11,570 research outputs found
Virtual Evidence: A Constructive Semantics for Classical Logics
This article presents a computational semantics for classical logic using
constructive type theory. Such semantics seems impossible because classical
logic allows the Law of Excluded Middle (LEM), not accepted in constructive
logic since it does not have computational meaning. However, the apparently
oracular powers expressed in the LEM, that for any proposition P either it or
its negation, not P, is true can also be explained in terms of constructive
evidence that does not refer to "oracles for truth." Types with virtual
evidence and the constructive impossibility of negative evidence provide
sufficient semantic grounds for classical truth and have a simple computational
meaning. This idea is formalized using refinement types, a concept of
constructive type theory used since 1984 and explained here. A new axiom
creating virtual evidence fully retains the constructive meaning of the logical
operators in classical contexts.
Key Words: classical logic, constructive logic, intuitionistic logic,
propositions-as-types, constructive type theory, refinement types, double
negation translation, computational content, virtual evidenc
A one-valued logic for non-one-sidedness
Does it make sense to employ modern logical tools for ancient philosophy? This well-known debate2 has been re-launched by the indologist Piotr Balcerowicz, questioning those who want to look at the Eastern school of Jainism with Western glasses. While plainly acknowledging the legitimacy of Balcerowicz's mistrust, the present paper wants to propose a formal reconstruction of one of the well-known parts of the Jaina philosophy, namely: the saptabhangi, i.e. the theory of sevenfold predication. Before arguing for this formalist approach to philosophy, let us return to the reasons to be reluctant at it
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
Truth and Theories of Truth
The concept of truth and competing philosophical theories on what truth amounts to have an important place in contemporary philosophy. The aim of this chapter is to give a synopsis of different theories of truth and the particular philosophical issues related to the concept of truth. The literature on this topic is vast, and we must necessarily be rather selective and very brief about complex questions of interpretation of various philosophers. The focus of the chapter is mainly on selected systematic issues and the most influential and well-established philosophical theories and key concepts
Quantum Non-Objectivity from Performativity of Quantum Phenomena
We analyze the logical foundations of quantum mechanics (QM) by stressing
non-objectivity of quantum observables which is a consequence of the absence of
logical atoms in QM. We argue that the matter of quantum non-objectivity is
that, on the one hand, the formalism of QM constructed as a mathematical theory
is self-consistent, but, on the other hand, quantum phenomena as results of
experimenter's performances are not self-consistent. This self-inconsistency is
an effect of that the language of QM differs much from the language of human
performances. The first is the language of a mathematical theory which uses
some Aristotelian and Russellian assumptions (e.g., the assumption that there
are logical atoms). The second language consists of performative propositions
which are self-inconsistent only from the viewpoint of conventional
mathematical theory, but they satisfy another logic which is non-Aristotelian.
Hence, the representation of quantum reality in linguistic terms may be
different: from a mathematical theory to a logic of performative propositions.
To solve quantum self-inconsistency, we apply the formalism of non-classical
self-referent logics
Identifying logical evidence
Given the plethora of competing logical theories of validity available, it’s understandable that there has been a marked increase in interest in logical epistemology within the literature. If we are to choose between these logical theories, we require a good understanding of the suitable criteria we ought to judge according to. However, so far there’s been a lack of appreciation of how logical practice could support an epistemology of logic. This paper aims to correct that error, by arguing for a practice-based approach to logical epistemology. By looking at the types of evidence logicians actually appeal to in attempting to support their theories, we can provide a more detailed and realistic picture of logical epistemology. To demonstrate the fruitfulness of a practice-based approach, we look to a particular case of logical argumentation—the dialetheist’s arguments based upon the self-referential paradoxes—and show that the evidence appealed to support a particular theory of logical epistemology, logical abductivism.publishedVersio
Noncomparabilities & Non Standard Logics
Many normative theories set forth in the welfare economics, distributive justice and cognate literatures posit noncomparabilities or incommensurabilities between magnitudes of various kinds. In some cases these gaps are predicated on metaphysical claims, in others upon epistemic claims, and in still others upon political-moral claims. I show that in all such cases they are best given formal expression in nonstandard logics that reject bivalence, excluded middle, or both. I do so by reference to an illustrative case study: a contradiction known to beset John Rawls\u27s selection and characterization of primary goods as the proper distribuendum in any distributively just society. The contradiction is avoided only by reformulating Rawls\u27s claims in a nonstandard form, which form happens also to cohere quite attractively with Rawls\u27s intuitive argumentation on behalf of his claims
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