X Workshop on Philosophical Logic:
Week III: Classical vs. Non-classical logics
August 19, 20 2021
Online Conference via ZOOM Meetings
Hartry Field (New York University)
Camillo Fiore (University of Buenos Aires)
Ulf Hlobil (Concordia University)
Luca Incurvati (University of Amsterdam)
Carlo Nicolai (King’s College London)
Lucas Rosenblatt (University of Buenos Aires)
Gillian Russell (Australian Catholic University)
Jack Woods (University of Leeds)
“Kripke and Lukasiewicz: A Synthesis”
In classical logic the naïve theory of truth and satisfaction is inconsistent. Kripke provided a well-known partial solution to the paradoxes in a non-classical logic. But it has a big limitation: it doesn’t work for logics with serious conditionals, or restricted universal quantification.
Another partial non-classical solution is given by Lukasiewicz continuum-valued logic. It allows naïve truth for sentences containing a rather natural conditional. But it has a different limitation: it doesn’t work for all sentences containing even unrestricted quantifiers. (Kripke’s partial solution handled those.)
So neither result handles restricted quantifiers. It would be nice to synthesize the two: to have an account which handled both unrestricted quantifiers and a Lukasiewicz-like conditional. (And to do so in “essentially” the way that Lukasiewicz and Kripke did.) It will thereby also handle restricted universal quantification, which is interdefinable with the conditional given unrestricted quantification.
I’ll show how to do so in this talk. The synthesized approach improves on my previous work on conditonals and restricted quantifiers, in essentially preserving the attractive features of the Lukasiewicz resolution of the quantifier-free semantic paradoxes, including the easy calculation of solutions.
Camillo Fiore and Lucas Rosenblatt:
“Recapture Results and Classical Logic”
The discussion on whether non-classical approaches to the semantic paradoxes can recover classical reasoning for non-problematic statements (what is sometimes referred to in the literature as ‘classical recapture’) is of the utmost importance for non-classical logicians. To show that classical logic can be recovered in certain contexts—and thus bolster the case for the suitability of their favored logic—these theorists sometimes prove so-called ‘recapture results’. Roughly, these results are intended to establish the availability of classical logic when it is needed. One problem that has been identified with this strategy is that in proving recapture results non-classical theorists typically employ classical principles that they themselves reject. In view of this, one might be tempted to conclude that these theorists are not entitled to those results. What they need to show, the objection goes, is that, by the standards of their own theory, classical recapture is possible. In this paper we will try to counter this objection by suggesting that it relies on a certain ambiguity, one that can be made visible only if one focuses on the specific instances of the classical principles that are needed to obtain recapture results. We argue that, once the ambiguity is removed, there is no problem with the recapture strategy.
“A Realist Motivation for the Nontransitive Approach to Paradox”
There are two main advantages of the nontransitive approach to semantic paradoxes: First, the nontransitive logic ST is, at the inferential level, at least as strong as classical logic. Second, there is an intuitive motivation for ST supplied by a bilateralist conception of consequence, namely that paradoxical sentences can neither be coherently asserted nor coherently denied. An opponent may aim to undermine the first advantage by arguing that ST’s weakness at the meta-inferential level is a problem. And they may try to undermine the second advantage by saying that the bilateralist motivation doesn’t allow us to understand what goes wrong in paradoxical reasoning because it concerns only norms of discourse and not the reality that paradoxical sentences purport to describe. I hope to make some progress on both points by presenting a realist motivation for the nontransitive approach. To do so, I give a truth-maker semantics for ST. On this basis, I will argue that the meta-inferential weakness of ST is a virtue and not a vice because it allows us to keep track of the hyperintensional structure of truth-makers. Regarding the second point, I will suggest that what goes wrong in paradoxical reasoning is that we overlook that the world is necessarily gappy, in the sense that there is neither a truth-maker nor a falsity-maker for paradoxical sentences that is a possible state. Against this idea, one may raise revenge worries by using sentences like “This sentence has neither a possible truth-maker nor a possible falsity-maker.” Perhaps this worry can be solved by allowing for sentences to be correct without having truth-makers.
Deflationists about truth hold that the function of the truth predicate is to enable us to make certain assertions we could not otherwise make. Pragmatists claim that the utility of negation lies in its role in registering incompatibility. The pragmatist insight about negation has been successfully incorporated into bilateral theories of content, which take the meaning of negation to be inferentially explained in terms of the speech act of rejection. In this talk, I will implement the deflationist insight in a bilateral theory by taking the meaning of the truth predicate to be explained by its inferential relation to assertion. This account of the meaning of the truth predicate is combined with a new diagnosis of the Liar Paradox: its derivation requires the truth rules to preserve evidence, but these rules only preserve commitment. The result is a novel inferential deflationist theory of truth. The theory solves the Liar Paradox in a principled manner and deals with a purported revenge paradox in the same way. If time permits, I will show how the theory and simple extensions thereof have the resources to axiomatise the internal logic of several supervaluational hierarchies, thereby solving open problems of Halbach (2011) and Horsten (2011). This is joint work with Julian Schlöder.
If the concept of truth is characterized by the T-schema (‘A’ is true iff A), the two sides of the biconditional are not only materially equivalent, but they are necessarily equivalent (although the notion of necessity employed varies substantially in the literature). Paradox constraints the precise formulation of the T-schema and its modal extensions. I will first consider classical options, including Halbach’s Modalized Disquotationalism. The classical proposal, under suitable assumptions, leads to inconsistency. I will then study a nonclassical proposal, based on Leitgeb’s logic HYPE (i.e. an extension of FDE with an intuitionistic conditional). The consistency of the nonclassical modal T-schema will be obtained by means of a possible worlds fixed-point construction. I will then analyze the proof-theoretic properties of the nonclassical modal T-schema. I will conclude by comparing classical and nonclassical alternatives on the basis of standard theory choice criteria.
“Barriers to Entailment”
A Barrier to Entailment says that you can’t get certain kinds of conclusion from certain kinds of premises, for example: you can’t get normative conclusions from descriptive premises or you can’t get conclusions about the future from premises about the past or universal conclusions from particular premises.
The literature in logic on the is/ought barrier is often a response to the counterexamples proposed by A.N. Prior in “The Autonomy of Ethics” (1960) but more recent criticisms (due to e.g. Peter Vranas and Gerhard Schurz) of proofs that avoid Prior’s worries in simple deontic logics have emphasised the importance and difficulty of extending such proofs to complex logics.
This paper proposes a new way to handle such counterexamples to the is/ought barrier and shows how to generalise it to analogous counterexamples to other barrier theses.
Lucas Rosenblatt (UBA/IIF-SADAF-CONICET)
We are thankful for the support provided by CONICET and MINCYT