: A node is an ordered 3-tuple whose members are: a wff of g ‘T’ or %‘,

and an integer. A tree is an ordered pair W, DD), where N is a set of nodes and DD is a

relation from N to N, such that

Journal of Philosophical Logic 8 (1979) 289-2%.0022-3611/79/0083-0289$00.80 Copyright 0 1979 by D. Retdel AtbWing Co., Dotdrecht, Holknd, and Boston. (I.SA.

290 LAWRENCE DAVIS

(0

00

N is not empty,

for every n and m, distinct members of N, there is a sequence of members of N(n, nr , . . . ,ni,m)suchthatDDn,nr or DDnl,n,DDn,,nz orDDnz,nl ,... ,DDni,morDDm,nf,

(iii) DD is irreflexive, and

(iv) there is no sequence of members of N(ni . . . ni) such that D&,n2,. . . DDniSl, ni and DDni, nl .

(Trees are often represented in a form which is visually more perspicuous, as displays in which, for each n and m in N, if DDn, m, n is written above m with a line connecting the two. In this alternate notation, the tree ({(p vq),F, lo), Q&F, 21, (q,F, 3% ((<p vq,F, 101, (p,F, 2)), ((p vq, F, IO), (4, F, 3)))) would be represented in this form:

((P vd, F, 10)

<pF& (::F3) 9 t ,, .

It requires less effort to grasp the relationships among nodes when the visual display format is used, but the precision of the set-theoretical notation makes it more amenable to use here.)

If n and m are in N and DDn, m then n directly dominates m. If there is a sequence of members of N(nr . . . ni ) such that each member

directly dominates the next, n 1 dominates nl. Any sequence of nodes in N(nr , n2, . . . > such that (i) each node directly

dominates the next, and (2) every node which directly dominates another node of N has a successor in the sequence is a dominance path from n, .

Any node of N which dominates every other node of N is the topmost node of the tree.

A subtree t’ of a free t is a tree such that (i) the first member of t’ is a subset of the first member oft, (ii) t’ has a topmost node, and (iii) the set of dominance paths from the topmost node oft’ is identical to the set of dominance paths from that node in t.

A semantic tree for L? based on an interpretation (D, V) is a tree t such that

(9 t has a topmost node,

(ii) for every node n oft, where i,j, and k are integers,

ALTERNATE FORMULATION OF KRIPKE’S THEORY 291

Dl if n is of the form (Pkol . . . mr, T, i) or (P&z1 . . . a,,F,i),nisaterminalnode.

D2 ifnisoftheform(-p,T,i),or(-p,F,i),thenn directly dominates a single node of the form (p, F, j> or (p, T, j) respectively.

D3 if n is of the form (@ vq), T, i) then n directly dominates a single node of the form (p, T, j> or (4, T,i). if n is of the form <@ v q), F, i ) then n directly dominates exactly two nodes of the form (p, F, j) and (q,F,k).

D4 if n is of the form ((Ex)(pX), T, i > then n directly dominates a single node of the form (pa, T, j), where a is a name of LZ if n is of the form ((Ex)(AE), F, i > then n directly dominates all nodes of the form (Pa, F, -4, where a is a constant of A!’ and Ais an integer, not necessarily the same for each node.

(Note that this rule employs a substitutional interpretation of the quantifier rather than Kripke’s objectuai interpretation. It is required that Z contain names of all its wffs, as Kripke stipulates, for the Downward semantics to be equivalent to the K semantics. For the equivalence to hold in other languages which do not have names for every wff, a more compli- cated semantics would need to be given.)

D5 if n is of the form (Ta, T, i > then n directly dominates a single node of the form (p, T, j >, where a names p, unless p is not a wff of L%. if n is of the form (Tu, F, i 1 then n directly dominates a single node of the form (p, F, j ), where a names p, unless p is not a wff of L?? if n is of the form ( Ta, T, i ) or (Tu, F, i > and (I names an entity which is not a wff of 2, then n is a terminal node.

D6 relationships of direct dominance not specified by Dl- D5 are not ahowed on a semantic tree based on an interpretation CD, Y>.

