Implicational Paradoxes and the Meaning of Logical Constants

Analysis, 2008

European Review of Philosophy, The Nature of Logic ( …, 1999

Thought, 2020

Work on the nature and scope of formal logic has focused unduly on the distinction between logical and extra-logical vocabulary; which argument forms a logical theory countenances depends not only on its stock of logical terms, but also on its range of grammatical categories and modes of composition. Furthermore, there is a sense in which logical terms are unnecessary. Alexandra Zinke has recently pointed out that propositional logic can be done without logical terms. By defining a logical-term-free language with the full expressive power of first-order logic with identity, I show that this is true of logic more generally. Furthermore, having, in a logical theory, non-trivial valid forms that do not involve logical terms is not merely a technical possibility. As the case of adverbs shows, issues about the range of argument forms logic should countenance can quite naturally arise in such a way that they do not turn on whether we countenance certain terms as logical.

Notre Dame Journal of Formal Logic, 1981

Journal of Philosophical Logic, 1983

The paper proposes a semantic value for the logical constants (connectives and quantifiers) within the framework of proof-theoretic semantics, basic meaning on the introduction rules of a meaning conferring natural deduction proof system. The semantic value is defined based on Frege's Context Principle, by taking "contributions" to sentential meanings as determined by the functionargument structure as induced by a type-logical grammar. In doing so, the paper proposes a novel proof-theoretic interpretation of the semantic types, traditionally interpreted in Henkin models. The compositionality of the resulting attribution of semantic values is discussed. Elsewhere, the same method was used for defining proof-theoretic meaning of subsentential phrases in a fragment of natural language. Doing the same for (the simpler and clearer case of) logic sheds more light on the proposal.

A many-valued logic in 8 truth-values based upon Classical logic termed ‘Universal Logic’, denoted U8, provides a correspondence to the ‘if-then’ implication meaning of natural language. Truth tables of implication and equivalence for U8 will be given, expanding the definition of validity. Accordingly, a new analysis of the ‘paradoxes of material implication’ will be undertaken. Material implication will be found to be identical with Universal logic when Boolean assignments are employed signifying that Classical is a subset of U8 logic. However, when 2 truth-values {true, false} are employed, denoted U2, implication resembles material equivalence and so validity is amended. As illustration of this approach five major ‘paradoxes of material implication’ will be analysed in terms of U2 validity. Remarkable results will be elucidated showing two ‘paradoxes’ to be affirmed while three were denied providing evidence that ‘Universal Logic’ offers an intuitive inductive logic.

Synthese, 2011

Bolzano's definition of consequence in effect associates with each set X of symbols (in a given interpreted language) a consequence relation ⇒X. We present this in a precise and abstract form, in particular studying minimal sets of symbols generating ⇒X. Then we present a method for going in the other direction: extracting from an arbitrary consequence relation ⇒ its associated set C⇒ of constants. We show that this returns the expected logical constants from familiar consequence relations, and that, restricting attention to sets of symbols satisfying a strong minimality condition, there is an isomorphism between the set of strongly minimal sets of symbols and the set of corresponding consequence relations (both ordered under inclusion). * This is a much revised version of one part of my talk at the 2008 Philosophy of Logical Consequence Conference in Uppsala. I thank the audience there, in particular Stephen Read and Göran Sundholm, as well as the referee for this paper, for helpful remarks. The paper presents the beginnings of ongoing work on logical constants now done jointly with Denis Bonnay-our discussions have been helpful also for this paper-supported by the ESF-funded project 'Logic for Interaction' (LINT; http://www.illc.uva.nl/lint/index.php), a Collaborative Research Project under the Eurocores program LogICCC. My work on the paper was also supported by a grant from the Swedish Research Council. 1 The referee pointed out that some algebraic approaches, where the consequence relation is a partial order and logical constants are definable in terms of it (e.g. as suprema, infima, and complement), may be exceptions to this rule. 2 See MacFarlane (2009) for a recent survey of approaches to the issue of logicality.

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Godel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have pride of place. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, et al. Perhaps, by Godel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Godel's incompleteness theorems will have ramifications in other areas involving logic. The paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics.

Journal of Philosophical Logic

Despite the tendency to be otherwise, some non-classical logics are known to validate formulas that are invalid in classical logic. A subclass of such systems even possesses pairs of a formula and its negation as theorems, without becoming trivial. How should these provable contradictions be understood? The present paper aims to shed light on aspects of this phenomenon by taking as samples the constructive connexive logic C, which is obtained by a simple modification of a system of constructible falsity, namely N4, as well as its non-constructive extension C3. For these systems, various observations concerning provable contradictions are made, using mainly a proof-theoretic approach. The topics covered in this paper include: how new contradictions are found from parts of provable contradictions; how to characterise provable contradictions in C3 that are constructive; how contradictions can be seen from the relative viewpoint of strong implication; and as an appendix an attempt at ge...