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Decidability and Undecidability Results for Propositional Schemata
"... We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions (e.g., pi) and iterated connectives ∨ or ∧ ranging over intervals parameterized by arithmetic variables (e.g., ∧n i=1 pi, where n is a parameter). The satisfiability problem is shown ..."
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We define a logic of propositional formula schemata adding to the syntax of propositional logic indexed propositions (e.g., pi) and iterated connectives ∨ or ∧ ranging over intervals parameterized by arithmetic variables (e.g., ∧n i=1 pi, where n is a parameter). The satisfiability problem is shown to be undecidable for this new logic, but we introduce a very general class of schemata, called boundlinear, for which this problem becomes decidable. This result is obtained by reduction to a particular class of schemata called regular, for which we provide a sound and complete terminating proof procedure. This schemata calculus (called stab) allows one to capture proof patterns corresponding to a large class of problems specified in propositional logic. We also show that the satisfiability problem becomes again undecidable for slight extensions of this class, thus demonstrating that boundlinear schemata represent a good compromise between expressivity and decidability. 1.
Sentence, proposition, judgment, statement, and fact, The Many Sides of Logic
"... friend, collaborator, codiscoverer of the Truthset Principle, and cocreator of the Classical Logic of Variablebinding Term Operatorson his eightieth birthday. abstract. The five ambiguous words—sentence, proposition, judgment, statement, and fact—each have meanings that are vague in the sense ..."
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friend, collaborator, codiscoverer of the Truthset Principle, and cocreator of the Classical Logic of Variablebinding Term Operatorson his eightieth birthday. abstract. The five ambiguous words—sentence, proposition, judgment, statement, and fact—each have meanings that are vague in the sense of admitting borderline cases. This paper discusses several senses of these and related words used in logic. It focuses on a constellation of recommended primary senses. A judgment is a private epistemic act that results in a new belief; a statement is a public pragmatic event involving an utterance. Each is executed by a unique person at a unique time and place. Propositions and sentences are timeless and placeless abstractions. A proposition is an intensional entity; it is a meaning composed of concepts. A sentence is a linguistic entity. A written sentence is a string of characters. A sentence can be used by a person to express meanings, but no sentence is intrinsically meaningful. Only propositions are properly said to be true or to be false—in virtue of facts, which are subsystems of the universe. The fact that two is even is timeless; the fact that Socrates was murdered is semieternal; the most general facts of physics—in virtue of which propositions of physics are true or false—are eternal. As suggested by the title, this paper is meant to be read aloud. 1
NonStandard Models of Arithmetic: a Philosophical and Historical perspective
, 2010
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1 Existential import today: New metatheorems; historical, philosophical, and pedagogical misconceptions
"... Willard V. O. Quine, inspiring teachers who taught far more than what they were taught. Contrary to common misconceptions, today‘s logic is not devoid of existential import: the universalized conditional x [S(x) P(x)] implies its corresponding existentialized conjunction x [S(x) & P(x)], not ..."
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Willard V. O. Quine, inspiring teachers who taught far more than what they were taught. Contrary to common misconceptions, today‘s logic is not devoid of existential import: the universalized conditional x [S(x) P(x)] implies its corresponding existentialized conjunction x [S(x) & P(x)], not in all cases, but in some. We characterize the proexamples by proving the ExistentialImport Equivalence: x [S(x) P(x)] implies x [S(x) & P(x)] iff x S(x) is logically true. The antecedent S(x) of the universalized conditional alone determines whether the universalized conditional has existential import, i.e., whether it implies its corresponding existentialized conjunction. A predicate is an open formula having only x free. An existentialimport predicate Q(x) is one whose existentialization, x Q(x), is logically true; otherwise, Q(x) is existentialimportfree or simply importfree. How abundant or widespread is existential import? How abundant or widespread are existentialimport predicates in themselves or in comparison to importfree predicates? We show that existentialimport predicates are quite abundant, and no less so than importfree predicates. Existentialimport implications hold as widely as they fail. Existential import is not an isolated phenomenon. As documented below, these results correct false or misleading passages even in respected logic texts.