Category Theory as a Framework for an in re Interpretation of Mathematical Structuralism (original) (raw)

Intuitionistic Logic Considered As An Extension of Classical Logic: Some Critical Remarks

appeared in _Philosophia Scientiae_ 5 (2001), 2, pp. 27-50

In this paper we analyze the consideration of intuitionistic logic as an extension of classical logic. This -at first sight surprising- point of view has been sustained explicitly by Jan Lukasiewicz on the basis of a mapping of classical propositional logic into intuitionistic propositional logic by Kurt Gödel in 1933. Simultaneously with Gödel, Gerhard Gentzen had proposed another mapping of Peano´s arithmetic into Heyting´s arithmetic. We shall discuss these mappings in connection with the problem of determining what are the logical symbols that properly express the idiosyncracy of intuitionistic logic. Many philosophers and logicians do not seem to be sufficiently aware of the difficulties that arise when classical logic is considered as a subsystem of intuitionistic logic. As an outcome of the whole discussion these difficulties will be brought out. The notion of logical translation will play an essential role in the argumentation and some consequences related to the meaning of logical constants will be drawn.

Intuitionistic Epistemic Logic

The Review of Symbolic Logic, 2016

We outline an intuitionistic view of knowledge which maintains the original Brouwer-Heyting-Kolmogorov semantics of intuitionism and is consistent with Williamson's suggestion that intuitionistic knowledge be regarded as the result of verification. We argue that on this view co-reflection A → KA is valid and reflection KA → A is not; the latter is a distinctly classical principle, too strong as the intuitionistic truth condition for knowledge which can be more adequately expressed by other modal means, e.g. ¬A → ¬KA "false is not known." We introduce a system of intuitionistic epistemic logic, IEL, codifying this view of knowledge, and support it with an explanatory possible worlds semantics. From this it follows that previous outlines of intuitionistic knowledge are insufficiently intuitionistic: by endorsing KA → A they implicitly adopt a classical view of knowledge, by rejecting A → KA they reject the constructivity of truth. Within the framework of IEL, the knowability paradox is resolved in a constructive manner which, as we hope, reflects its intrinsic meaning.

Panu Raatikainen (Helsinki University) : Intuitionistic Logic and Its Philosophy

The informal notion of proof on which intuitionistic logic is founded is critically scrutinized. It is demonstrated that the traditional idea that we must actually possess a proof is inconsistent with intuitionistic logic. It is then argued that the more idealized notion of provability, that is, the possibility of proof, required by intuitionistic logic, is more unclear than is often recognized and difficult to make more specific without circularity. Its abstractness and inaccessibility is also highlighted.

The information in intuitionistic logic

Synthese, 2009

Issues about information spring up wherever one scratches the surface of logic. Here is a case that raises delicate issues of 'factual' versus 'procedural' information, or 'statics' versus 'dynamics'. What does intuitionistic logic, perhaps the earliest source of informational and procedural thinking in contemporary logic, really tell us about information? How does its view relate to its 'cousin' epistemic logic? We discuss connections between intuitionistic models and recent protocol models for dynamic-epistemic logic, as well as more general issues that emerge.

Intuitionistic logic and its philosophy

2013

Panu Raatikainen Intuitionistic Logic and Its Philosophy Formally, intuitionistic logic differs from classical logic in its denial of the universal validity of the law of the excluded middle, in short LEM (or the rule of double negation, which amounts to the same). This difference is based on the specific proof-interpretation, or BHK-interpretation (BHK stands for Brouwer-Heyting-Kolmogorov) of the meanings of logical connectives. It explains the meaning of the logical operators by describing the proofs of logically compound statements in terms of the proofs of their immediate subformulas. The BHK-interpretation of the sentential connectives goes as follows: 1 (1) There is a proof of  A  there is a procedure for transforming any proof of A into a proof of  ('absurdity' or 'the contradiction'). (2) There is a proof of A  B  there is a proof of A and there is a proof of B. (3) There is a proof of A  B  either there is a proof of A or there is a proof of B. (4) There is a proof of A  B  there is a procedure for transforming any proof of A into a proof of B. Now under such an interpretation of logical constants, LEM apparently fails to be valid. Namely, a proof of a disjunction A  A requires a proof of A or a proof of A. But for some A, there may not exist a proof of either. 1 I'll focus here only on propositional logic.

Reference and perspective in intuitionistic logics

Journal of Logic, Language and Information, 2006

What an intuitionist may refer to with respect to a given epistemic state depends not only on that epistemic state itself but on whether it is viewed concurrently from within, in the hindsight of some later state, or ideally from a standpoint "beyond" all epistemic states (though the latter perspective is no longer strictly intuitionistic). Each of these three perspectives has a different-and, in the last two cases, a novel-logic and semantics. This paper explains these logics and their semantics and provides soundness and completeness proofs. It provides, moreover, a critique of some common versions of Kripke semantics for intuitionistic logic and suggests ways of modifying them to take account of the perspective-relativity of reference.

Some remarks on the validity of the principle of explosion in intuitionistic logic (Revised version)

Seminário Lógica no Avião: 2013-2018, 2019

The formal system proposed by Heyting (1930, 1956) became the standard formulation of intuitionistic logic. The inference called ex falso quodlibet, or principle of explosion, according to which anything follows from a contradiction, holds in intuitionistic logic. However, it is not clear that explosion is in accordance with Brouwer's views on the nature of mathematics and its relationship with logic. Indeed, van Atten (2009) argues that a formal system in line with Brouwer's ideas should be a relevance logic. We agree that explosion should not hold in intuitionistic logic, but a relevance logic requires more than the invalidity of explosion. The principle known as ex quodlibet verum, according to which a valid formula follows from anything, should also be rejected by a relevantist. Given ex quodlibet verum, the inference we call weak explosion, according to which any negated proposition follows from a contradiction, is proved in a few steps. Although the same argument against explosion can be also applied against weak explosion, rejecting the latter requires the rejection of ex quodlibet verum. The result is the loss of at least one among reflexivity, monotonicity, and the deduction theorem in a Brouwerian intuitionistic logic, which seems to be an undesirable result.