10,262 research outputs found
An Objection to Naturalism and Atheism from Logic
I proffer a success argument for classical logical consequence. I articulate in what sense that notion of consequence should be regarded as the privileged notion for metaphysical inquiry aimed at uncovering the fundamental nature of the world. Classical logic breeds necessitism. I use necessitism to produce problems for both ontological naturalism and atheism
Decidability of quantified propositional intuitionistic logic and S4 on trees
Quantified propositional intuitionistic logic is obtained from propositional
intuitionistic logic by adding quantifiers \forall p, \exists p over
propositions. In the context of Kripke semantics, a proposition is a subset of
the worlds in a model structure which is upward closed. Kremer (1997) has shown
that the quantified propositional intuitionistic logic H\pi+ based on the class
of all partial orders is recursively isomorphic to full second-order logic. He
raised the question of whether the logic resulting from restriction to trees is
axiomatizable. It is shown that it is, in fact, decidable. The methods used can
also be used to establish the decidability of modal S4 with propositional
quantification on similar types of Kripke structures.Comment: v2, 9 pages, corrections and additions; v1 8 page
Undecidability of first-order modal and intuitionistic logics with two variables and one monadic predicate letter
We prove that the positive fragment of first-order intuitionistic logic in
the language with two variables and a single monadic predicate letter, without
constants and equality, is undecidable. This holds true regardless of whether
we consider semantics with expanding or constant domains. We then generalise
this result to intervals [QBL, QKC] and [QBL, QFL], where QKC is the logic of
the weak law of the excluded middle and QBL and QFL are first-order
counterparts of Visser's basic and formal logics, respectively. We also show
that, for most "natural" first-order modal logics, the two-variable fragment
with a single monadic predicate letter, without constants and equality, is
undecidable, regardless of whether we consider semantics with expanding or
constant domains. These include all sublogics of QKTB, QGL, and QGrz -- among
them, QK, QT, QKB, QD, QK4, and QS4.Comment: Pre-final version of the paper published in Studia
Logica,doi:10.1007/s11225-018-9815-
Computer Science and Metaphysics: A Cross-Fertilization
Computational philosophy is the use of mechanized computational techniques to
unearth philosophical insights that are either difficult or impossible to find
using traditional philosophical methods. Computational metaphysics is
computational philosophy with a focus on metaphysics. In this paper, we (a)
develop results in modal metaphysics whose discovery was computer assisted, and
(b) conclude that these results work not only to the obvious benefit of
philosophy but also, less obviously, to the benefit of computer science, since
the new computational techniques that led to these results may be more broadly
applicable within computer science. The paper includes a description of our
background methodology and how it evolved, and a discussion of our new results.Comment: 39 pages, 3 figure
Logical consequence in modal logic II: Some semantic systems for S4
ABSTRACT: This 1974 paper builds on our 1969 paper (Corcoran-Weaver [2]). Here we present three (modal, sentential) logics which may be thought of as partial systematizations of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of these three logics coincide with one another and with those of standard formalizations of Lewis's S5. These logics, when regarded as logistic systems (cf. Corcoran [1], p. 154), are seen to be equivalent; but, when regarded as consequence systems (ibid., p. 157), one diverges from the others in a fashion which suggests that two standard measures of semantic complexity may not be as closely linked as previously thought.
This 1974 paper uses the linear notation for natural deduction presented in [2]: each two-dimensional deduction is represented by a unique one-dimensional string of characters. Thus obviating need for two-dimensional trees, tableaux, lists, and the like—thereby facilitating electronic communication of natural deductions.
The 1969 paper presents a (modal, sentential) logic which may be thought of as a partial systematization of the semantic and deductive properties of a sentence operator which expresses certain kinds of necessity. The logical truths [sc. tautologies] of this logic coincides those of standard formalizations of Lewis’s S4. Among the paper's innovations is its treatment of modal logic in the setting of natural deduction systems--as opposed to axiomatic systems.
The author’s apologize for the now obsolete terminology. For example, these papers speak of “a proof of a sentence from a set of premises” where today “a deduction of a sentence from a set of premises” would be preferable.
1. Corcoran, John. 1969. Three Logical Theories, Philosophy of Science 36, 153–77. J P R
2. Corcoran, John and George Weaver. 1969. Logical Consequence in Modal Logic: Natural Deduction in S5 Notre Dame Journal of Formal Logic 10, 370–84. MR0249278 (40 #2524).
3. Weaver, George and John Corcoran. 1974. Logical Consequence in Modal Logic: Some Semantic Systems for S4, Notre Dame Journal of Formal Logic 15, 370–78. MR0351765 (50 #4253)
Actuality and the a priori
We consider a natural-language sentence that cannot be formally represented in a first-order language for epistemic two-dimensional semantics. We also prove this claim in the “Appendix” section. It turns out, however, that the most natural ways to repair the expressive inadequacy of the first-order language render moot the original philosophical motivation of formalizing a priori knowability as necessity along the diagonal
Questions and Answers about Oppositions
A general characterization of logical opposition is given in the present paper, where oppositions are defined by specific answers in an algebraic question-answer game. It is shown that opposition is essentially a semantic relation of truth values between syntactic opposites, before generalizing the theory of opposition from the initial Apuleian square to a variety of alter- native geometrical representations.
In the light of this generalization, the famous problem of existential import is traced back to an ambiguous interpretation of assertoric sentences in Aristotle's traditional logic. Following Abelard’s distinction between two alternative readings of the O-vertex: Non omnis and Quidam non, a logical difference is made between negation and denial by means of a more fine- grained modal analysis.
A consistent treatment of assertoric oppositions is thus made possible by an underlying abstract theory of logical opposition, where the central concept is negation. A parallel is finally drawn between opposition and consequence, laying the ground for future works on an abstract operator of opposition that would characterize logical negation just as does Tarski’s operator of consequence for logical truth
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