Philos Logic Research Articles

Wright’s First-Order Logic of Strict Finitism

AbstractA classical reconstruction of Wright’s first-order logic of strict finitism is presented. Strict finitism is a constructive standpoint of mathematics that is more restrictive than intuitionism. Wright sketched the semantics of said logic in Wright (Realism, Meaning and Truth, chap 4, 2nd edition in 1993. Blackwell Publishers, Oxford, Cambridge, pp.107–75, 1982), in his strict finitistic metatheory. Yamada (J Philos Log. https://doi.org/10.1007/s10992-022-09698-w, 2023) proposed, as its classical reconstruction, a propositional logic of strict finitism under an auxiliary condition that makes the logic correspond with intuitionistic propositional logic. In this paper, we extend the propositional logic to a first-order logic that does not assume the condition. We will provide a sound and complete pair of a Kripke-style semantics and a natural deduction system, and show that if the condition is imposed, then the logic exhibits natural extensions of Yamada (2023)’s results.

Saving logic from paradox via nonclassical recapture

The Liar paradox arguably shows that a coherent and self-applicable notion of truth is governed by nonclassical logic. It then seems natural to conclude that classical logic is inadequate for defining a truth theory. In this article, we argue that this is not the case. In the spirit of Reinhardt (Math Logic Formal Syst 94:227, 1985; J Philos Logic 15:219–251, 1986), and in analogy with Hilbert’s program for the foundation of classical mathematics, we will articulate an instrumentalist justification for the use classical logic: it will be argued that classical reasoning is a useful but dispensable instrument, which can yield philosophically adequate truth theories.

Compositional truth with propositional tautologies and quantifier-free correctness

In Cieśliński (J Philos Logic 39:325–337, 2010), Cieśliński asked whether compositional truth theory with the additional axiom that all propositional tautologies are true is conservative over Peano Arithmetic. We provide a partial answer to this question, showing that if we additionally assume that truth predicate agrees with arithmetical truth on quantifier-free sentences, the resulting theory is as strong as Δ0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta _0$$\\end{document}-induction for the compositional truth predicate, hence non-conservative. On the other hand, it can be shown with a routine argument that the principle of quantifier-free correctness is itself conservative.

Notes on Models of (Partial) Kripke–Feferman Truth

This article investigates models of axiomatizations related to the semantic conception of truth presented by Kripke (J Philos 72(19):690–716, 1975), the so-called fixed-point semantics. Among the various proof systems devised as a proof-theoretic characterization of the fixed-point semantics, in recent years two alternatives have received particular attention: classical systems (i.e., systems based on classical logic) and nonclassical systems (i.e., systems based on some nonclassical logic). The present article, building on Halbach and Nicolai (J Philos Log 47(2):227–257, 2018), shows that there is a sense in which classical and nonclassical theories (in suitable variants) have the same models.

Radical Pooling and Imprecise Probabilities

This paper focuses on radical pooling, or the question of how to aggregate credences when there is a fundamental disagreement about which is the relevant logical space for inquiry. The solution advanced is based on the notion of consensus as common ground (Levi in Synthese 62:3–11, 1985), where agents can find it by suspending judgment on logical possibilities. This is exemplified with cases of scientific revolution. On a formal level, the proposal uses algebraic joins and imprecise probabilities; which is shown to be compatible with the principles of marginalization, rigidity, reverse bayesianism, and minimum divergence commonly endorsed in these contexts. Furthermore, I extend results from previous work by (Stewart & Ojea Quintana in J Philos Logic 47:17–45, 2016; Erkenntnis 83:369–389, 2018) to show that pooling sets of imprecise probabilities can satisfy important pooling axioms.

On the Costs of Classical Logic

This article compares classical (or KF-like) and nonclassical (or PKF-like) axiomatisations of the fixed-point semantics developed by Kripke (J Philos 72(19): 690–716, 1975). Following the line of investigation of Halbach and Nicolai (J Philos Logic 47(2): 227–257, 2018), we do not compare KF and PKF qua theories of truth simpliciter, but rather qua axiomatisations of the Kripkean conception of truth. We strengthen the central results of Halbach and Nicolai (2018) and Nicolai (Stud Log 106(1): 101–130, 2018), showing that, on the one hand, there is a stronger sense in which some variants of KF and some variants of PKF can be taken to be, truth-theoretically, equivalent. On the other hand, we show that this truth-theoretical equivalence is not preserved by some other variants of KF and PKF, arguing that the PKF variants are more adequate axiomatisations of the fixed-point semantics than the corresponding KF variants.

