Knowledge Representation and Classical Logic
... clear that F is equivalent to G if and only if F ↔ G is a tautology. A set Γ of formulas is satisfiable if there exists an interpretation satisfying all formulas in Γ. We say that Γ entails a formula F (symbolically, Γ |= F ) if every interpretation satisfying Γ satisfies F .2 To represent knowledge ...
... clear that F is equivalent to G if and only if F ↔ G is a tautology. A set Γ of formulas is satisfiable if there exists an interpretation satisfying all formulas in Γ. We say that Γ entails a formula F (symbolically, Γ |= F ) if every interpretation satisfying Γ satisfies F .2 To represent knowledge ...
Knowledge Representation and Classical Logic
... clear that F is equivalent to G if and only if F ↔ G is a tautology. A set Γ of formulas is satisfiable if there exists an interpretation satisfying all formulas in Γ. We say that Γ entails a formula F (symbolically, Γ |= F ) if every interpretation satisfying Γ satisfies F .2 To represent knowledge ...
... clear that F is equivalent to G if and only if F ↔ G is a tautology. A set Γ of formulas is satisfiable if there exists an interpretation satisfying all formulas in Γ. We say that Γ entails a formula F (symbolically, Γ |= F ) if every interpretation satisfying Γ satisfies F .2 To represent knowledge ...
A Cut-Free Calculus for Second
... cut-free fragment of) HIF with these rules. Our main result is that HIF2 is sound and complete for secondorder Gödel logic. Since we do not include the cut rule in HIF2 , this automatically implies the admissibility of cut, which makes this calculus a suitable possible basis for automated theorem p ...
... cut-free fragment of) HIF with these rules. Our main result is that HIF2 is sound and complete for secondorder Gödel logic. Since we do not include the cut rule in HIF2 , this automatically implies the admissibility of cut, which makes this calculus a suitable possible basis for automated theorem p ...
Chapter 2
... Proof by example: not so much. Works only if you are able to consider every case that the conjecture applies to. This leads to the following definition: Exhaustive enumeration (proof by exhaustion) is a proof technique that can be applied to conjectures that are applied to a finite set of objects. ...
... Proof by example: not so much. Works only if you are able to consider every case that the conjecture applies to. This leads to the following definition: Exhaustive enumeration (proof by exhaustion) is a proof technique that can be applied to conjectures that are applied to a finite set of objects. ...
Tableau-based decision procedure for the full
... in [7] had been developed until quite recently, even for the systems with a relatively low known lower bounds. The only exception is [11], where a top-down tableau-style decision procedure for the logic ATEL, which subsumes the basic branching-time logic considered in [7] and this paper, was present ...
... in [7] had been developed until quite recently, even for the systems with a relatively low known lower bounds. The only exception is [11], where a top-down tableau-style decision procedure for the logic ATEL, which subsumes the basic branching-time logic considered in [7] and this paper, was present ...
The logic of negationless mathematics
... that a certain proposition can be proved as soon as A has been proved and also as soon as B has been proved. And then we might say that A v B implies C. But in the system itself we shall never progress from A (or from B ) to another proposition before A (or B) has been proved. So in the system itsel ...
... that a certain proposition can be proved as soon as A has been proved and also as soon as B has been proved. And then we might say that A v B implies C. But in the system itself we shall never progress from A (or from B ) to another proposition before A (or B) has been proved. So in the system itsel ...
Cylindric Modal Logic - Homepages of UvA/FNWI staff
... to it. The aim that we set ourselves here is the converse: to devise and study a modal formalism which is equally expressive as first-order logic itself. In fact, in this paper we will show how the above-mentioned gap vanishes if we implement the following idea: we can restrict the syntax of first-o ...
... to it. The aim that we set ourselves here is the converse: to devise and study a modal formalism which is equally expressive as first-order logic itself. In fact, in this paper we will show how the above-mentioned gap vanishes if we implement the following idea: we can restrict the syntax of first-o ...
An Introduction to Prolog Programming
... A Prolog program corresponds to a set of formulas, all of which are assumed to be true. This restricts the range of possible interpretations of the predicate and function symbols appearing in these formulas. The formulas in the translated program may be thought of as the premises in a proof. If Prol ...
