Tarski on Logical Consequence

Fixed‐ versus Variable‐domain Interpretations of Tarski’s Account of Logical Consequence

“…Whereas Sher and Ray did not exclude the possibility that Tarski (1936) might have defended a fixed‐domain conception of logical consequence, they argued on ground of charity that it would be very unlikely that Tarski could have held such a view, for it would be in conflict with other aspects of Tarski’s work and it would imply an inability to see the importance of domain variation in determining logical consequence. Gómez‐Torrente (1996) pursues a different strategy. While agreeing that ‘the supposition that Tarski did not contemplate domain variation is nearly impossible to reconcile with his contemporary work’ (145), he did not stop there and offered an exegetical interpretation meant to establish that Tarski ‘did contemplate domain variation, but was not sufficiently explicit about it’ (145).…”

“…In the wake of Gómez‐Torrente’s (1996) article it became important to look in detail at Tarski’s (1936) article, especially in the context of the other articles he wrote during that period. Bays (2001) carried out such an analysis with the aim of showing (i) that Tarski’s conception of model in 1936 is a fixed‐domain conception and (ii) that such a conception does not have the devastating consequences alleged by Sher, Ray, and Gómez‐Torrente:…”

“…Recall that Mancosu (2006) had argued that Tarski’s practice in Einführung in die mathematische Logik (1937) did not conform to the conventions proposed by Gómez‐Torrente (1996). Mancosu referred to a theory A discussed by Tarski which contains a nonlogical predicate Zl(x).…”

Alfred Tarski: Philosophy of Language and Logic

“…One either (i) argues that Tarski just is assuming that the sequences that form his models are of variable length without mentioning it ( [Sher, 1996], [Ray, 1996] (endorsed by [Hanson, 1999, 610] and Stroinska and Hitchcock's introduction to [Tarski, 2002, 170]), or (ii) argues that though Tarski's sequences are all infinite, suitable use of implicit restriction of the range of the variables gives Tarski the effect of variable domains within the confines of the conception more clearly on the surface of [Tarski, 2002] ([Gómez-Torrente, 1996). Our way with (i) can be brief: as Etchemendy notes [Etchemendy, 2008, 280], there is simply no evidence that Tarski assumes that the domain varies in [Tarski, 2002] and the reading is incompatible with Tarski's claim that logical consequence degenerates to material consequence if all terms are treated as logical.…”

“…As many commentators have pointed out (e.g. [Gómez-Torrente, 1996], ]) Tarski's other discussions of the ω-rule assume that the inferences in question are formulated in STT with numerical terms defined in the logicist manner. As these commentators note, the ω-rule is valid so formulated in STT and the selection of constants involved isn't ad hoc.…”