Notes on Leitgeb’s What Truth Depends on

In Hannes Leitgeb’s article What truth depends on (Leitgeb in J Philos Logic 34:155–192, 2005) the author provides a formally correct and materially adequate truth definition for the set of all grounded sentences, defined as the least fixed point of a monotone operator of semantic dependence. In this paper we will focus on the mathematical aspects of Leitgeb’s notions of dependence, grounding and truth, recasting Leitgeb’s construction in a functional setting in which we establish some new facts about these notions.

Strategic Conversations Under Imperfect Information: Epistemic Message Exchange Games

This paper refines the game theoretic analysis of conversations in Asher et al. (J Philos Logic 46:355–404, 2017) by adding epistemic concepts to make explicit the intuitive idea that conversationalists typically conceive of conversational strategies in a situation of imperfect information. This ‘epistemic’ turn has important ramifications for linguistic analysis, and we illustrate our approach with a detailed treatment of linguistic examples.

Weak speech reports

Indirect speech reports can be true even if they attribute to the speaker the saying of something weaker than what she in fact expressed, yet not all weakenings of what the speaker expressed yield true reports. For example, if Anna utters ‘Bob and Carla passed the exam’, we can accurately report her as having said that Carla passed the exam, but we can not accurately report her as having said that either it rains or it does not, or that either Carla passed the exam or pandas are cute. This paper offers an analysis of speech reports that distinguishes weakenings of what the speaker expressed that yield true reports from weakenings that do not. According to this analysis, speech reports are not only sensitive to the informational content of what the speaker expressed, but also to the possibilities a speaker raises in making an utterance. As I argue, this analysis has significant advantages over its most promising competitors, including views based on work by Barwise and Perry (J Philos 78(11): 668–691, 1981), views appealing to recent work on the notion of content parthood by Fine (J Philos Log 45(2):199–226, 2016) and Yablo (Aboutness. Princeton University Press, Princeton, 2014), and Richard’s (Mind Lang 13(4): 605–616, 1998) proposal appealing to structured propositions.

What paradoxes depend on

This paper gives a definition of self-reference on the basis of the dependence relation given by Leitgeb (J Philos Logic 34(2):155–192, 2005), and the dependence digraph by Beringer and Schindler (Reference graphs and semantic paradox, 2015. https://www.academia.edu/19234872/Reference_Graphs_and_Semantic_Paradox). Unlike the usual discussion about self-reference of paradoxes centering around Yablo’s paradox and its variants, I focus on the paradoxes of finitary characteristic, which are given again by use of Leitgeb’s dependence relation. They are called ‘locally finite paradoxes’, satisfying that any sentence in these paradoxes can depend on finitely many sentences. I prove that all locally finite paradoxes are self-referential in the sense that there is a directed cycle in their dependence digraphs. This paper also studies the ‘circularity dependence’ of paradoxes, which was introduced by Hsiung (Logic J IGPL 22(1):24–38, 2014). I prove that the locally finite paradoxes have circularity dependence in the sense that they are paradoxical only in the digraph containing a proper cycle. The proofs of the two results are based directly on Konig’s infinity lemma. In contrast, this paper also shows that Yablo’s paradox and its $$\forall \exists $$-unwinding variant are non-self-referential, and neither McGee’s paradox nor the $$\omega $$-cycle liar has circularity dependence.

Model-theoretic semantics and revenge paradoxes

Revenge arguments purport to show that any proposed solution to the semantic paradoxes generates new paradoxes that prove that solution to be inadequate. In this paper, I focus on revenge arguments that employ the model-theoretic semantics of a target theory and I argue, contra the current revenge-theoretic wisdom, that they can constitute genuine expressive limitations. I consider the anti-revenge strategy elaborated by Field (J Philos Log 32:139–177, 2003; Revenge of the Liar, Oxford University Press, Oxford, pp 53–144, 2007; Saving truth from paradox, Oxford University Press, Oxford, 2008, §§21–23) and argue that it does not offer a way out of the revenge problem. More generally, I argue that the difference between ‘standard’ and ‘revenge’ paradoxes is ill-conceived and should be abandoned. This will contribute to show that the theories that provide a uniform account of truth and other semantic notions are the ones best equipped to avoid the paradoxes altogether—‘standard’ and ‘revenge’ alike.

