admissible and derivable rules in intuitionistic logic
... (disjunction and existence property) but appear also in modal logics. We study here a particular case of this phenomenon, admissible rules in propositional calculus. G.E.Mints in [Mi 72] give sufficients conditions for admissible rules to be derivable. H.Friedman in [Fr 75] states the problem of the ...
... (disjunction and existence property) but appear also in modal logics. We study here a particular case of this phenomenon, admissible rules in propositional calculus. G.E.Mints in [Mi 72] give sufficients conditions for admissible rules to be derivable. H.Friedman in [Fr 75] states the problem of the ...
Guarded negation
... Several answers to these questions have been proposed. The first one is to consider the two variable fragment of first-order logic, which is decidable and has the finite model property [12]. Unfortunately, this observation does not go very far towards explaining the robust decidability of modal logi ...
... Several answers to these questions have been proposed. The first one is to consider the two variable fragment of first-order logic, which is decidable and has the finite model property [12]. Unfortunately, this observation does not go very far towards explaining the robust decidability of modal logi ...
The Complete Proof Theory of Hybrid Systems
... that it is often inadvertently forsaken. In logic, we can simply ensure soundness by checking it locally per proof rule. More intriguingly, however, our logical setting also enables us to ask the converse: is the proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s i ...
... that it is often inadvertently forsaken. In logic, we can simply ensure soundness by checking it locally per proof rule. More intriguingly, however, our logical setting also enables us to ask the converse: is the proof calculus complete, i.e., can it prove all that is true? A corollary to Gödel’s i ...
Logic and Resolution - Institute for Computing and Information
... Logic and Resolution One of the earliest formalisms for the representation of knowledge is logic. The formalism is characterized by a well-defined syntax and semantics, and provides a number of inference rules to manipulate logical formulas on the basis of their form in order to derive new knowledge ...
... Logic and Resolution One of the earliest formalisms for the representation of knowledge is logic. The formalism is characterized by a well-defined syntax and semantics, and provides a number of inference rules to manipulate logical formulas on the basis of their form in order to derive new knowledge ...
Clausal Logic and Logic Programming in Algebraic Domains*
... sets of clauses conjunctively as theories. We prove our representation theorem (Theorem 3.2) using the Hofmann-Mislove theorem [HM81]. This proof makes clear the basic Galois connection (duality) between theories in the clausal logic, and sets of models as Scott-compact saturated sets. The main resu ...
... sets of clauses conjunctively as theories. We prove our representation theorem (Theorem 3.2) using the Hofmann-Mislove theorem [HM81]. This proof makes clear the basic Galois connection (duality) between theories in the clausal logic, and sets of models as Scott-compact saturated sets. The main resu ...
Epsilon Substitution for Transfinite Induction
... in place of the usual induction axiom ∀x(φ[x] → φ[Sx]) → ∀x As a preliminary step, he proves termination for first order arithmetic with the complete induction axiom using ordinal assignments in the style of [Ackermann, 1940]. This has the added advantage of allowing the consideration of more powerf ...
... in place of the usual induction axiom ∀x(φ[x] → φ[Sx]) → ∀x As a preliminary step, he proves termination for first order arithmetic with the complete induction axiom using ordinal assignments in the style of [Ackermann, 1940]. This has the added advantage of allowing the consideration of more powerf ...
Temporal Here and There - Computational Cognition Lab
... and a pair of connections with other logics based on HT [5] are known. In this paper we deal with two problems that remained open in THT. The first problem consists in determining whether modal operators are interdefinable or not while the second problem corresponds to the definition of a sound an comp ...
... and a pair of connections with other logics based on HT [5] are known. In this paper we deal with two problems that remained open in THT. The first problem consists in determining whether modal operators are interdefinable or not while the second problem corresponds to the definition of a sound an comp ...
LOGIC AND p-RECOGNIZABLE SETS OF INTEGERS 1
... whose p-ary expansions are recognizable by a finite automaton. Various characterizations of p-recognizability are related in Theorem 4.1 : iterated uniform morphisms, algebraic formal power series, and definability by first-order formulae. Section 4 is centered around these four models of p-recogniz ...
... whose p-ary expansions are recognizable by a finite automaton. Various characterizations of p-recognizability are related in Theorem 4.1 : iterated uniform morphisms, algebraic formal power series, and definability by first-order formulae. Section 4 is centered around these four models of p-recogniz ...
Chapter 4. Logical Notions This chapter introduces various logical
... at least two cats, so there is at least one cat is formally valid the logician may paraphrase there are at least two cats as there is an x such that there is a y such that x is a cat and y is cat and it is not the case that x is identical to y. The numerical sentence and its "identity" paraphrase a ...
