Proof translation for CVC3
... Tcvc 3 is translated into Hol light that produces a theorem Thol Are Thol and Tcvc 3 the same theorem? ...
... Tcvc 3 is translated into Hol light that produces a theorem Thol Are Thol and Tcvc 3 the same theorem? ...
Unification in Propositional Logic
... • check whether A is projective or not; • in the negative case, compute a projective approximation of A. The explicit computation of mgus or of complete sets of unifiers seems to be less important (see the application to admissible rules) and, in any case, it is only a question of writing down expli ...
... • check whether A is projective or not; • in the negative case, compute a projective approximation of A. The explicit computation of mgus or of complete sets of unifiers seems to be less important (see the application to admissible rules) and, in any case, it is only a question of writing down expli ...
Ramsey`s Theorem and Compactness
... We will use this to produce a certain finitely satisfiable set of sentences Γ. From the compactness theorem, we will conclude that Γ is satisfiable, and from infinitary Ramsey’s theorem, we will conclude that Γ is not satisfiable. This contradiction proves finitary Ramsey’s theorem. First we define ...
... We will use this to produce a certain finitely satisfiable set of sentences Γ. From the compactness theorem, we will conclude that Γ is satisfiable, and from infinitary Ramsey’s theorem, we will conclude that Γ is not satisfiable. This contradiction proves finitary Ramsey’s theorem. First we define ...
When is Metric Temporal Logic Expressively Complete?
... Given an N -bounded FOK formula with one free variable x, we show that it is equivalent to a N 0 -bounded formula (over a possibly larger set of monadic predicates, suitably interpreted) in which the unary functions are only applied to x. We can remove occurrences of unary functions within the scope ...
... Given an N -bounded FOK formula with one free variable x, we show that it is equivalent to a N 0 -bounded formula (over a possibly larger set of monadic predicates, suitably interpreted) in which the unary functions are only applied to x. We can remove occurrences of unary functions within the scope ...
CHAPTER 5 SOME EXTENSIONAL SEMANTICS
... If T is the only designated value, the third value ⊥ corresponds to some notion of incomplete information, like undefined or unknown and is often denoted by the symbol U or I. If, on the other hand, ⊥ corresponds to inconsistent information, i.e. its meaning is something like known to be both true a ...
... If T is the only designated value, the third value ⊥ corresponds to some notion of incomplete information, like undefined or unknown and is often denoted by the symbol U or I. If, on the other hand, ⊥ corresponds to inconsistent information, i.e. its meaning is something like known to be both true a ...
Speaking Logic - SRI International
... compactness by replacing Con(Γ) by finite satisfiability. Then the set Γ̂ yields a complete finitely satisfiable set and hence it does not contain both A and ¬A for any A. Since Γ̂ is a superset of Γ, the latter is also satisfiable. Alternately, we can use completeness so show that if Γ is unsatisfi ...
... compactness by replacing Con(Γ) by finite satisfiability. Then the set Γ̂ yields a complete finitely satisfiable set and hence it does not contain both A and ¬A for any A. Since Γ̂ is a superset of Γ, the latter is also satisfiable. Alternately, we can use completeness so show that if Γ is unsatisfi ...
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... first-order case, however, we have to be a bit more careful. We know that because of γ-formulas proofs may have infinite branches. But that is not the main problem, since Hintikka’s lemma also works for infinite sets. However, not every infinite branch in a tableau is automatically a Hintikka set. C ...
... first-order case, however, we have to be a bit more careful. We know that because of γ-formulas proofs may have infinite branches. But that is not the main problem, since Hintikka’s lemma also works for infinite sets. However, not every infinite branch in a tableau is automatically a Hintikka set. C ...
M131-Tutorial_3-Integers-Division
... • let a be an integer and d a positive integer, then there exist unique integers q and r such that: a = d ×q + r ,0r
... • let a be an integer and d a positive integer, then there exist unique integers q and r such that: a = d ×q + r ,0r
notes
... 3 Natural Deduction (Gentzen, 1943) Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgments of the form φ1 , . . . , φn ⊢ φ, which means that φ can be derived from the assumptions φ1 , . . . , φn . If ⊢ φ without assumptions, then φ is a theorem of intu ...
... 3 Natural Deduction (Gentzen, 1943) Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgments of the form φ1 , . . . , φn ⊢ φ, which means that φ can be derived from the assumptions φ1 , . . . , φn . If ⊢ φ without assumptions, then φ is a theorem of intu ...