... A Prolog program corresponds to a set of formulas, all of which are assumed to be true. This restricts the range of possible interpretations of the predicate and function symbols appearing in these formulas. The formulas in the translated program may be thought of as the premises in a proof. If Prol ...
Notes on First Order Logic
... Induction Step Suppose that ϕ is (∀y)ψ. Since τ is substitutable for x in ϕ we have two cases: 1. x does not occur free in ψ. Then ((∀y)ψ)[x/τ ] is the same as (∀y)ψ. Furthermore s and s[x/τ ] agree on all free variables in (∀y)ψ. By Theorem ??, we have A, s |= (∀y)ψ[x/τ ] iff A, s |= (∀y)ψ iff A, ...
... Induction Step Suppose that ϕ is (∀y)ψ. Since τ is substitutable for x in ϕ we have two cases: 1. x does not occur free in ψ. Then ((∀y)ψ)[x/τ ] is the same as (∀y)ψ. Furthermore s and s[x/τ ] agree on all free variables in (∀y)ψ. By Theorem ??, we have A, s |= (∀y)ψ[x/τ ] iff A, s |= (∀y)ψ iff A, ...
CSE 20 - Lecture 14: Logic and Proof Techniques
... faculty and at least one noble laureate.” There is an university in USA where every department has less than 20 faculty and at least one noble laureate. All universitis in USA where every department has at least 20 faculty and at least one noble laureate. For all universities in USA there is a depar ...
... faculty and at least one noble laureate.” There is an university in USA where every department has less than 20 faculty and at least one noble laureate. All universitis in USA where every department has at least 20 faculty and at least one noble laureate. For all universities in USA there is a depar ...
First-Order Proof Theory of Arithmetic
... great interest because some weak theories of arithmetic have very close connection of feasible computational classes. Because of Gödel’s second incompleteness theorem that the theory of numbers is not recursive, there is no good proof theory for the complete theory of numbers; therefore, proof-theo ...
... great interest because some weak theories of arithmetic have very close connection of feasible computational classes. Because of Gödel’s second incompleteness theorem that the theory of numbers is not recursive, there is no good proof theory for the complete theory of numbers; therefore, proof-theo ...
Applications of Automated Reasoning Nr. 9/2007 Arbeitsberichte
... so called Hypertableaux, which are introduced in [BFN96] and which is used in KRHyper, a theorem prover, which is the basis throughout our applications. The calculus is a clause normal form tableau calculus and hence we start constructing a tableau from a given set of clauses, which are regarded as ...
... so called Hypertableaux, which are introduced in [BFN96] and which is used in KRHyper, a theorem prover, which is the basis throughout our applications. The calculus is a clause normal form tableau calculus and hence we start constructing a tableau from a given set of clauses, which are regarded as ...
The Gödelian inferences - University of Notre Dame
... and distinguish from this theorem the two inferences that Gödel drew from it: first, that S does not prove its own consistency, and second, that no proof of the consistency of S can be formalized in S. These are the Gödelian inferences. I now turn to the problem of their justification. 3. Extensio ...
... and distinguish from this theorem the two inferences that Gödel drew from it: first, that S does not prove its own consistency, and second, that no proof of the consistency of S can be formalized in S. These are the Gödelian inferences. I now turn to the problem of their justification. 3. Extensio ...
Lecture - 04 (Logic Knowledge Base)
... Entailment and Proof • To clarify the difference between entailment and proof: • Entailment: if we have a set of formulae which are true, then as a logical consequence of this, some partic ...
... Entailment and Proof • To clarify the difference between entailment and proof: • Entailment: if we have a set of formulae which are true, then as a logical consequence of this, some partic ...
Contextual Reasoning - Homepages of UvA/FNWI staff
... This thesis is the result of a graduation project carried out at the Institute for Scientific and Technological Research (IRST) in Trento, Italy. Ever since its outset in the late 1980’s, fundamental AI research on contextual reasoning has always been pursued most notably by two prominent research g ...
... This thesis is the result of a graduation project carried out at the Institute for Scientific and Technological Research (IRST) in Trento, Italy. Ever since its outset in the late 1980’s, fundamental AI research on contextual reasoning has always been pursued most notably by two prominent research g ...