Na\xefve validity

Beall and Murzi (J Philos 110(3):143–165, 2013) introduce an object-linguistic predicate for naïve validity, governed by intuitive principles that are inconsistent with the classical structural rules (over sufficiently expressive base theories). As a consequence, they suggest that revisionary approaches to semantic paradox must be substructural. In response to Beall and Murzi, Field (Notre Dame J Form Log 58(1):1–19, 2017) has argued that naïve validity principles do not admit of a coherent reading and that, for this reason, a non-classical solution to the semantic paradoxes need not be substructural. The aim of this paper is to respond to Field’s objections and to point to a coherent notion of validity which underwrites a coherent reading of Beall and Murzi’s principles: grounded validity. The notion, first introduced by Nicolai and Rossi (J Philos Log. doi:10.1007/s10992-017-9438-x, 2017), is a generalisation of Kripke’s notion of grounded truth (J Philos 72:690–716, 1975), and yields an irreflexive logic. While we do not advocate the adoption of a substructural logic (nor, more generally, of a revisionary approach to semantic paradox), we take the notion of naïve validity to be a legitimate semantic notion that points to genuine expressive limitations of fully structural revisionary approaches.

On unidirectionality in precisification

This paper provides a formal pragmatic analysis of (im)precision which accounts for its essential properties, but also for Lewis’s (J Philos Logic 8(1):339–359, 1979) observation of asymmetry in how standards of precision may shift due to normal discourse moves: Only up, not down. I propose that shifts of the kind observed and discussed by Lewis are in fact cases of underlying disagreement about the standard of precision, which is only revealed when one interlocutor uses an expression which signals their adherence to a higher standard than the one adhered to by the other interlocutor(s). This paper shows that a modest formal pragmatic analysis along the lines of many prior optimality-theoretic and game-theoretic accounts can easily capture the natural asymmetry in standard-signaling that gives rise to Lewis’s observation, so long as such an account is dynamic and enriched with a notion of relevance.

Stability and Scepticism in the Modelling of Doxastic States: Probabilities and Plain Beliefs

There are two prominent ways of formally modelling human belief. One is in terms of plain beliefs (yes-or-no beliefs, beliefs simpliciter), i.e., sets of propositions. The second one is in terms of degrees of beliefs, which are commonly taken to be representable by subjective probability functions. In relating these two ways of modelling human belief, the most natural idea is a thesis frequently attributed to John Locke: a proposition is or ought to be believed (accepted) just in case its subjective probability exceeds a contextually fixed probability threshold $$t<1$$t<1. This idea is known to have two serious drawbacks: first, it denies that beliefs are closed under conjunction, and second, it may easily lead to sets of beliefs that are logically inconsistent. In this paper I present two recent accounts of aligning plain belief with subjective probability: the Stability Theory of Leitgeb (Ann Pure Appl Log 164(12):1338---1389, 2013; Philos Rev 123(2):131---171, 2014; Proc Aristot Soc Suppl Vol 89(1):143---185, 2015a; The stability of belief: an essay on rationality and coherence. Oxford University Press, Oxford, 2015b) and the Probalogical Theory (or Tracking Theory) of Lin and Kelly (Synthese 186(2):531---575, 2012a; J Philos Log 41(6):957---981, 2012b). I argue that Leitgeb's theory may be too sceptical for the purposes of real life.

Rigged lotteries: a diachronic problem for reducing belief to credence

Lin and Kelly (J Philos Log 41(6):957–981, 2012) and Leitgeb (Ann Pure Appl Log 164(12):1338–1389, 2013, Philos Rev 123(2):131–171, 2014), offer similar solutions to the Lottery Paradox, defining acceptance rules which determine a rational agent’s beliefs in terms of broader features of her credal state than just her isolated credences in individual propositions. I express each proposal as a method for obtaining an ordering over a partition from a credence function, and then a belief set from the ordering. Although these proposals avoid the original Lottery Paradox, I raise a diachronic case which illustrates that neither satisfies both (i) Lin and Kelly’s constraint that the update on orderings track the update on credence functions, and (ii) the intuitive constraint that credence of at least 0.5 is necessary for rational belief. I conclude by suggesting that we reformulate these proposals in terms of orderings over entire algebras based on partitions rather than orderings just over the partitions themselves. Reformulating both rules in this way yields acceptance rules which avoid the Lottery Paradox while satisfying both the tracking and likeliness constraints.

