On Natural Deduction in Classical First-Order Logic: Curry
... If one wants to understand how is it possible that a classical proof has any computational content in the first place, the concept of learning is essential. It was a discovery by Hilbert that from classical proofs one can extract approximation processes, that learn how to construct non-effective obj ...
... If one wants to understand how is it possible that a classical proof has any computational content in the first place, the concept of learning is essential. It was a discovery by Hilbert that from classical proofs one can extract approximation processes, that learn how to construct non-effective obj ...
6. Truth and Possible Worlds
... The first assumption says that thought is capable, at least, of capturing reality. It would be rather depressing if every possible world were false. The second assumption is grounded in the fact that the possible worlds are mutually inconsistent, so that only one can be believed. If two or more of t ...
... The first assumption says that thought is capable, at least, of capturing reality. It would be rather depressing if every possible world were false. The second assumption is grounded in the fact that the possible worlds are mutually inconsistent, so that only one can be believed. If two or more of t ...
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... 5. If ∆ is consistent, then at least one of ∆ ∪ {A} or ∆ ∪ {¬A} is consistent for any wff A. 6. If there is a truth-valuation v such that v(A) = 1 for all A ∈ ∆, then ∆ is consistent. 7. If 6` A, and ∆ contains the schema based on A, then ∆ is not consistent. Remark. The converse of 6 is also true; ...
... 5. If ∆ is consistent, then at least one of ∆ ∪ {A} or ∆ ∪ {¬A} is consistent for any wff A. 6. If there is a truth-valuation v such that v(A) = 1 for all A ∈ ∆, then ∆ is consistent. 7. If 6` A, and ∆ contains the schema based on A, then ∆ is not consistent. Remark. The converse of 6 is also true; ...
full text (.pdf)
... between first-order interpreted reasoning to justify the premises p1 = q1 ∧ · · · ∧ pn = qn and purely propositional reasoning to establish that the conclusion p = q follows from the premises. Unfortunately, the Horn theory is computationally more complex than the equational theory. The general Horn ...
... between first-order interpreted reasoning to justify the premises p1 = q1 ∧ · · · ∧ pn = qn and purely propositional reasoning to establish that the conclusion p = q follows from the premises. Unfortunately, the Horn theory is computationally more complex than the equational theory. The general Horn ...
3.1 Syntax - International Center for Computational Logic
... smallest set, if there is no proper subset satisfying the the conditions. Obviously, there are sets satisfying the conditions. The intersection of two sets satisfying the conditions satisfies the conditions again. Hence, the smallest set is the intersection of all sets satisfying the two conditions ...
... smallest set, if there is no proper subset satisfying the the conditions. Obviously, there are sets satisfying the conditions. The intersection of two sets satisfying the conditions satisfies the conditions again. Hence, the smallest set is the intersection of all sets satisfying the two conditions ...
URL - StealthSkater
... measurement resolution as a property of dynamics itself. The string word sheets are uniquely identified as inverse images under imbedding map of space-time surface to H=M4×CP2 of homologically non-trivial geodesic sphere of CP2 defining homological magnetic monopole. Holography in its strongest sens ...
... measurement resolution as a property of dynamics itself. The string word sheets are uniquely identified as inverse images under imbedding map of space-time surface to H=M4×CP2 of homologically non-trivial geodesic sphere of CP2 defining homological magnetic monopole. Holography in its strongest sens ...
A Contraction-free and Cut-free Sequent Calculus for
... From the point of view of Hilbert systems, propositional dynamic logic is well-defined. Indeed, there are several equivalent axiomatisations of P DL (see for example [4, 7]), each of which is obtained by adding to classical propositional logic: (i) the distribution axiom schema, that now has the form ...
... From the point of view of Hilbert systems, propositional dynamic logic is well-defined. Indeed, there are several equivalent axiomatisations of P DL (see for example [4, 7]), each of which is obtained by adding to classical propositional logic: (i) the distribution axiom schema, that now has the form ...
WHAT IS THE RIGHT NOTION OF SEQUENTIALITY? 1. Introduction
... for coding formulas and the like. We can start with a bare minimum and build everything we need in addition from that. For example, we do not need addition and multiplication for our numbers since we can define these operations using the sequences to code the recursive mechanisms. Well, but couldn’t ...
... for coding formulas and the like. We can start with a bare minimum and build everything we need in addition from that. For example, we do not need addition and multiplication for our numbers since we can define these operations using the sequences to code the recursive mechanisms. Well, but couldn’t ...
Roland HINNION ULTRAFILTERS (WITH DENSE ELEMENTS
... 5.6. Remarks about δE -complete ultrafilters. As usually, "δE -complete" means "closed under δE -finite intersections". It is easy to see that, when δE is infinite, SD(E) has the δE -FIP (i.e. any δE -finite intersection of elements of SD(E) is non-empty), but has not the ...
