8,523 research outputs found
Linear logic for constructive mathematics
We show that numerous distinctive concepts of constructive mathematics arise
automatically from an interpretation of "linear higher-order logic" into
intuitionistic higher-order logic via a Chu construction. This includes
apartness relations, complemented subsets, anti-subgroups and anti-ideals,
strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We
also explain the constructive bifurcation of classical concepts using the
choice between multiplicative and additive linear connectives. Linear logic
thus systematically "constructivizes" classical definitions and deals
automatically with the resulting bookkeeping, and could potentially be used
directly as a basis for constructive mathematics in place of intuitionistic
logic.Comment: 39 page
A Galois connection between classical and intuitionistic logics. I: Syntax
In a 1985 commentary to his collected works, Kolmogorov remarked that his
1932 paper "was written in hope that with time, the logic of solution of
problems [i.e., intuitionistic logic] will become a permanent part of a
[standard] course of logic. A unified logical apparatus was intended to be
created, which would deal with objects of two types - propositions and
problems." We construct such a formal system QHC, which is a conservative
extension of both the intuitionistic predicate calculus QH and the classical
predicate calculus QC.
The only new connectives ? and ! of QHC induce a Galois connection (i.e., a
pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying
posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation
translation of propositions into problems extends to a retraction of QHC onto
QH; whereas Goedel's provability translation of problems into modal
propositions extends to a retraction of QHC onto its QC+(?!) fragment,
identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic
modal logic, whose modality !? is a strict lax modality in the sense of Aczel -
and thus resembles the squash/bracket operation in intuitionistic type
theories.
The axioms of QHC attempt to give a fuller formalization (with respect to the
axioms of intuitionistic logic) to the two best known contentual
interpretations of intiuitionistic logic: Kolmogorov's problem interpretation
(incorporating standard refinements by Heyting and Kreisel) and the proof
interpretation by Orlov and Heyting (as clarified by G\"odel). While these two
interpretations are often conflated, from the viewpoint of the axioms of QHC
neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a
simplified version of Isabelle's meta-logic
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
Betting on the Outcomes of Measurements: A Bayesian Theory of Quantum Probability
We develop a systematic approach to quantum probability as a theory of
rational betting in quantum gambles. In these games of chance the agent is
betting in advance on the outcomes of several (finitely many) incompatible
measurements. One of the measurements is subsequently chosen and performed and
the money placed on the other measurements is returned to the agent. We show
how the rules of rational betting imply all the interesting features of quantum
probability, even in such finite gambles. These include the uncertainty
principle and the violation of Bell's inequality among others. Quantum gambles
are closely related to quantum logic and provide a new semantics to it. We
conclude with a philosophical discussion on the interpretation of quantum
mechanics.Comment: 21 pages, 2 figure
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
A predicative variant of a realizability tripos for the Minimalist Foundation.
open2noHere we present a predicative variant of a realizability tripos validating
the intensional level of the Minimalist Foundation extended with Formal Church
thesis.the file attached contains the whole number of the journal including the mentioned pubblicationopenMaietti, Maria Emilia; Maschio, SamueleMaietti, MARIA EMILIA; Maschio, Samuel
Dialectica Interpretation with Marked Counterexamples
Goedel's functional "Dialectica" interpretation can be used to extract
functional programs from non-constructive proofs in arithmetic by employing two
sorts of higher-order witnessing terms: positive realisers and negative
counterexamples. In the original interpretation decidability of atoms is
required to compute the correct counterexample from a set of candidates. When
combined with recursion, this choice needs to be made for every step in the
extracted program, however, in some special cases the decision on negative
witnesses can be calculated only once. We present a variant of the
interpretation in which the time complexity of extracted programs can be
improved by marking the chosen witness and thus avoiding recomputation. The
achieved effect is similar to using an abortive control operator to interpret
computational content of non-constructive principles.Comment: In Proceedings CL&C 2010, arXiv:1101.520
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