On the Strict–Tolerant Conception of Truth

Abstract

We discuss four distinct semantic consequence relations which are based on Strong Kleene theories of truth and which generalize the notion of classical consequence to 3-valued logics. Then we set up a uniform signed tableau calculus(the strict–tolerant calculus), which we show to be sound and complete with respect to each of the four semantic consequence relations. The signs employed by our calculus are , , and , which indicate a strict assertion, strict denial, tolerant assertion and tolerant denial respectively. Recently, Ripley applied the strict–tolerant account of assertion and denial (originally developedby Cobreros et al. [2012] to bear on vagueness) to develop a new approach to truth and alethic paradox, which we call the Strict–Tolerant Conceptionof Truth (STCT). The paper aims to contribute to our understanding of STCT in at least three ways. First, by developing the strict–tolerant calculus.Second, by developing a semantic version of the strict–tolerant calculus(assertoric semantics), which informs us about the (strict–tolerant) assertoric possibilities relative to a fixed ground model. Third, by showing that the strict–tolerant calculus and assertoric semantics jointly suggest that STCT's claim that the strict and tolerant can be understood in terms of one another has to be reconsidered.

Keywords:

• truth
• paradox
• assertion
• denial
• bilateralism
• non-transitive consequence

Notes

¹ Here is a name for sentence σ and ↔ is the material biconditional of classical logic. Tarski [1944] famously imposed the assertibility of all instances of the T-schema as an adequacy condition on theories of truth.

² The Liar sentence valuates as ½ in every fixed point and, in particular, is not a theorem of K3T. On the other hand, is a theorem of LPT.

³ However, see Soames [1999] for a defence of K3T and see Horsten [2009] for a defence of a proof-theoretic representation of a logic according to which an inference is valid when it is both K3T and LPT valid.

⁴ To see that (5) holds, plug in the Liar for α and take ‘snow is black’ for β.

⁵ Observe that (6) testifies that, unlike K3T and LPT, the relation of strict–tolerant consequence is not understood in terms of ‘preservation of designated value’.

⁶ The fact that satisfies the transparency of truth is a non-distinguishing property: also K3T, LPT and the logics advocated by, amongst others, Field [2008] and Beall [2009] satisfy the transparency of truth. Most notably, the logics associated with the revision theory of truth of Gupta and Belnap [1993] do not satisfy the transparency of truth.

⁷ Especially Field [2008] and Beall [2009] argue for the central importance of the transparency of truth.

⁸ However, note that the negation of a T-biconditional that is instantiated with a paradoxical sentence such as the Liar is also a theorem of .

⁹ A sentence is ‘paradoxical’ just in case it receives value ½ in every SK fixed point.

¹⁰ In particular, it is shown that satisfies reflexivity, monotonicity, classical reduction, proof by cases and a deduction theorem.

¹¹ It is out of bounds to assert the Liar as ST derives and the Liar is a theorem as ST derives .

¹² We use ‘→’ to express material implication, defined as usual.

¹³ Another way to model such sentences in the present context—see e.g. Ripley [forthcoming] for this approach—is to work with a distinguished set of constant symbols and to assume that a symbol in this set denotes the same sentence in all ground models. We think that it is a little more elegant not to rely on such a denotation function and hence will follow the approach of Kremer [1988].

¹⁴ The phrase ‘u fresh’ means that u is a term of which has not been previously used in the tableau construction.

¹⁵ In a truth-free formula, the truth predicate may only occur within a quotational constant; is a truth-free sentence but is not.

¹⁶ To be sure, this notion of closure (closure under downwards applications of the tableau rules) has nothing to do with the notion of closure discussed earlier (closure under the closure conditions of the strict–tolerant calculus).

¹⁷ Ripley [forthcoming] presents 3-sided sequent calculi, which are sound and complete with respect to each of the four relations, from which compactness follows. See Wintein [2012b] for an independent proof of compactness and also or a detailed proof of Theorem 2.

¹⁸ Assertoric semantics was first developed by Wintein [2012a], where it was used to argue that ‘self-referential truth has computational power’. The semantic valuation function that was used by Wintein [2012a] to argue for his claim coincides with the strict valuation function as defined below.

¹⁹ As a consequence the assertoric rules manipulate (only) sentences of .

²⁰ Observe that (11) implies that, with B a branch of some tolerant assertoric tree , B is closed_M just in case B is ground_M-closed.

²¹ Here, is used as in the definition of , i.e. * swaps s for t and vice versa.

²² If one thinks that, given a ground model M, there is exactly one correct fixed-point valuation , it would make sense to define a consequence relation by quantifying over the class of all valuations. Thus, the fact that STCT defines its privileged consequence relation by quantifying over all fixed-point valuations testifies of its commitment to the fixed-point conception of truth.

²³ An anonymous referee suggested this line of defence, together with the objection to it due to cases of potential ignorance.

²⁴ Something along the following lines might work: if agent A refuses agent B's tolerant assertion of σ, then A is committed to a strict denial of σ. Currently, I am exploring such a multi-agent characterization of in terms of (strict/tolerant) assertion, denial, refusal and acceptance.

²⁵ I am grateful to Paul Egré, Reinhard Muskens, Dave Ripley and Harrie de Swart for helpful discussions, and to anonymous AJP referees for their feedback and suggestions for improvement of the paper.