LAWRENCEDAVIS 292

(iii) Every dominance path oft contains an acceptable terminal node. (if n is of the form Pial, , . . , I,, T, i), n is acceptable iff <V(u,). . . V(a,)> E V(Pi). if n is of the form (Pzal, . . , , u,, F, i), n is acceptable iff <V(d), . . . ) V(a,)> 4 v(P;). if n is of the form (Ta, F, i ), n is acceptable iff V(u) is not a wff of 2

No other terminal nodes are acceptable.)

Given these definitions, the Downward semantics for 9’ is as follows: A wff s of P’is true on an interpretation (D, V) if there is a semantic

tree for Phased on (D, V) with topmost node of the form (s, T, i), where i is any integer.

A wff s of P’is false on an interpretation (D, V) if there is a semantic tree for 9 based on (D, V) with topmost node of the form 0, F, i), where i is any integer.

Otherwise, a wff s of P’is undetermined on (D, V).

2.THEEQtiIVALENCEOFTHEDOWNWARD AND K SEMANTICS

Given any interpretation of 2’ the Downward semantics assigns each wff of 9’the same truth-values as the K semantics. An outline of the proof of this result follows.

THEOREM 1. If there is a semantic tree for 9 based on an interpretation (D, V) with topmost node of the form (s, T, i > (or (s, F, i )) then s is true (or false) according to the K semantics.

DEFINITION. 7%e length ofa semantic tree for 9 based on an interpretation (D, V> is the number of nodes on the longest dominance path from the topmost node oft.

The theorem is proved by induction on 1, the length of semantic trees. For the base case, the length oft is one and t consists of a single, acceptable terminal node n. The conditions for acceptable nodes in the D semantics are the same as the truth-conditions in the K semantics.


For the induction case, assume the theorem true for all semantic trees of length less than 1. We notice that any subtree of a semantic tree for 9 based on (D, V> is a semantic tree for A? based on (D, V), since no acceptable nodes of a subtree are deleted when a subtree is formed, and all dominance paths are preserved. Thus, the subtrees oft created by deleting the topmost node of t from N are semantic trees of length less than 1. We consider the form oft and its relation to the subtrees obtained by deleting its topmost node from n case by case. For example, suppose the topmost node of c is of the form <(jr vq), F, i). The two subtrees oft with topmost nodes of the form (p, F, j > and (4, F, k) are semantic trees. By hypothesis, the K semantics rules p and 4 false. K3 requires that @ v 4) also be false. Hence in this case the two semantics agree.

The other cases are worked out in a similar way and the theorem is proved.

THEOREM 2. Given a wff s of 9’and an interpretation (D, V), ifs is true (or false) according to the K semantics, then s is true (or false) according to the Downward semantics.

Roof. Assume s true (or false) in the K semantics. Then, as Kripke proved, s first becomes a member of the Sli (or ,#Q) at some stage a in the hierarchy -Epi(Sli, 5’2i). We construct a semantic tree t for 9’ based on (D, V) in the following way. First we construct a tree t’ with nodes whose members are ordered 4-tuples. The first three members of each node are as in the Downward semantics. The fourth member is the subscript of the earliest language for which the first member of the node gets a truth value in the K semantics.

We construct t ’ recursively, by specifying that each of its nonterminal nodes (s, TV, i, j) directly dominates nodes whose first two members are wffs and truth-values guaranteed in the K semantics ifs is to have truth value TV. We use the following convention: For any wff s of 2 which first receives a truth value TV at stage a, the associated node of s is (s, TV, k, a). The recursive rules sometimes specify that certain associated nodes are in f ’ It is understood that such nodes are new nodes; that is, when the existence of a node of this form is guaranteed, k is to be chosen so that, for all nodes alreadyinNoftheform(s,TV,m,a),k$m.

The tree t’ is (N, DD ), where N and DD are the smallest set and relation such that

LAWRENCE DAVIS 294

(9

(ii)

ifs is true in the K semantics, (s, T, 1, a) EN, and ifs is false in the K semantics, (s, F, 1, a) E N, where a is the ordinal of the earliest stage at which s was true or false.

for each n EN,

Cl ifnisoftheform(P$zr...a,,T,i,j)or Vial. . . a,, F, i, j) then n is a terminal node.

