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Basic logic: reflection, symmetry, visibility
 Journal of Symbolic Logic
, 1997
"... Abstract We introduce a sequent calculus B for a new logic, named basic logic. The aim of basic logic is to find a structure in the space of logics. Classical, intuitionistic, quantum and nonmodal linear logics, are all obtained as extensions in a uniform way and in a single framework. We isolate t ..."
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Abstract We introduce a sequent calculus B for a new logic, named basic logic. The aim of basic logic is to find a structure in the space of logics. Classical, intuitionistic, quantum and nonmodal linear logics, are all obtained as extensions in a uniform way and in a single framework. We isolate three properties, which characterize B positively: reflection, symmetry and visibility. A logical constant obeys to the principle of reflection if it is characterized semantically by an equation binding it with a metalinguistic link between assertions, and if its syntactic inference rules are obtained by solving that equation. All connectives of basic logic satisfy reflection. To the control of weakening and contraction of linear logic, basic logic adds a strict control of contexts, by requiring that all active formulae in all rules are isolated, that is visible. From visibility, cutelimination follows. The full, geometric symmetry of basic logic induces known symmetries of its extensions, and adds a symmetry among them, producing the structure of a cube.
From Constructivism to Computer Science
, 1996
"... This paper is an expanded version of a lecture presented november 15, 1996, at the Technische Universitat Munchen, on the occassion of receiving the F.L. BauerPreis. I am indebted to K.R. Apt for detailed comments on an earlier draft of this paper. heated debate, but attracted few actual followers ..."
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This paper is an expanded version of a lecture presented november 15, 1996, at the Technische Universitat Munchen, on the occassion of receiving the F.L. BauerPreis. I am indebted to K.R. Apt for detailed comments on an earlier draft of this paper. heated debate, but attracted few actual followers. But even those who did not agree with Brouwer's ideas, such as D. Hilbert, were influenced by them and by the debate generated by Brouwer's views. Ideas developed in the context of intuitionism turned out to have a relevance which transcends the original setting. Brouwer's ideas on the foundations of mathematics were embedded in a highly personal and rather extreme version of idealistic philosophy, but it would carry us too far to go into this here. The first exposition of these ideas on the foundations of mathematics is found in his thesis from 1907 ("On the Foundations of Mathematics"). Briefly, Brouwer viewed mathematics as the activity of building constructions in the mind (of an ideal mathematician); mathematics is about such mental constructions, not about objects in some outside reality. For Brouwer, there is no platonistic universe of abstract ideas existing somewhere quite independently of human cognition. In his thesis, Brouwer had not yet realized the effect of his views on logic; but in a paper which appeared a year later, in 1908 ("On the unreliability of the logical principles") he did see the consequences. He demonstrated that from an intuitionistic point of view, we cannot assume that a mathematical statement is either true or false, independently of human knowledge; we can assert "A or not A" only in case we either have a proof of A or an argument showing that any attempt at constructing a proof of A must fail. As a result, if A represents an open mathemat...
Logically possible worlds and counterpart semantics . . .
, 2005
"... The paper reviews the technical results from modal logic as well as their philosophical significance. It focuses on possible worlds semantics in general and on the notion of a possible world, of accessibility, and object. ..."
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The paper reviews the technical results from modal logic as well as their philosophical significance. It focuses on possible worlds semantics in general and on the notion of a possible world, of accessibility, and object.
Syntax and Semantics
"... The year is 2002 and here we are at a symposium on Foundations ..."
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The year is 2002 and here we are at a symposium on Foundations
History of Constructivism in the 20th Century
"... notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providi ..."
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notions, such as `constructive proof', `arbitrary numbertheoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifierfree statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 18871963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifierfree expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...
Understanding Intuitionism
, 1997
"... This paper is an attempt to understand the differences between them. I am grateful to Mitsuru Yasuhara for stimulating discussions of this material and for pinpointing errors and obscurities in earlier drafts. I also wish to thank Simon Kochen and Per MartinLof for helpful comments. ..."
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This paper is an attempt to understand the differences between them. I am grateful to Mitsuru Yasuhara for stimulating discussions of this material and for pinpointing errors and obscurities in earlier drafts. I also wish to thank Simon Kochen and Per MartinLof for helpful comments.
Individual Choice Sequences in the Work of L.E.J. Brouwer
, 2002
"... Choice sequences are sequences not completely determined by a law. We state that the introduction of particular choice sequences by Brouwer in the late twenties was not recognised as such. We claim that their later use in the method of the creative subject was not traced back to this original use of ..."
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Choice sequences are sequences not completely determined by a law. We state that the introduction of particular choice sequences by Brouwer in the late twenties was not recognised as such. We claim that their later use in the method of the creative subject was not traced back to this original use of them and has been misinterpreted. We show where these particular choice sequences appear in the work of Brouwer and we show how they should be handled.
FEASIBILITY IN LOGIC
"... “ A proof is, not an object, but an act (…), and the act is primarily the act as it is being performed, only secondarily, and irrevocably, does it become the act that has been performed.� ” (MartinLöf) Introduction. One of the most disputed issues in contemporary philosophy is the following: should ..."
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“ A proof is, not an object, but an act (…), and the act is primarily the act as it is being performed, only secondarily, and irrevocably, does it become the act that has been performed.� ” (MartinLöf) Introduction. One of the most disputed issues in contemporary philosophy is the following: should we admit a vericonditional theory of meaning, which makes room for truthconditions that possibly transcend our ability to know whether they are fulfilled, or rather, as the antirealist suggests, should we replace these conditions by assertabilityconditions whose fulfillment, when obtaining, cannot fail to be recognized�? I will not declare myself in this
Epistemic truth and excluded middle*
"... Abstract: Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemi ..."
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Abstract: Can an epistemic conception of truth and an endorsement of the excluded middle (together with other principles of classical logic abandoned by the intuitionists) cohabit in a plausible philosophical view? In PART I I describe the general problem concerning the relation between the epistemic conception of truth and the principle of excluded middle. In PART II I give a historical overview of different attitudes regarding the problem. In PART III I sketch a possible holistic solution. Part I The Problem §1. The epistemic conception of truth. The epistemic conception of truth can be formulated in many ways. But the basic idea is that truth is explained in terms of epistemic notions, like experience, argument, proof, knowledge, etc. One way of formulating this idea is by saying that truth and knowability coincide, i.e. for every statement S
The irreflexivity of Brouwer’s philosophy ∗
, 2000
"... I argue that Brouwer’s general philosophy cannot account for itself, and, a fortiori, cannot lend justification to mathematical principles derived from it. Thus it cannot ground intuitionism, the job Brouwer had intended it to do. The strategy is to ask whether that philosophy actually allows for th ..."
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I argue that Brouwer’s general philosophy cannot account for itself, and, a fortiori, cannot lend justification to mathematical principles derived from it. Thus it cannot ground intuitionism, the job Brouwer had intended it to do. The strategy is to ask whether that philosophy actually allows for the kind of knowledge that such an account of itself would amount to. Brouwer tried to go ‘from philosophy to mathematics ’ and grounded his intuitionistic mathematics in a more general philosophy. 1 This background philosophy can be characterized as a transcendental one. That is, it purports to explain how a nonmundane subject builds up its world in consciousness. It is a radical transcendental philosophy in that this ‘world ’ does not contain just physical objects but everything, including abstract objects and the mundane subject (the subject as part of the world). From the empirical point of view, such a nonmundane subject is an idealized one. Like fellow transcendentalists