Axiomatization of a Basic Logic of Logical Bilattices

A sequential axiomatization is given for the 16-valued logic that has been proposed by Shramko-Wansing (J Philos Logic 34:121–153, 2005) as a candidate for the basic logic of logical bilattices.

Anti-exceptionalism about logic

Logic isn’t special. Its theories are continuous with science; its method continuous with scientific method. Logic isn’t a priori, nor are its truths analytic truths. Logical theories are revisable, and if they are revised, they are revised on the same grounds as scientific theories. These are the tenets of anti-exceptionalism about logic. The position is most famously defended by Quine, but has more recent advocates in Maddy (Proc Address Am Philos Assoc 76:61–90, 2002), Priest (Doubt truth to be a liar, OUP, Oxford, 2006a, The metaphysics of logic, CUP, Cambridge, 2014, Log et Anal, 2016), Russell (Philos Stud 171:161–175, 2014, J Philos Log 0:1–11, 2015), and Williamson (Modal logic as metaphysics, Oxford University Press, Oxford, 2013b, The relevance of the liar, OUP, Oxford, 2015). Although these authors agree on many methodological issues about logic, they disagree about which logic anti-exceptionalism supports. Williamson uses an anti-exceptionalist argument to defend classical logic, while Priest claims that his anti-exceptionalism supports nonclassical logic. This paper argues that the disagreement is due to a difference in how the parties understand logical theories. Once we reject Williamson’s deflationary account of logical theories, the argument for classical logic is undercut. Instead an alternative account of logical theories is offered, on which logical pluralism is a plausible supplement to anti-exceptionalism.

Gentzenization of Trilattice Logics

Sequent calculi for trilattice logics, including those that are determined by the truth entailment, the falsity entailment and their intersection, are given. This partly answers the problems in Shramko-Wansing (J Philos Logic 34:121---153, 2005).

Probabilities of conditionals in context

The Ramseyan thesis that the probability of an indicative conditional is equal to the corresponding conditional probability of its consequent given its antecedent is both widely confirmed and subject to attested counterexamples (e.g., McGee, in Analysis 60(1):107–111, 2000; Kaufmann, in J Philos Logic 33:583–606, 2004). This raises several puzzling questions. For instance, why are there interpretations of conditionals that violate this Ramseyan thesis in certain contexts, and why are they otherwise very rare? In this paper, I raise some challenges to Stefan Kaufmann’s account of why the Ramseyan thesis sometimes fails, and motivate my own theory. On my theory, the proposition expressed by an indicative conditional is partially determined by a background partition, and hence its probability depends on the choice of such a partition. I hold that this background partition is contextually determined, and in certain conditions is set by a salient question under discussion in the context. I argue that the resulting theory offers compelling answers to the puzzling questions raised by failures of the Ramseyan thesis.

Possibilities regained: neo-Lewisian contextualism and ordinary life

According to David Lewis, the predicate ‘knows’ is context-sensitive in the sense that its truth conditions vary across conversational contexts, which stretch or compress the domain of error possibilities to be eliminated by the subject’s evidence (Lewis, Aust J Philos 74:549–567, 1996; Lewis, J Philos Log 8:339–359, 1979). Our concern in this paper is to thematize, assess, and overcome within a neo-Lewisian contextualist project two important mismatches between our use of ‘know’ in ordinary life and the use of ‘know’ by ‘Lewisian’ ordinary speakers. The first mismatch is that Lewisian contextualism still overgenerates the error possibilities which cannot be ignored in a given context, since it is oblivious to the distinction between ‘invented’ and ‘discovered’ possibilities. The second mismatch is a full-scale one: an adequate account of knowledge attribution is not exhausted by the subject’s negative capacity of pruning branches off the tree of counterpossibilities. We therefore introduce a new vector of value, which explains how ‘know’ comes in degrees: the satisfaction of ‘know better’ is made to depend on the capacity of imagining (actualized) possibilities connected in a relevant way with the subject’s (true) beliefs.