... at least two cats, so there is at least one cat is formally valid the logician may paraphrase there are at least two cats as there is an x such that there is a y such that x is a cat and y is cat and it is not the case that x is identical to y. The numerical sentence and its "identity" paraphrase a ...
pdf
... We consider two sound and complete axiomatizations for the HMS model, that differ with respect to the language used and the notion of validity. One axiomatization captures weak validity: a formula is weakly valid if it is never false (although it may be undefined). In the single-agent case, this axi ...
... We consider two sound and complete axiomatizations for the HMS model, that differ with respect to the language used and the notion of validity. One axiomatization captures weak validity: a formula is weakly valid if it is never false (although it may be undefined). In the single-agent case, this axi ...
The Gödelian inferences - University of Notre Dame
... consistency? This depends on whether the unprovability in S of the formula Con S suffices to show that all proper formalizations of ‘S ` ⊥’ are unprovable in S and also that all proper formalizations of other consistency statements are unprovable in S. One way to justify the Gödelian inference is t ...
... consistency? This depends on whether the unprovability in S of the formula Con S suffices to show that all proper formalizations of ‘S ` ⊥’ are unprovable in S and also that all proper formalizations of other consistency statements are unprovable in S. One way to justify the Gödelian inference is t ...
Logic for Computer Science. Lecture Notes
... 1 And it is often desirable and even necessary to follow both methodologies, assuming that they lead to compatible results. ...
... 1 And it is often desirable and even necessary to follow both methodologies, assuming that they lead to compatible results. ...
On Provability Logic
... (which is provable in PA) can be read the number three is a prime. The term S(S(S(0))) is denoted 3. More generally, the n-th numeral is defined as the term S(S . . (0) . .) with n occurrence of the symbol S. As an exercise we suggest the reader to formulate the fact that there are infinitely many p ...
... (which is provable in PA) can be read the number three is a prime. The term S(S(S(0))) is denoted 3. More generally, the n-th numeral is defined as the term S(S . . (0) . .) with n occurrence of the symbol S. As an exercise we suggest the reader to formulate the fact that there are infinitely many p ...
On Herbrand`s Theorem - UCSD Mathematics
... instances of equality axioms only). In the case where B does not contain the equality sign, then (2) is equivalent to B being a tautology. Let T be a first-order theory. A sequence of terms is said to witness A over T if the above conditions hold except with condition (2) replaced by the weaker cond ...
... instances of equality axioms only). In the case where B does not contain the equality sign, then (2) is equivalent to B being a tautology. Let T be a first-order theory. A sequence of terms is said to witness A over T if the above conditions hold except with condition (2) replaced by the weaker cond ...
Introduction to Mathematical Logic
... collection of constant, relation, and function symbols with their fixed arities. Types are usually denoted by τ . The cardinality of τ , denoted by |τ | , is the cardinality of its symbols. Whenever we define a type we usually list the symbols only; their sorts and arities are determined by the cont ...
... collection of constant, relation, and function symbols with their fixed arities. Types are usually denoted by τ . The cardinality of τ , denoted by |τ | , is the cardinality of its symbols. Whenever we define a type we usually list the symbols only; their sorts and arities are determined by the cont ...
Sequent calculus for predicate logic
... classical logic any formula ϕ can be brought in prenex normal form, that is, it can be rewritten as: ∃x0 ∀y0 ∃x1 ∀y1 . . . ∃xn ∀yn ψ(x0 , . . . , xn , y0 , . . . , yn ), with ψ quantifier-free. And a formula of this form is a tautology if and only if ∃x0 . . . ∃xn ψ(x0 , . . . , xn , f0 (x0 ), f1 (x ...
... classical logic any formula ϕ can be brought in prenex normal form, that is, it can be rewritten as: ∃x0 ∀y0 ∃x1 ∀y1 . . . ∃xn ∀yn ψ(x0 , . . . , xn , y0 , . . . , yn ), with ψ quantifier-free. And a formula of this form is a tautology if and only if ∃x0 . . . ∃xn ψ(x0 , . . . , xn , f0 (x0 ), f1 (x ...
PDF
... In their paper, they construct a class of graphs and give a PTIME query on this class not expressible in FO + IFP + C. We will not go into their proof in this thesis, but we will use their construction in one of our results. Now is a good time to make precise the notion of a logic capturing PTIME, a ...
... In their paper, they construct a class of graphs and give a PTIME query on this class not expressible in FO + IFP + C. We will not go into their proof in this thesis, but we will use their construction in one of our results. Now is a good time to make precise the notion of a logic capturing PTIME, a ...
