Philosophers who write about the meaning of life are few nowadays. Thesubject has lost its attractiveness. Perceived from a viewpoint of logical positivism or language philosophy, the whole issue of meaningfulness seems rather pointless. It is often considered to be related to metaphysics, making it less suitable for philosophical inquiry. The topic of meaningfulness seems too intangible. Indeed, the few philosophers that have embarked on examining meaningfulness have proven to be well aware of the challenges this poses. At times they (...) acknowledge that the more they concentrate on the subject, the more it seems to fall apart into unintelligible pieces about whichnothing of philosophical value can be said. (shrink)

This paper’s aim is to help winnow out some ideas about the role of formal logic in human doings at large. I start by discussing some metaphysical presuppositions of logical theory; specifically, I attempt to work towards a clearer understanding of the role of modalities, together with the notions of meaning and truth, in mainstream logical theory. I then appeal to a modal formal semantics (Brandom, 2007a) in order to outline the cognitive role of logical constants in general. From (...) these discussions I conclude that any view of formal logic, and its core concepts, which does not involve a grasp of what we do with them will be found to be distorted. (shrink)

This note motivates a logic for a theory that can express its own notion of logical consequence—a ‘syntactically closed’ theory of naive validity. The main issue for such a logic is Curry’s paradox, which is averted by the failure of contraction. The logic features two related, but different, implication connectives. A Hilbert system is proposed that is complete and non-trivial.

The logic E4 is related to Brady’s BN4 in a similar way to which Anderson and Belnap’s logic of entailment E is related to their logic of the relevant implication R. In ‘A companion to Brady’s 4-valued relevant logic: the 4-valued logic of entailment E4’, quoted in this paper, three alternatives to BN4 and another three to E4 are summarily introduced in a couple of pages as the only alternatives containing Routley and Meyer’s basic (...) class='Hi'>logic B, provided some conditions are fulfilled. The aim of this note is to prove BN4 and its three alternatives are the same logic up to some point, since they are functionally equivalent to each other, as it is the case with E4 and its three alternatives, which are also functionally equivalent to each other; to some extent, E4 is superior to BN4, since the latter is functionally included in the former, but not conversely. (shrink)

hese notes don’t reach any conclusions. Their purpose is to point to issues one needs to think through seriously when thinking about logic teaching. They indicate some of the relevant literature where some of these issues are addressed, but they also raise points that seem to have been overlooked. They aim to promote informed discussion. That indeed was their origin: they are descended from an internal discussion document prepared a few years ago when the then Department of Philosophy (...) at the University of Edinburgh was reviewing its logic teaching. (shrink)

A propositional logic has the variable sharing property if φ → ψ is a theorem only if φ and ψ share some propositional variable. In this note, I prove that positive semilattice relevance logic and its extension with an involution negation have the variable sharing property. Typical proofs of the variable sharing property rely on ad hoc, if clever, matrices. However, in this note, I exploit the properties of rather more intuitive arithmetical structures to establish the variable sharing (...) property for the systems discussed. (shrink)

It is an unusual property for a logic to prove a formula and its negation without ending up in triviality. Some systems have nonetheless been observed to satisfy this property: one group of such non-trivial negation inconsistent logics has its archetype in H. Wansing’s constructive connexive logic, whose negation-implication fragment already proves contradictions. N. Francez and Y. Weiss subsequently investigated relevant subsystems of this fragment, and Weiss in particular showed that they remain negation inconsistent. In this note, we (...) take a closer look at this phenomenon in the systems of Francez and Weiss, and point out two types of necessary conditions, one proof-theoretic and one relevant, which any contradictory formula must satisfy. As a consequence, we propose a nine-fold classification of provable contradictions for the logics. (shrink)

This brief note corrects an error in one of the reduction steps in my paper 'Normalisation for Bilateral Classical Logic with some Philosophical Remarks' published in the Journal of Applied Logics 8/2 (2021): 531-556.