Version 1.5 - Trent University
... of the reasons for studying mathematical logic, which is also pursued for its own sake and finding new tools to use in the rest of mathematics and in related fields. In any case, mathematical logic is concerned with formalizing and analyzing the kinds of reasoning used in the rest of mathematics. Th ...
... of the reasons for studying mathematical logic, which is also pursued for its own sake and finding new tools to use in the rest of mathematics and in related fields. In any case, mathematical logic is concerned with formalizing and analyzing the kinds of reasoning used in the rest of mathematics. Th ...
An Automata Theoretic Decision Procedure for the Propositional Mu
... Propositional versions of the mu-calculus have been proposed by Pratt (1981) and Kozen (1982). These logics use a least lixpoint construct to increase the expressive power of propositional dynamic logic (PDL) of Fischer and Ladner (1979). Kozen’s formulation captures the infinite looping construct o ...
... Propositional versions of the mu-calculus have been proposed by Pratt (1981) and Kozen (1982). These logics use a least lixpoint construct to increase the expressive power of propositional dynamic logic (PDL) of Fischer and Ladner (1979). Kozen’s formulation captures the infinite looping construct o ...
Interpreting Lattice-Valued Set Theory in Fuzzy Set Theory
... by W. C. Powell [12] and R. J. Grayson [7], which is apparent, among other things, in its spelling of axioms—in a weak setting (such as that of an intermediate logic), different but classically equivalent versions of axioms are no longer equivalent; some are too strong, so that they strengthen the l ...
... by W. C. Powell [12] and R. J. Grayson [7], which is apparent, among other things, in its spelling of axioms—in a weak setting (such as that of an intermediate logic), different but classically equivalent versions of axioms are no longer equivalent; some are too strong, so that they strengthen the l ...
Point-free geometry, Approximate Distances and Verisimilitude of
... equip the class of possible scientific theories with a structure of approximate metric space and we define the verisimilitude of a theory as a function of its approximate distance from the truth (see also the definition of verisimilitude proposed in [5]). The motivation of such a choice is that usua ...
... equip the class of possible scientific theories with a structure of approximate metric space and we define the verisimilitude of a theory as a function of its approximate distance from the truth (see also the definition of verisimilitude proposed in [5]). The motivation of such a choice is that usua ...
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... Call a wff of FO(Σ) quasi-atom if it is either atomic, or of the form ∀xA, where A is a wff of FO(Σ). Let Γ be the set of all quasi-atoms of FO(Σ). Proposition 1. Every wff of FO(Σ) can be uniquely built up from Γ using only logical connectives → and ¬. Proof. Induction on the complexity of wff. For ...
... Call a wff of FO(Σ) quasi-atom if it is either atomic, or of the form ∀xA, where A is a wff of FO(Σ). Let Γ be the set of all quasi-atoms of FO(Σ). Proposition 1. Every wff of FO(Σ) can be uniquely built up from Γ using only logical connectives → and ¬. Proof. Induction on the complexity of wff. For ...
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... lemma, one giving a direct construction of a Hintikka set, and one using Lindenbaum’s construction, extending S to a maximally consistent set, which turned out to be a proof set. In first-order logic the question of compactness leads to a spin-off question. Can we extend Löwenheim’s theorem to sets ...
... lemma, one giving a direct construction of a Hintikka set, and one using Lindenbaum’s construction, extending S to a maximally consistent set, which turned out to be a proof set. In first-order logic the question of compactness leads to a spin-off question. Can we extend Löwenheim’s theorem to sets ...
Homogeneous structures, ω-categoricity and amalgamation
... Definition 1.9. Suppose L is a countable language and M is a countably infinite L-structure. We say that M (or T h(M )) is ω-categorical if every countable model of T h(M ) is isomorphic to M . Proposition 1.10. Suppose L is countable relational language and M is a countably infinite homogeneous L- ...
... Definition 1.9. Suppose L is a countable language and M is a countably infinite L-structure. We say that M (or T h(M )) is ω-categorical if every countable model of T h(M ) is isomorphic to M . Proposition 1.10. Suppose L is countable relational language and M is a countably infinite homogeneous L- ...
The First Incompleteness Theorem
... standard; some are used in importantly different ways by different authors; while some natural ideas seem to have no commonly used labels at all. It might be helpful, then, if I star my own non-standard terminology when it is first defined. You can safely re-use unstarred jargon without comment; but ...