... 5.6. Remarks about δE -complete ultrafilters. As usually, "δE -complete" means "closed under δE -finite intersections". It is easy to see that, when δE is infinite, SD(E) has the δE -FIP (i.e. any δE -finite intersection of elements of SD(E) is non-empty), but has not the ...
Introduction to Logic for Computer Science
... The early logicians of the nineteenth and twentieth centuries hoped to establish formal logic as a foundation for mathematics, though that never really happened. But mathematics does rest on one firm foundation, namely set theory. But Set theory itself has been expressed in first order logic. What r ...
... The early logicians of the nineteenth and twentieth centuries hoped to establish formal logic as a foundation for mathematics, though that never really happened. But mathematics does rest on one firm foundation, namely set theory. But Set theory itself has been expressed in first order logic. What r ...
Modal Reasoning
... nice properties; in particular, if Σ is a maximally consistent set: (i) ¬φ ∈ Σ if and only if not φ ∈ Σ (ii) φ ∧ ψ ∈ Σ if and only if φ ∈ Σ and ψ ∈ Σ It follows easily that they are also closed under K-derivable formulas. (iii) 3φ ∈ Σ if and only if there is some ∆ with ΣR∆ and φ ∈ ∆ ...
... nice properties; in particular, if Σ is a maximally consistent set: (i) ¬φ ∈ Σ if and only if not φ ∈ Σ (ii) φ ∧ ψ ∈ Σ if and only if φ ∈ Σ and ψ ∈ Σ It follows easily that they are also closed under K-derivable formulas. (iii) 3φ ∈ Σ if and only if there is some ∆ with ΣR∆ and φ ∈ ∆ ...
Chapter 4, Mathematics
... effective. Effectiveness is extremely important, for in everyday argument it is often hard to be sure whether an argument is valid or not. As I have already remarked in chapter 2 it can be especially hard to be sure that an argument is invalid, as that may require showing that there is no valid form ...
... effective. Effectiveness is extremely important, for in everyday argument it is often hard to be sure whether an argument is valid or not. As I have already remarked in chapter 2 it can be especially hard to be sure that an argument is invalid, as that may require showing that there is no valid form ...
Interpreting and Applying Proof Theories for Modal Logic
... conditions of modal statements (in Kripke models with points and accessibility relations with different properties, underwriting different principles governing and and their interaction), we have no such consensus on what the basic items of deduction in modal vocabulary are. If our goal is to f ...
... conditions of modal statements (in Kripke models with points and accessibility relations with different properties, underwriting different principles governing and and their interaction), we have no such consensus on what the basic items of deduction in modal vocabulary are. If our goal is to f ...
Logic and Resolution - Institute for Computing and Information
... with the two quantifiers ∀ and ∃ just introduced. As was done for propositional logic, we now define the notion of a well-formed formula in predicate logic. The following definition also introduces the additional notions of free and bound variables. Definition A.8 A well-formed formula in predicate ...
... with the two quantifiers ∀ and ∃ just introduced. As was done for propositional logic, we now define the notion of a well-formed formula in predicate logic. The following definition also introduces the additional notions of free and bound variables. Definition A.8 A well-formed formula in predicate ...
Continuous and random Vapnik
... ITAÏ BEN YAACOV Abstract. Nous démontrons que si T est une théorie dépendante, sa randomisée de Keisler T R l’est aussi. Pour faire cela nous généralisons la notion d’une classe de Vapnik-Chervonenkis à des familles de fonctions à valeurs dans [0, 1] (une classe de Vapnik-Chervonenkis conti ...
... ITAÏ BEN YAACOV Abstract. Nous démontrons que si T est une théorie dépendante, sa randomisée de Keisler T R l’est aussi. Pour faire cela nous généralisons la notion d’une classe de Vapnik-Chervonenkis à des familles de fonctions à valeurs dans [0, 1] (une classe de Vapnik-Chervonenkis conti ...
The interpretation of the Einstein-Rupp experiments and their
... radiation. 5 Given these contexts, and Einstein’s initial expectations and gradual turn-around, one should expect that he had a dual wave-particle picture of light when the canal ray experiments were under discussion in the spring of 1926. But what could the details of that dual picture have looked ...
... radiation. 5 Given these contexts, and Einstein’s initial expectations and gradual turn-around, one should expect that he had a dual wave-particle picture of light when the canal ray experiments were under discussion in the spring of 1926. But what could the details of that dual picture have looked ...
THE MODEL CHECKING PROBLEM FOR INTUITIONISTIC
... [21]. Recall that every fragment of classical propositional logic with a fixed number of variables has an NC1 -complete validity problem (follows from [2]). The most common semantics for intuitionistic logic are Heyting semantics [11] and Kripke semantics [14, 13]—see also [22, Chap. 2]. The Heyting ...
... [21]. Recall that every fragment of classical propositional logic with a fixed number of variables has an NC1 -complete validity problem (follows from [2]). The most common semantics for intuitionistic logic are Heyting semantics [11] and Kripke semantics [14, 13]—see also [22, Chap. 2]. The Heyting ...