Mathematical pluralism notes that there are many different kinds of pure mathematical structures—notably those based on different logics—and that, qua pieces of pure mathematics, they are all equally good. Logical pluralism is the view that there are different logics, which are, in an appropriate sense, equally good. Some, such as Shapiro, have argued that mathematical pluralism entails logical pluralism. In this brief note I argue that this does not follow. There is a crucial distinction to be drawn between the (...) preservation of truth and the preservation of truth-in-a-structure; and once this distinction is drawn, this suffices to block the argument. The paper starts by clarifying the relevant notions of mathematical and logical pluralism. It then explains why the argument from the first to the second does not follow. A final section considers a few objections. (shrink)

Zermelo’s Theorem that the axiom of choice is equivalent to the principle that every set can be well-ordered goes through in third-order logic, but in second-order logic we run into expressivity issues. In this note, we show that in a natural extension of second-order logic weaker than third-order logic, choice still implies the well-ordering principle. Moreover, this extended second-order logic with choice is conservative over ordinary second-order logic with the well-ordering principle. We also discuss (...) a variant choice principle, due to Hilbert and Ackermann, which neither implies nor is implied by the well-ordering principle. (shrink)

Fuzzy logics are systems of logic with infinitely many truth values. Such logics have been claimed to have an extremely wide range of applications in linguistics, computer technology, psychology, etc. In this note, we canvass the known results concerning infinitely many valued logics; make some suggestions for alterations of the known systems in order to accommodate what modern devotees of fuzzy logic claim to desire; and we prove some theorems to the effect that there can be no fuzzy (...) logic which will do what its advocates want. Finally, we suggest ways to accommodate these desires in finitely many valued logics. (shrink)

We can think of functional completeness in systems of propositional logic as a form of expressive completeness: while every logical constant in such system expresses a truth-function of finitely many arguments, functional completeness garantees that every truth-function of finitely many arguments can be expressed with the constants in the system. From this point of view, a functionnaly complete system of propositionnal logic can thus be seen as one where no logical constant is missing. Can a similar question be (...) formulated for quantified first-order logics ? How to make sense of the question whether, e.g., ordinary first-order logic is "functionaly" complete or have no logical constant missing ? In this note, we build on a suggestive proposal made by Bonnay(2006) and shows that it is equivalent to the criterion that a first-order logic L be functionaly complete if and only if every class of structures closed under L-elementary equivalence is L-elementary. Ordinary first-order logic is not complete in this sense. We raise the question whether any logic can be. -/- . (shrink)

The present note revisits the joint work of Leonard Goddard and Richard Routley on significance logics with the aim of shedding new light on their understanding by studying them under the lens of recent semantic developments, such as the plurivalent semantics developed by Graham Priest. These semantics allow sentences to receive one, more than one, or no truth-value at all from a given carrier set. Since nonsignificant sentences are taken to be neither true nor false, i.e. truth-value gaps, in this (...) essay we show that with the aid of plurivalent semantics it is possible to straightforwardly instantiate Goddard and Routley’s understanding of how the connectives should work within significance logics. (shrink)

Discussive logic was introduced by Jaskowski as a logic of discussion. In this note we show that some natural translation-based formalizations of discussive logic in modal logic do not yield a paraconsistent logic but rather classical logic. Some alternative modal formalizations of discussive logic that avoid the collapse into classical logic are put forward.

This paper consists of some lecture notes in which conditional logic is treated as an extension of modal logic. Completeness and filtration theorems are provided for some basis systems.

Brief note explaining the content, importance, and historical context of my joint translation of Quine's The Significance of the New Logic with my single-authored historical-philosophical essay 'Willard Van Orman Quine's Philosophical Development in the 1930s and 1940s'.

In Schwichtenberg, Schwichtenberg fine-tuned Tait’s technique so as to provide a simplified version of Gentzen’s original cut-elimination procedure for first-order classical logic. In this note we show that, limited to the case of classical propositional logic, the Tait–Schwichtenberg algorithm allows for a further simplification. The procedure offered here is implemented on Kleene’s sequent system G4. The specific formulation of the logical rules for G4 allows us to provide bounds on the height of cut-free proofs just in terms of (...) the logical complexity of their end-sequent. (shrink)

A succinct introduction to mathematical logic and set theory, which together form the foundations for the rigorous development of mathematics. Suitable for all introductory mathematics undergraduates, Notes on Logic and Set Theory covers the basic concepts of logic: first-order logic, consistency, and the completeness theorem, before introducing the reader to the fundamentals of axiomatic set theory. Successive chapters examine the recursive functions, the axiom of choice, ordinal and cardinal arithmetic, and the incompleteness theorems. Dr. Johnstone (...) has included numerous exercises designed to illustrate the key elements of the theory and to provide applications of basic logical concepts to other areas of mathematics. (shrink)

In his 2010 paper ‘Grounding and Truth-Functions’, Fabrice Correia has developed the first and so far only proposal for a logic of ground based on a worldly conception of facts. In this paper, we show that the logic allows the derivation of implausible grounding claims. We then generalize these results and draw some conclusions concerning the structural features of ground and its associated notion of relevance, which has so far not received the attention it deserves.