Intuitionistic Logic - Institute for Logic, Language and Computation
... is best seen as a particular manner of implementing the idea of constructivism in mathematics, a manner due to the Dutch mathematician Brouwer and his pupil Heyting. Constructivism is the point of view that mathematical objects exist only in so far they have been constructed and that proofs derive t ...
... is best seen as a particular manner of implementing the idea of constructivism in mathematics, a manner due to the Dutch mathematician Brouwer and his pupil Heyting. Constructivism is the point of view that mathematical objects exist only in so far they have been constructed and that proofs derive t ...
Logic and Computation Lecture notes Jeremy Avigad Assistant Professor, Philosophy
... The word “logic” derives from the Greek word “logos,” or reason, and logic can be broadly construed as the study of the principles of reasoning. Understood this way, logic is the subject of this course. I should qualify this remark, however. In everyday life, we use different modes of reasoning in d ...
... The word “logic” derives from the Greek word “logos,” or reason, and logic can be broadly construed as the study of the principles of reasoning. Understood this way, logic is the subject of this course. I should qualify this remark, however. In everyday life, we use different modes of reasoning in d ...
A Mathematical Introduction to Modal Logic
... We have two proof rules. The first one, modus ponens, is a familiar one: If ` ϕ and ` ϕ → ψ, then ` ψ. The second one is unique to modal models, and called the necessitation rule: If ` ϕ, then ` ϕ. A brief explanation for the necessitation rule is in order here. The assumption ` ϕ means that the fo ...
... We have two proof rules. The first one, modus ponens, is a familiar one: If ` ϕ and ` ϕ → ψ, then ` ψ. The second one is unique to modal models, and called the necessitation rule: If ` ϕ, then ` ϕ. A brief explanation for the necessitation rule is in order here. The assumption ` ϕ means that the fo ...
The Logic of Provability
... If φ is a true Σ sentence, then PA ` φ Proof Sketch. First, prove that if t is a closed term and t denotes i, then ` t = i using induction on the construction of t. Next, prove that if t, t0 are closed terms and t = t0 is true, then ` t = t0 by leveraging the previous result. Finally, run an inducti ...
... If φ is a true Σ sentence, then PA ` φ Proof Sketch. First, prove that if t is a closed term and t denotes i, then ` t = i using induction on the construction of t. Next, prove that if t, t0 are closed terms and t = t0 is true, then ` t = t0 by leveraging the previous result. Finally, run an inducti ...
Formal deduction in propositional logic
... syntactic one, that is, we give formal rules for deduction which are purely syntactical. • We want to define a relation called formal deducibility to allow us to mechanically check the validity of a proof. • The significance of the word formal will be explained later. The important point is that for ...
... syntactic one, that is, we give formal rules for deduction which are purely syntactical. • We want to define a relation called formal deducibility to allow us to mechanically check the validity of a proof. • The significance of the word formal will be explained later. The important point is that for ...
Towards NP−P via Proof Complexity and Search
... have been combinatorial, algebraic, or probabilistic; this includes, for instance switching lemmas, or polynomial approximations, and a host of results about randomization. In this paper, we survey instead various “logico-algorithmic” attempts to solve the P versus NP problem. This is motivated by t ...
... have been combinatorial, algebraic, or probabilistic; this includes, for instance switching lemmas, or polynomial approximations, and a host of results about randomization. In this paper, we survey instead various “logico-algorithmic” attempts to solve the P versus NP problem. This is motivated by t ...
cl-ch9
... denotations, but an interpretation must still specify a domain, and that specification makes a difference as to truth for closed formulas involving =. For instance, ∃x∃y ∼ x = y will be true if the domain has at least two distinct elements, but false if it has only one.) Closed formulas, which are a ...
... denotations, but an interpretation must still specify a domain, and that specification makes a difference as to truth for closed formulas involving =. For instance, ∃x∃y ∼ x = y will be true if the domain has at least two distinct elements, but false if it has only one.) Closed formulas, which are a ...
First-Order Extension of the FLP Stable Model
... The syntax of an aggregate expression considered in Ferraris and Lifschitz [2010] is more general. The results in this paper can be extended to the general syntax, which we omit for simplicity. ...
... The syntax of an aggregate expression considered in Ferraris and Lifschitz [2010] is more general. The results in this paper can be extended to the general syntax, which we omit for simplicity. ...