A Note on Carnap’s Result and the Connectives

Carnap’s result about classical proof-theories not ruling out non-normal valuations of propositional logic formulae has seen renewed philosophical interest in recent years. In this note I contribute some considerations which may be helpful in its philosophical assessment. I suggest a vantage point from which to see the way in which classical proof-theories do, at least to a considerable extent, encode the meanings of the connectives (not by determining a range of admissible valuations, but in their own way), and I demonstrate a kind of converse to Carnap’s result

An inferentialist approach to paraconsistency

This paper develops and motivates a paraconsistent approach to semantic paradox from within a modest inferentialist framework. I begin from the bilateralist theory developed by Greg Restall, which uses constraints on assertions and denials to motivate a multiple-conclusion sequent calculus for classical logic, and, via which, classical semantics can be determined. I then use the addition of a transparent truth-predicate to motivate an intermediate speech-act. On this approach, a liar-like sentence should be "weakly asserted", involving a commitment to the sentence and its negation, without rejecting the sentence. From this, I develop a proof-theory, which both determines a typical paraconsistent model theory, and also gives us a nice way to understand classical recapture

A General Schema for Bilateral Proof Rules

Bilateral proof systems, which provide rules for both affirming and denying sentences, have been prominent in the development of proof-theoretic semantics for classical logic in recent years. However, such systems provide a substantial amount of freedom in the formulation of the rules, and, as a result, a number of different sets of rules have been put forward as definitive of the meanings of the classical connectives. In this paper, I argue that a single general schema for bilateral proof rules has a reasonable claim to inferentially articulating the core meaning of all of the classical connectives. I propose this schema in the context of a bilateral sequent calculus in which each connective is given exactly two rules: a rule for affirmation and a rule for denial. Positive and negative rules for all of the classical connectives are given by a single rule schema, harmony between these positive and negative rules is established at the schematic level by a pair of elimination theorems, and the truth-conditions for all of the classical connectives are read off at once from the schema itself

Negation in context

The present essay includes six thematically connected papers on negation in the areas of the philosophy of logic, philosophical logic and metaphysics. Each of the chapters besides the first, which puts each the chapters to follow into context, highlights a central problem negation poses to a certain area of philosophy. Chapter 2 discusses the problem of logical revisionism and whether there is any room for genuine disagreement, and hence shared meaning, between the classicist and deviant's respective uses of 'not'. If there is not, revision is impossible. I argue that revision is indeed possible and provide an account of negation as contradictoriness according to which a number of alleged negations are declared genuine. Among them are the negations of FDE (First-Degree Entailment) and a wide family of other relevant logics, LP (Priest's dialetheic "Logic of Paradox"), Kleene weak and strong 3-valued logics with either "exclusion" or "choice" negation, and intuitionistic logic. Chapter 3 discusses the problem of furnishing intuitionistic logic with an empirical negation for adequately expressing claims of the form 'A is undecided at present' or 'A may never be decided' the latter of which has been argued to be intuitionistically inconsistent. Chapter 4 highlights the importance of various notions of consequence-as-s-preservation where s may be falsity (versus untruth), indeterminacy or some other semantic (or "algebraic") value, in formulating rationality constraints on speech acts and propositional attitudes such as rejection, denial and dubitability. Chapter 5 provides an account of the nature of truth values regarded as objects. It is argued that only truth exists as the maximal truthmaker. The consequences this has for semantics representationally construed are considered and it is argued that every logic, from classical to non-classical, is gappy. Moreover, a truthmaker theory is developed whereby only positive truths, an account of which is also developed therein, have truthmakers. Chapter 6 investigates the definability of negation as "absolute" impossibility, i.e. where the notion of necessity or possibility in question corresponds to the global modality. The modality is not readily definable in the usual Kripkean languages and so neither is impossibility taken in the broadest sense. The languages considered here include one with counterfactual operators and propositional quantification and another bimodal language with a modality and its complementary. Among the definability results we give some preservation and translation results as well

Categorical Quantification

Due to Gӧdel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments –Warren (2020), Murzi and Topey (2021)– for the idea that the natural deduction rules for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s (2013) local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e. non-compact. Consequently, I reconsider the use of the ω-rule and I show that the addition of the ω-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement

Classical Harmony and Separability

According to logical inferentialists, the meanings of logical expressions are fully determined by the rules for their correct use. Two key proof-theoretic requirements on admissible logical rules, harmony and separability, directly stem from this thesisrequirements, however, that standard single-conclusion and assertion-based formalizations of classical logic provably fail to satisfy (Dummett in The logical basis of metaphysics, Harvard University Press, Harvard, MA, 1991; Prawitz in Theoria, 43:140, 1977; Tennant in The taming of the true, Oxford University Press, Oxford, 1997; Humberstone and Makinson in Mind 120(480):10351051, 2011). On the plausible assumption that our logical practice is both single-conclusion and assertion-based, it seemingly follows that classical logic, unlike intuitionistic logic, cant be accounted for in inferentialist terms. In this paper, I challenge orthodoxy and introduce an assertion-based and single-conclusion formalization of classical propositional logic that is both harmonious and separable. In the framework I propose, classicality emerges as a structural feature of the logic.(VLID)283667

Intuitionism and logical revision.

The topic of this thesis is logical revision: should we revise the canons of classical reasoning in favour of a weaker logic, such as intuitionistic logic? In the first part of the thesis, I consider two metaphysical arguments against the classical Law of Excluded Middle-arguments whose main premise is the metaphysical claim that truth is knowable. I argue that the first argument, the Basic Revisionary Argument, validates a parallel argument for a conclusion that is unwelcome to classicists and intuitionists alike: that the dual of the Law of Excluded Middle, the Law of Non-Contradiction, is either unknown, or both known and not known to be true. As for the second argument, the Paradox of Knowability, I offer new reasons for thinking that adopting intuitionistic logic does not go to the heart of the matter. In the second part of the thesis, I motivate an inferentialist framework for assessing competing logics-one on which the meaning of the logical vocabulary is determined by the rules for its correct use. I defend the inferentialist account of understanding from the contention that it is inadequate in principle, and I offer reasons for thinking that the inferentialist approach to logic can help model theorists and proof-theorists alike justify their logical choices. I then scrutinize the main meaning-theoretic principles on which the inferentialist approach to logic rests: the requirements of harmony and separability. I show that these principles are motivated by the assumption that inference rules are complete, and that the kind of completeness that is necessary for imposing separability is strictly stronger than the completeness needed for requiring harmony. This allows me to reconcile the inferentialist assumption that inference rules are complete with the inherent incompleteness of higher-order logics-an apparent tension that has sometimes been thought to undermine the entire inferentialist project. I finally turn to the question whether the inferentialist framework is inhospitable in principle to classical logical principles. I compare three different regimentations of classical logic: two old, the multiple-conclusions and the bilateralist ones, and one new. Each of them satisfies the requirements of harmony and separability, but each of them also invokes structural principles that are not accepted by the intuitionist logician. I offer reasons for dismissing multiple-conclusions and bilateralist formalizations of logic, and I argue that we can nevertheless be in harmony with classical logic, if we are prepared to adopt classical rules for disjunction, and if we are willing to treat absurdity as a logical punctuation sign