Edited in collaboration with FoLLI, the Association of Logic, Language and Information this book constitutes the refereed proceedings of the 22nd Workshop on Logic, Language, Information and Computation, WoLLIC 2015, held in the campus of Indiana University, Bloomington, IN, USA in July 2015. The 14 contributed papers, presented together with 8 invited lectures and 4 tutorials, were carefully reviewed and selected from 44 submissions. The focus of the workshop was on interdisciplinary research involving formal logic, computing and (...) programming theory, and natural language and reasoning. (shrink)

This note follows up an earlier paper in which a possibility of defining logical constants within abstract logical frameworks was discussed, by using duals as a general method of applying the idea of invariance under replacement as a criterion for logicality. In the present note, this approach is applied to the discussion on logicality of generalized quantifiers. It is demonstrated that generalized quantifiers are logical constants by this criterion.

These notes contain the material covered in the second level logic course which has been offered at the Institut für Maschinelle Sprachverarbeitung of the University of Stuttgart on an annual basis since 1992. The course is aimed at students who are familiar with the notation and use of the first order predicate calculus but have had little or no previous exposure to metamathematics.

This volume provides the first ever extensive translation of the notes and fragments that survived Kant's death in 1804. These include marginalia, lecture notes, and sketches and drafts for his published works. They are important as an indispensable resource for understanding Kant's intellectual development and published works, casting new light on Kant's conception of his own philosophical methods and his relations to his predecessors, as well as on central doctrines of his work such as the theory of space, (...) time and categories, the refutations of skepticism and metaphysical dogmatism, the theory of the value of freedom and the possibility of free will, the conception of God, the theory of beauty, and much more. (shrink)

A note correcting some technical terminology from linguistics found in ‘Distributive Terms, Truth, and The Port Royal Logic’, this journal, Jan. 17, 2013, 133–54.

This note gives a new proof of the 'operational extensionality' property of Abramsky's lazy lambda calculus-namely the coincidence of contextual equivalence with a co-inductively defined notion of 'applicative bisimilarity'. This purely syntactic results is here proved using a logical relation between the syntax and its denotational semantics. The proof exploits a mixed inductive/coinductive characterisation of the logical relation recently discovered by the author.

The paper explains how Gettier’s conclusion can be reached on general theoretical grounds within the framework of epistemic logic, without reliance on thought experiments. It extends the argument to permissive conceptions of justification that invalidate principles of multi-premise closure and require neighbourhood semantics rather than semantics of a more standard type. The paper concludes by recommending a robust methodology that aims at convergence in results between thought experimentation and more formal methods. It also warns against conjunctive definitions as sharing (...) several of the drawbacks of disjunctive definitions. (shrink)

The present note offers a proof for separating the truth-values of an arbitrary finitely many valued Łukasiewicz logic by making use of Gray codes.

We present and show topological completeness for LB, the logic of the topological border. LB is also a logic of epistemic ignorance. Also, we present and show completeness for LUT, the logic of unknown truths. A simple topological completeness proof for S4 is also presented using a T1 space.

In this note, I show how Christian List's modal logic of republican freedom (as published in this journal in 2006) can be extended (1) to grasp the differences between liberal freedom (noninterference) and republican freedom (non-domination) in terms of two purely logical axioms and (2) to cover a more recent definition of republican freedom in terms of `arbitrary interference' that gains popularity in the literature.

These short notes are intended to supplement the lectures and text ntroduce some of the basic concepts of Modal Logic. The primary goal is to provide students in Philosophy 151 at Stanford University with a study guide that will complement the lectures on modal logic. There are many textbooks that you can consult for more information. The following is a list of some texts (this is not a complete list, but a pointer to books that I have (...) found particularly useful). (shrink)

It is proved that all bimodal tense logics which contain the logic of the weak orderings and have unbounded depth do not have the interpolation property.

Logical consequence in first-order predicate logic is defined substitutionally in set theory augmented with a primitive satisfaction predicate: an argument is defined to be logically valid if and only if there is no substitution instance with true premises and a false conclusion. Substitution instances are permitted to contain parameters. Variants of this definition of logical consequence are given: logical validity can be defined with or without identity as a logical constant, and quantifiers can be relativized in substitution instances or (...) not. It is shown that the resulting notions of logical consequence are extensionally equivalent to versions of first-order provability and model-theoretic consequence. Every model-theoretic interpretation has a substitutional counterpart, but not vice versa. In particular, in contrast to the model-theoretic account, there is a trivial intended interpretation on the substitutional account, namely, the homophonic interpretation that does not substitute anything. Applications to free logic, and theories and languages other than set theory are sketched. (shrink)

This short note considers the formulation of Full Intuitionistic Linear Logic given by Hyland and de Paiva . Unfortunately the formulation is not closed under the process of cut elimination. This note proposes an alternative formulation based on the notion of patterns.

