Verification, Falsification, and Cancellation in KT - Project Euclid
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Verification, Falsification, and Cancellation in KT - Project Euclid
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Verification, Falsification, and Cancellation in KT TIMOTHY WILLIAMSON* Abstract The main result of this paper is that KT (=T) is closed under a cancellation principle (if LA is provably equivalent to LB and MA is provably equivalent to MB then A is provably equivalent to B). This result extends to KTG1, but it does not extend to modal systems associated with the provability interpretation of L, such as KW (=G) and KT4Grz (=S4Grz). Following Williamson, these results are applied to philosophical concerns about the proper form for theories of meaning, via the interpretation of L as some kind of veriflability. The cancellation principle can then be read as saying that verif lability conditions and falsiflability conditions jointly determine truth conditions.
A modal logic S has the single cancellation property just in case A B is a thesis of S whenever LA /?, has the double (but not the single) cancellation property. Although these properties are mathematically quite natural, there is also a philosophical reason for investigating them, which it may be worthwhile to mention. One can give a nonstandard interpretation of a modal logic S by reading L as an operator expressive of some kind of verifiability rather than of necessity. One could then say that formulas A and B have the same verifiability conditions according to S if and only if LA LB is a thesis of S. By the redundancy property of truth, one can also say that A and B have the same truth conditions according to S if and only if A «-> B is a thesis of S. Thus S has the single cancellation property just in case formulas with the same verifiability conditions (ac*I would like to thank Lloyd Humberstone and the referee for comments on an earlier draft. Received May 9, 1988; revised September 29, 1988
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cording to 5) have the same truth conditions (according to S). More briefly: the single cancellation property expresses the claim that verifiability conditions determine truth conditions. In the same vein, A and B would have the same falsiflability conditions according to S if and only if L~A *+L~B were a thesis of S, given that falsification is verification of the contradictory; if M is equivalent to ~ L~ and the underlying propositional logic is classical, L~A «-> L~B is equivalent in 5 to MA (ML/? -* Lp) + LMp is a model of KT, so by hypothesis V~(LA