C2 ifnisoftheform(-p,T,i,j)then(p,F,k,j)EN and n directly dominates it. ifnisoftheform(-p,F,i,j)then(p,T,k,j)EN and n directly dominates it.

C3 if n is of the form (@ vq), T, i, j> then (p, T, k, m ) or (4, T, k, n > is a member of N and n directly dominates it. (We know from K3 that at least one of p and q is true. Choose the one which gets its truth value at the lowest level and add its associated node to N. If both get their truth-value at the same level, add the node associated with p.) if n is of the form (@ vq), F, i,j> then (p, F, k, m) and (p, F, k, n) are members of N and n directly dominates them.

C4 if n is of the form ((Ex)(Ar), T, i, j) by K4 there is at least one wff of the form PQ, which is true. Choose the one which gets its truth-value at the lowest level in the hierarchy of languages. (If there are several, choose the one with the lowest index on the constant). Add its associated node to N. It is directly dominated by m. if n is of the form ((Ex)(Ac), F, i, j) by K4 all wffs of the form Pa,, have value Fat stage j or lower. Add the associated node of each to N. Each is directly dominated by n.

CS if n is of the form (Tu, T, i, j > and V(u) is not a wff of 9, n is terminal. if n is of the form (Ta, T, i, j> (or (Ta, F, i, j>) and V(a) is a wff of 3 then (V(a), T, k, m > (or (V(a), F, k, m>) is a member of N and n directly dominates it.


We establish by inspection that any tree t formed by the application of Cl -C5, with the last member of each node deleted, satisfies clauses (i) and (ii) of the definition of a semantic tree for 9 based on (D, V). It remains to show that every dominance path of such a tree contains an acceptable terminal node.

Assume not. Then there is a dominance path p of t which contains no acceptable terminal node. The fourth member of each node on p is the same as the fourth member of the node which preceded it or it less than the fourth member of the node which preceded it (by inspection of Cl -C5 and the conditions on the Kripke hierarchy). There is some ordinal i which is the lowest ordinal which appears as fourth member of a node in p. Consider thesetofnodesnl,n2... which have i as fourth member. We know that this set is of length m + 1, where m is the number of logical operators in the wff of n, , by inspection of Cl-CL These are the only clauses which allow direct dominance of nodes with identical fourth members, and they require that the node directly dominated have one less logical operator than the node which directly dominates. Hence there is a node nf which is the last node of p. It must be a terminal node (by inspection of Cl -C5). By hypoth. esis it is not acceptable. But this is impossible. The second member of ni is T (or F) iff the first member of ni is true (or false) in the K semantics. But the conditions for truth and falsehood in the K semantics are the same as the conditions for acceptability in the Downward semantics. Hence the assumption is false.

(Note that it is crucial to this proof that the fourth members of the nodes of N be the ordinals of the earliest stages at which their first members receive truth values. A wff of the form ‘Ta’ receives its truth value at some stage after the wff named by a receives its truth value. If this were not the case, it would not be demonstrable in the way employed here that the number of nodes on a dominance path with identical fourth member is fmite.)

There is one difference between the semantics in its K and Downward forms which seems worth noting. The natural definition of the level of a wff in the Downward system is the length of the semantic tree in which it is first member of the topmost node. On this defmition no wff will be of level greater than omega, for no dominance path contains two nodes with the same first member (each first member has as last member the level at which it first acquired its truth value; no two nodes with same last member

296 LAWRENCE DAVIS

in a dominance path have first members of same length.) There are omega wffs in 2 Hence no dominance path has more than omega members.

University of Hawaii at Manoa

NOTE

* A prototype of this paper was written for a seminar on the Liar paradoxes taught at the University of Massachusetts at Amherst by Terence Parsons. My understanding of the Kripke paper is based to a large extent on Parsons’ lucid instruction.

Thanks to a JPL referee, this paper lacks some obscure passages and mistakes which infected an earlier &aft.

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An alternate formulation of Kripke's theory of truth