On Horn envelopes and hypergraph transversals
... For (b) we reduce the NP-complete problem C L I Q U E to the problem of maximum eardinality ._M_M.Given a graph G = (V, E) and an integer k we construct a set of models M C_ {0, 1}IEI (equivalently, subsets of E), as follows: M contains all singletons {e}, and, for each vertex v E V, the set {e C E ...
... For (b) we reduce the NP-complete problem C L I Q U E to the problem of maximum eardinality ._M_M.Given a graph G = (V, E) and an integer k we construct a set of models M C_ {0, 1}IEI (equivalently, subsets of E), as follows: M contains all singletons {e}, and, for each vertex v E V, the set {e C E ...
Transfinite progressions: A second look at completeness.
... than an extension by n-reflection, unless the same formula is used to define the axioms of T in both extensions. (This is a consequence of the fact, which will emerge below, that definitions φ and of the axioms of T can be chosen so that T + REF0 (φ) proves the consistency of T + REFn ().) In the ca ...
... than an extension by n-reflection, unless the same formula is used to define the axioms of T in both extensions. (This is a consequence of the fact, which will emerge below, that definitions φ and of the axioms of T can be chosen so that T + REF0 (φ) proves the consistency of T + REFn ().) In the ca ...
CHAPTER 1 The main subject of Mathematical Logic is
... negation as A → ⊥. To embed classical logic, we need to go further and add as an axiom schema the principle of indirect proof, also called stability (∀~x (¬¬R~x → R~x ) for relation symbols R), but then it is appropriate to restrict to the language based on →, ∀, ⊥ and ∧. The reason for this restric ...
... negation as A → ⊥. To embed classical logic, we need to go further and add as an axiom schema the principle of indirect proof, also called stability (∀~x (¬¬R~x → R~x ) for relation symbols R), but then it is appropriate to restrict to the language based on →, ∀, ⊥ and ∧. The reason for this restric ...
The Mole
... over 9 miles. • If we were able to count atoms at the rate of 10 million per second, it would take about 2 billion years to count the atoms in one mole. ...
... over 9 miles. • If we were able to count atoms at the rate of 10 million per second, it would take about 2 billion years to count the atoms in one mole. ...
Introduction to Predicate Logic
... Proof of ∀x[¬P (x)] ⇔ ¬∃x[P (x)]. (i) Proof that ∀x[¬P (x)] entails ¬∃x[P (x)] – Assume that [[∀x[¬P (x)]]]M,g = 1 for any model M and any assignment g. – For all d ∈ U , [[¬P (x)]]M,g[d/x] = 1, by the semantics for ∀. – For all d ∈ U , [[P (x)]]M,g[d/x] = 0, by the semantics for ¬. – There is no d ...
... Proof of ∀x[¬P (x)] ⇔ ¬∃x[P (x)]. (i) Proof that ∀x[¬P (x)] entails ¬∃x[P (x)] – Assume that [[∀x[¬P (x)]]]M,g = 1 for any model M and any assignment g. – For all d ∈ U , [[¬P (x)]]M,g[d/x] = 1, by the semantics for ∀. – For all d ∈ U , [[P (x)]]M,g[d/x] = 0, by the semantics for ¬. – There is no d ...
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... is satisfiable, then it is satisfiable in a small finite model. The standard way of proving such results for modal logics is to "collapse" a (possibly infinite) model by identifying states according to an equivalence relation of small finite index, and then showing that the resulting finite quotient ...
... is satisfiable, then it is satisfiable in a small finite model. The standard way of proving such results for modal logics is to "collapse" a (possibly infinite) model by identifying states according to an equivalence relation of small finite index, and then showing that the resulting finite quotient ...
CHAPTER 9 Two Proofs of Completeness Theorem 1 Classical
... lies in a fact that they can be applied in an extended version to the proof of completeness for classical predicate logic and many non-classical propositional and predicate logics. The second proof is based on the fact that it provides a method of a construction of a counter-model for a formula A ba ...
... lies in a fact that they can be applied in an extended version to the proof of completeness for classical predicate logic and many non-classical propositional and predicate logics. The second proof is based on the fact that it provides a method of a construction of a counter-model for a formula A ba ...
Chapter 9 Propositional Logic Completeness Theorem
... lies in a fact that they can be applied in an extended version to the proof of completeness for classical predicate logic and many non-classical propositional and predicate logics. The second proof is based on the fact that it provides a method of a construction of a counter-model for a formula A ba ...
... lies in a fact that they can be applied in an extended version to the proof of completeness for classical predicate logic and many non-classical propositional and predicate logics. The second proof is based on the fact that it provides a method of a construction of a counter-model for a formula A ba ...