Temporal and atemporal truth in intuitionistic mathematics

Martino, Enrico; Usberti, Gabriele

doi:10.1007/bf00763507

Cited by 13 publications

(10 citation statements)

References 4 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Hart's negative appraisal notwithstanding, some commentators have proposed that intuitionists ought to adopt such a resolution-e.g. Martino and Usberti (1994). 9 Our ultimate view on these matters is in fact slightly more complex.…”

Section: Knowledge Proof and Explicit Modal Logicmentioning

confidence: 98%

From the Knowability Paradox to the existence of proofs

Dean

Kurokawa

2009

Synthese

View full text Add to dashboard Cite

The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Löf using the sort of propositional modal operators which are employed in the original derivation of the Paradox. Instead we propose that the central tenet of verificationism is most accurately formulated as follows: if ϕ is true, then there exists a proof of ϕ. Building on the work of Artemov (Bull Symb Log 7(1): 1-36, 2001), a system of explicit modal logic with proof quantifiers is introduced to reason about such statements. When the original reasoning of the Paradox is developed in this setting, we reach not a contradiction, but rather the conclusion that there must exist non-constructed proofs. This outcome is evaluated relative to the controversy between Dummett and Prawitz about proof existence and bivalence.

show abstract

Section: Knowledge Proof and Explicit Modal Logicmentioning

confidence: 98%

From the Knowability Paradox to the existence of proofs

Dean

Kurokawa

2009

Synthese

View full text Add to dashboard Cite

show abstract

“…Rather, a key requirement on constructivism is typically that it admits a constructive account of truth, such for a proposition  to be "true" consists in there existing a constructive proof of  , and that, once true it remains so (monotonicity). However, pinning exactly what this requires is, notoriously tricky, resulting in a bifurcation of the understanding of what is meant by proof in the literature [17], [23], [27] into:  Actualism (there actually exists a proof of  )  Potentialism ( is potentially provable) 2 . Actualism is a tensed, time-bound, notion of proof given in a specific moment, and is typically thought to have been held by Brouwer and Heyting, see e.g.…”

Section: Constructivism and Undetermined Statementsmentioning

confidence: 99%

“…[20]), but unlike standard Kripke-structures, there is no clause for negation since that can only ever be involved in an indirect proof (refutation). 17 Definition 9 (Interpretation for compound formulas for conclusive proofs and refutations):…”

Section: Interpreting Problems and Solutionsmentioning

confidence: 99%

Co-constructive Logics for Proofs and Refutations

Trafford

2015

Studia Humana

View full text Add to dashboard Cite

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.

show abstract

“…[2, pp. 18-19], and touched upon by some authors, D. Pravitz [17], P. Martin-Löf [14] and W. Rabinowicz [19] among them (see also 1994 issue of Topoi [29] devoted to intuitionistic truth, especially the contribution by E. Martino and G. Usberti [15]). The opposition of "actualist" and "possibilist" approaches to intuitionistic truth has been also considered in [25, pp.…”

Section: Intuitionistic Conception Of Truth and Its Semantic Implemenmentioning

confidence: 99%

“…But that is as it should be, since the antirealist must, of course, refuse any intuition of a realistic nature. [15, p. 88-89] It has been repeatedly observed in the literature (see, e.g., [15,18]) that an orthodox intuitionism, as conceived and advocated by Brouwer and Heyting, expressly asserts the temporal (and thus, actualist) approach to the main mathematical and logical notions, and this temporality reflects the very essence of intuitionist philosophy of mathematics 1 .…”

Section: Intuitionistic Conception Of Truth and Its Semantic Implemenmentioning

confidence: 99%