Intertranslatability, Theoretical Equivalence, and Perversion

Abstract

I investigate syntactic notions of theoretical equivalence between logical theories and a recent objection thereto. I show that this recent criticism of syntactic accounts, as extensionally inadequate, is unwarranted by developing an account which is plausibly extensionally adequate and more philosophically motivated. This is important for recent anti-exceptionalist treatments of logic since syntactic accounts require less theoretical baggage than semantic accounts.Keywords philosophy of logic; theoretical equivalence; anti-exceptionalism; revising logic; intuitionistic logic; non-classical logic DOI:10.1002/tht3.269It's increasingly common to claim that logic isn't special, that a logical theory is just a theory like any other (Hjortland 2017; Priest 2014; Russell 2014; Williamson 2017) These anti-exceptionalist views use theoretical virtues of alternative logics-simplicity, strength, etc.-in order to evaluate them analogously to how we evaluate scientific theories. 1 In doing so, it's helpful to know when two seemingly distinct theories are really differently formulated versions of the same theory, saying the same things in different ways. In other words, it's helpful to have an account of theoretical equivalence between logics.For example, axiomatic and natural deduction presentations of classical propositional logic initially look quite different even though they're intuitively theoretically equivalent. Now, natural deduction is less metatheoretically tractable while axiomatic accounts do worse at revealing obvious entailments. Neither disadvantage is problematic once we're justified in treating them as theoretically equivalent; perceived disadvantages of a logical theory that are merely presentational can be ignored in calculating its overall virtues.We have options when developing a notion of theoretical equivalence. One standard way uses semantic properties, like having the same set of models, to account for when two logical theories are equivalent. However, as the required model theory can only be developed by appealing to significant logical resources, many of which are in contention when choosing between logics, it's natural for anti-exceptionalists to prefer less theoretically taxing syntactic accounts of theoretical equivalence. Syntactic accounts require lighter metatheoretic resources (roughly a basic theory of syntax) which can often be developed

Individuating Logics

“…Based on this argument, I suggested a purely semantic conception of logical theories, understanding them in terms of categories of models. In response, Jack Woods () has proposed a stronger syntactic account of theoretical equivalence, according to which classical and intuitionistic logic are not equivalent. Based on this proposal, Woods argues that a purely syntactic account of logical theories remains plausible.…”

“…To summarize the arguments in Wigglesworth () and Woods (), take a theory to be a set of sentences closed under a relation of logical consequence. Given a set of sentences, S , the theory that S generates by closing S under a logic ℒ is the set .…”

“…Woods () argues that there is a stronger syntactic criterion that is able to distinguish between classical and intuitionistic logic. The stronger criterion is achieved by making three refinements to the intertranslatability account .…”

“…Institutions give us a unified approach that gets it right in cases where purely syntactic and purely semantics approaches fail. The arguments presented in Wigglesworth () and Woods () are given in the context of a view known as logical anti‐exceptionalism. Anti‐exceptionalism about logic takes logical theories to be continuous with scientific theories.…”

What theoretical equivalence could not be

“…(Though, as I will note shortly in the main text, in other places these latter authors make claims that presuppose more than purely mathematical ambitions for sentential criteria.) The diagnosis is also suggested by work applying sentential criteria to different logics, such asWigglesworth (2017),Dewar (2018), andWoods (2018).…”