... standard; some are used in importantly different ways by different authors; while some natural ideas seem to have no commonly used labels at all. It might be helpful, then, if I star my own non-standard terminology when it is first defined. You can safely re-use unstarred jargon without comment; but ...
On Probability of First Order Formulas in a Given Model
... in their works [ l ] , [2], [3]. This paper tries to be a positive contribution to the efforts of building up a probability apparatus over formulas of semantic logic, in order to use it then to form the base of information theory and to apply it in forming information systems theory. Nevertheless, o ...
... in their works [ l ] , [2], [3]. This paper tries to be a positive contribution to the efforts of building up a probability apparatus over formulas of semantic logic, in order to use it then to form the base of information theory and to apply it in forming information systems theory. Nevertheless, o ...
The superjump in Martin-Löf type theory
... In [11, 12] Martin-Löf considered an infinite, externally indexed tower of universes U0 ∈ U1 ∈ · · · ∈ Un ∈ . . . all of which are closed under the same standard ensemble of set forming operations. The next natural step was to implement a universe operator into type theory which takes a family of s ...
... In [11, 12] Martin-Löf considered an infinite, externally indexed tower of universes U0 ∈ U1 ∈ · · · ∈ Un ∈ . . . all of which are closed under the same standard ensemble of set forming operations. The next natural step was to implement a universe operator into type theory which takes a family of s ...
Proofs as Efficient Programs - Dipartimento di Informatica
... be obtained when trying to characterize complexity classes by way of logical systems. Completeness is always of an extensional nature: one proves that any function in the complexity class under consideration can be represented by a proof in the logical system, without taking care of the intensional ...
... be obtained when trying to characterize complexity classes by way of logical systems. Completeness is always of an extensional nature: one proves that any function in the complexity class under consideration can be represented by a proof in the logical system, without taking care of the intensional ...
Arindama Singh`s "Cantor`s Little Theorem"
... the set of all first order formulas having one free variable, up to equivalence. F1 consists of all formulas of the type P (x), ∀yQ(x, y), ∀xQ(x, y), ∃yQ(x, y), ∀x∃y(¬P (x)∧Q(y, z)), etc, having exactly one unquantified variable. Since F1 is taken up to equivalence, two formulas in F1 are equal iff ...
... the set of all first order formulas having one free variable, up to equivalence. F1 consists of all formulas of the type P (x), ∀yQ(x, y), ∀xQ(x, y), ∃yQ(x, y), ∀x∃y(¬P (x)∧Q(y, z)), etc, having exactly one unquantified variable. Since F1 is taken up to equivalence, two formulas in F1 are equal iff ...
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... In [2], Church proves that the following formulation of one-half of Leibniz’s characterization of equality holds in pure predicate calculus F1: Definition 1. Substitution of equals for equals: If S results from R by substitution of Q for P at one or more places in R (not necessarily at all occurrenc ...
... In [2], Church proves that the following formulation of one-half of Leibniz’s characterization of equality holds in pure predicate calculus F1: Definition 1. Substitution of equals for equals: If S results from R by substitution of Q for P at one or more places in R (not necessarily at all occurrenc ...
On the complexity of perfect models of logic programs 1
... Let J be a perfect model of P . We prove that for any i after i executions of the loop (2)–(12) for all x ∈ Done, the following condition is satisfied: J ≡x Ii and Ii ∩ S Ii (x) = ∅. It holds trivially before the start of the loop. Suppose, it holds before the i-th execution. Let an atom a be selec ...
... Let J be a perfect model of P . We prove that for any i after i executions of the loop (2)–(12) for all x ∈ Done, the following condition is satisfied: J ≡x Ii and Ii ∩ S Ii (x) = ∅. It holds trivially before the start of the loop. Suppose, it holds before the i-th execution. Let an atom a be selec ...
Jean Van Heijenoort`s View of Modern Logic
... quantification; but this could not be obtained till the traditional decomposition of the proposition into subject and predicate had been replaced by its analysis into function and argument(s). A preliminary accomplishment was the propositional calculus, with a truth-functional definition of the conn ...
... quantification; but this could not be obtained till the traditional decomposition of the proposition into subject and predicate had been replaced by its analysis into function and argument(s). A preliminary accomplishment was the propositional calculus, with a truth-functional definition of the conn ...