Principles for Object-Linguistic Consequence: from Logical to Irreflexive

Abstract

We discuss the principles for a primitive, object-linguistic notion of consequence proposed by (Beall and Murzi, Journal of Philosophy, 3 pp. 143-65 (2013)) that yield a version of Curry's paradox. We propose and study several strategies to weaken these principles and overcome paradox: all these strategies are based on the intuition that the object-linguistic consequence predicate internalizes whichever meta-linguistic notion of consequence we accept in the first place. To these solutions will correspond different conceptions of consequence. In one possible reading of these principles, they give rise to a notion of logical consequence: we study the corresponding theory of validity (and some of its variants) by showing that it is conservative over a wide range of base theories: this result is achieved via a well-behaved form of local reduction. The theory of logical consequence is based on a restriction of the introduction rule for the consequence predicate. To unrestrictedly maintain this principle, we develop a conception of object-linguistic consequence, which we call grounded consequence, that displays a restriction of the structural rule of reflexivity. (strong Kleene version). Grounded validity will be shown to satisfy several desirable principles for a naïve, self-applicable notion of consequence.

(I can’t get no) antisatisfaction

“… See Petersen (2000),Shapiro (2010) Zardini (2011), Beall and Murzi (2013 and Rosenblatt (2019) for non-contractive approaches;Weir (2005),Cobreros et al (2013) andRipley (2013) for non-transitive approaches;French (2016) andNicolai and Rossi (2018) for non-reflexive approaches and Da Ré (2020) for non-monotonic.…”

Logical Nihilism and the Logic of ‘prem’

“…While this framework is certainly a good starting point since it permits a change in context from premise to conclusion, it will deliver us logical nihilism in the sense discussed by Russell (2018). This is as expected since the trivalent valuations obtained by dropping c, the result of which are valuations known as strong Kleene, is the typical tool for obtaining theories of transparent truth and naive validity based on a non-reflexive logic, see for example (Fjellstad, 2017), but also (French, 2016), (Nicolai and Rossi, 2018) and (Murzi and Rossi, 2017). After all, one virtue with that approach is that one can use standard rules from a bilateral sequent calculus for classical logic to define → and ¬.…”

“…While this is perhaps not the immediate candidate for a logical law to question when things get tough, there is some recent research on the use of a non-reflexive logic to accommodate self-referential definitions in order to block the set-theoretic and truth-theoretic paradoxes. Examples include (Gilmore, 1986), (Greenough, 2001), (Schroeder-Heister, 2016), (French, 2016), (Fjellstad, 2017), (Nicolai and Rossi, 2018) and (Murzi and Rossi, 2017). The non-reflexive logics presented in the literature to that purpose share the feature that they do not have any valid inferences that support uniform substitution of any formulas for propositional variables.…”

A Unified Theory of Truth and Paradox

“…Liar sentences can be used to show that no classical evaluation satisfies NAÏVETÉ. For suppose that a classical evaluation e satisfies NAÏVETÉ, let λ be the sentence ¬Tr( λ ), Leitgeb, & Welch (2003), Halbach & Horsten (2006), Priest (2006), Cieśliński (2007), Field (2008), Beall (2009), Horsten (2009), Zardini (2011), Cobreros, Égré, Ripley, & van Rooij (2013), Field (2013), Nicolai & Rossi (2018), Murzi & Rossi (2019)). Moreover, the analysis of paradoxes has been instrumental to determine the expressive power of theories of truth (see e.g., Ketland (2003), Beall (2006Beall ( , 2007aBeall ( , 2007b, Cook (2007), Field (2007), Leitgeb (2007), Maudlin (2007), Priest (2007), Restall (2007), Scharp (2007), Simmons (2007), Shapiro (2011), Scharp (2013), Rossi (2019)).…”