An interactive approach to proof-theoretic semantics

In truth-functional semantics for propositional logics, categoricity and compositionality are unproblematic. This is not the case for proof-theoretic semantics, where failures of both occur for the semantics determined by monological entailment structures for classical and intuitionistic logic. This is problematic for inferentialists, where the meaning of logical constants is supposed to be determined by their rules. Recent attempts to overcome these issues have primarily considered symmetric entailment structures, but these are tricky to interpret. Here, I instead consider an entailment structure that combines provability with the dual notion of disproof (or refutation). This is interpreted as a dialogue structure between the roles of prover and denier, where an assertion of a statement involves a commitment to its defence, and a denial of the statement involves a commitment to its challenge. The interaction between the two is constitutive of a proof-theoretic semantics capable of dealing with the above issues

On the Truth of G\"odelian and Rosserian Sentences

There is a longstanding debate in the logico-philosophical community as to why the G\"odelian sentences of a consistent and sufficiently strong theory are true. The prevalent argument seems to be something like this: since every one of the G\"odelian sentences of such a theory is equivalent to the theory's consistency statement, even provably so inside the theory, the truth of those sentences follows from the consistency of the theory in question. So, G\"odelian sentences of consistent theories should be true. In this paper, we show that G\"odelian sentences of only sound theories are true; and there is a long road from consistency to soundness, indeed a hierarchy of conditions which are satisfied by some theories and falsified by others. We also study the truth of Rosserian sentences and provide necessary and sufficient conditions for the truth of Rosserian (and also G\"odelian) sentences of theories.Comment: 10 page

Co-constructive logics for proofs and refutations

This paper considers logics which are formally dual to intuition- istic logic in order to investigate a co-constructive logic for proofs and refu- tations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely- held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for state- ments for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems

Co-constructive logic for proofs and refutations

This paper considers logics which are formally dual to intuitionistic logic in order to investigate a co-constructive logic for proofs and refutations. This is philosophically motivated by a set of problems regarding the nature of constructive truth, and its relation to falsity. It is well known both that intuitionism can not deal constructively with negative information, and that defining falsity by means of intuitionistic negation leads, under widely-held assumptions, to a justification of bivalence. For example, we do not want to equate falsity with the non-existence of a proof since this would render a statement such as “pi is transcendental” false prior to 1882. In addition, the intuitionist account of negation as shorthand for the derivation of absurdity is inadequate, particularly outside of purely mathematical contexts. To deal with these issues, I investigate the dual of intuitionistic logic, co-intuitionistic logic, as a logic of refutation, alongside intuitionistic logic of proofs. Direct proof and refutation are dual to each other, and are constructive, whilst there also exist syntactic, weak, negations within both logics. In this respect, the logic of refutation is weakly paraconsistent in the sense that it allows for statements for which, neither they, nor their negation, are refuted. I provide a proof theory for the co-constructive logic, a formal dualizing map between the logics, and a Kripke-style semantics. This is given an intuitive philosophical rendering in a re-interpretation of Kolmogorov’s logic of problems

Truth vs. provability – philosophical and historical remarks

Since Plato, Aristotle and Euclid the axiomatic method was considered as the best method to justify and to organize mathematical knowledge. The first mature and most representative example of its usage in mathematics were Elements of Euclid. They established a pattern of a scientific theory and in particular a paradigm in mathematics

Pacifican, March 15, 1968

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Games and Logic

The idea behind these games is to obtain an alternative characterization of logical notions cherished by logicians such as truth in a model, or provability (in a formal system). We offer a quick survey of Hintikka\u27s evaluation games, which offer an alternative notion of truth in a model for first-order langauges. These are win-lose, extensive games of perfect information. We then consider a variation of these games, IF games, which are win-lose extensive games of imperfect information. Both games presuppose that the meaning of the basic vocabulary of the language is given. To give an account of the linguistic conventions which settle the meaning of the basic vocabulary, we consider signaling games, inspired by Lewis\u27 work. We close with IF probabilistic games, a strategic variant of IF games which combines semantical games with von Neumann\u27s minimax theorem

Standards probatórios e convicção do juiz : a operacionalização do juízo de fatos mediante a adoção de standards de prova

O presente estudo examina a operacionalização do princípio do livre convencimento no âmbito do juízo de fatos. A partir da premissa teórica de que a relação entre prova e verdade não é ontológica, tem-se que a convicção embasa-se em probabilidades. Com a finalidade de prevenir arbitrariedades, é possível adotar standards probatórios como meios de balizar o convencimento judicial para a adoção da narrativa mais adequada ao contexto probatório. Surge, então, o questionamento de como deve ser a operacionalização no aspecto procedimental e substantivo do juízo de fatos com a adoção dos standards. Do contrário, em não havendo uma efetiva justificação da convicção, pouco adiantaria fazê-lo, ao passo que a não definição prévia do standard implicaria violação do direito ao contraditório. Posta essa problemática, caracterizam-se os standards probatórios, bem como expõem-se possíveis modos de operacionalização formal e material, com a análise de entendimentos consolidados do STJ quanto à prova de acordo com o direito material. Para verificar a utilidade teórica dos resultados da pesquisa, estes analisam-se a partir de sua adequação ao princípio do contraditório como direito à participação no processo.The present study examines the operationalization of the principle of free conviction in the context of the judgment of facts. From the theoretical premise that the nexus between proof and truth is not ontological, one has that the conviction is based on probabilities. In order to prevent arbitrariness, it is possible to adopt evidential standards as a means to guide the judicial conviction for the adoption of the narrative best suited to the evidential context. The question then arises of how operationalization should be in the procedural and substantive aspect of the judgment of facts with the adoption of standards. Otherwise, in the absence of an effective justification of the conviction, it would be of little use to do so, whereas failure to pre-define the standard would imply a violation of the right to contradiction. Given this problem, the evidential standards are characterized, as well as possible modes of formal and material operationalization are exposed, with the analysis of the STJ's consolidated understandings regarding the proof in accordance with material law. To verify the theoretical usefulness of the research results, they are analyzed from their suitability to the contradictory principle as the right to participate in the process

Proving unprovability

This paper addresses the question: given some theory T that we accept, is there some natural, generally applicable way of extending T to a theory S that can prove a range of things about what it itself (i.e., S) can prove, including a range of things about what it cannot prove, such as claims to the effect that it cannot prove certain particular sentences (e.g., 0 = 1), or the claim that it is consistent? Typical characterizations of Gödel’s second incompleteness theorem, and its significance, would lead us to believe that the answer is ‘no’. But the present paper explores a positive answer. The general approach is to follow the lead of recent (and not so recent) approaches to truth and the Liar paradox

Game theoretical semantics for some non-classical logics

Paraconsistent logics are the formal systems in which absurdities do not trivialise the logic. In this paper, we give Hintikka-style game theoretical semantics for a variety of paraconsistent and non-classical logics. For this purpose, we consider Priest’s Logic of Paradox, Dunn’s First-Degree Entailment, Routleys’ Relevant Logics, McCall’s Connexive Logic and Belnap’s four-valued logic. We also present a game theoretical characterisation of a translation between Logic of Paradox/Kleene’s K3 and S5. We underline how non-classical logics require different verification games and prove the correctness theorems of their respective game theoretical semantics. This allows us to observe that paraconsistent logics break the classical bidirectional connection between winning strategies and truth values