We shed light on an old problem by showing that the logic LP cannot define a binary connective $\odot$ obeying detachment in the sense that every valuation satisfying $\varphi$ and $(\varphi\odot\psi)$ also satisfies $\psi$ , except trivially. We derive this as a corollary of a more general result concerning variable sharing.

Witnessed Gödel logics are based on the interpretation of () by minimum instead of supremum . Witnessed Gödel logics appear for many practical purposes more suited than usual Gödel logics as the occurrence of proper infima/suprema is practically irrelevant. In this note we characterize witnessed Gödel logics with absoluteness operator w.r.t. witnessed Gödel logics using a uniform translation.

In a paper from the 1980s, Byrd claims that the logic of "eventual permanence" for linear time is KD5. In this note we take up Byrd's novel argument for this and, treating the problem as one concerning translational embeddings, show that rather than KD5 the correct logic of "eventual permanence" is KD45.

After an introduction which demonstrates the failure of the equational analogue of Beth?s definability theorem, the first two sections of this paper are devoted to an elementary exposition of a proof that a functional constant is equationally definable in an equational theory iff every model of the set of those consequences of the theory that do not contain the functional constant is uniquely extendible to a model of the theory itself.Sections three, four and five are devoted to applications and extensions (...) of this result.Topics considered here include equational definability in first order logic, an extended notion of definability in equational logic and the synonymy of equational theories.The final two sections briefly review some of the history of equational logic. (shrink)

A short characterization is given of the contents and origin of the collection of books on logic donated by John Venn to the Cambridge University Library and of its relation to the bibliography in...

1, . . . , n | ≈ ψ ? Here 1, . . . , n, ψ are premisses of some formal language, such as a propositional language or a predicate language. | ≈ is an entailment relation: the entailment holds if all models of the premisses also satisfy the conclusion, where the logic provides some suitable notion of ‘model’ and ‘satisfy’. Proof theory is normally invoked to answer a question of this form: one tries to prove the (...) conclusion from the premisses in a finite sequence of steps, where at each step one invokes an axiom or applies a rule of inference. (shrink)

We here make preliminary investigations into the model theory of DeMorgan logics. We demonstrate that Łoś's Theorem holds with respect to these logics and make some remarks about standard model-theoretic properties in such contexts. More concretely, as a case study we examine the fate of Cantor's Theorem that the classical theory of dense linear orderings without endpoints is $\aleph_{0}$-categorical, and we show that the taking of ultraproducts commutes with respect to previously established methods of constructing nonclassical structures, namely, Priest's Collapsing (...) Lemma and Dunn's Theorem in 3-Valued Logic. (shrink)

A discussion of Crispin Wright's 'paradox of higher-order vagueness', I suggest that the paradox may be resolved by careful attention to the logical principles used in its formulation. In particular, I focus attention on the rule of inference that allows for the inference from A to 'Definitely A', and argue that this rule, though valid, may not be used in subordinate deductions, e.g., in the course of a conditional proof. Wright's paradox uses the rule (or its equivalent) in this way.

In ‘Semantic Foundations for the Logic of Preference’ (Rescher, ed.,The Logic of Decision and Action, University Press, Pittsburgh, 1967), Nicholas Rescher claims that, on the semantics developed in that paper, a certain principle - call it ‘Q’ turns out to be ‘unacceptable’. I argue, however, that, given certain assumptions that Rescher invokes in that same paper,Q can in fact be shown to be a ‘preference-tautology’, and henceQ should be classified as ‘acceptable’ on Rescher's theory.

Logic is formal in the sense that all arguments of the same form as logically valid arguments are also logically valid and hence truth-preserving. However, it is not known whether all arguments that are valid in the usual model-theoretic sense are truthpreserving. Tarski claimed that it could be proved that all arguments that are valid (in the sense of validity he contemplated in his 1936 paper on logical consequence) are truthpreserving. But he did not offer the proof. The question (...) arises whether the usual modeltheoretic sense of validity and Tarski's 1936 sense are the same. I argue in this paper that they probably are not, and that the proof Tarski had in mind, although unusable to prove that model-theoretically valid arguments are truth-preserving, can be used to prove that arguments valid in Tarski's 1936 sense are truth-preserving. (shrink)