Henkin`s Method and the Completeness Theorem
... sentence ϕ in the alphabet A, we will use the standard notation “` ϕ” for ϕ is provable in L (that is, ϕ is derivable from the axioms of L by the use of the inference rules of L); and “|= ϕ” for ϕ is valid (that is, ϕ is satisfied in every interpretation of L). The soundness theorem for L states tha ...
... sentence ϕ in the alphabet A, we will use the standard notation “` ϕ” for ϕ is provable in L (that is, ϕ is derivable from the axioms of L by the use of the inference rules of L); and “|= ϕ” for ϕ is valid (that is, ϕ is satisfied in every interpretation of L). The soundness theorem for L states tha ...
Completeness theorems and lambda
... only to form ΠX.T (X) if T (X) is built only with X and →?? Can one use the techniques of proof theory and give a predicative normalisation proof for this fragment? I. Takeuti gave such a proof in 1993, following G. Takeuti, and provided an ordinal analysis of normalisation, with ordinals < 0 In TL ...
... only to form ΠX.T (X) if T (X) is built only with X and →?? Can one use the techniques of proof theory and give a predicative normalisation proof for this fragment? I. Takeuti gave such a proof in 1993, following G. Takeuti, and provided an ordinal analysis of normalisation, with ordinals < 0 In TL ...
A Generic Modular Data Structure for Proof - Claus
... step inside another tree, we do not overemphasize reduction by this: – For purely generative abstract theory expansion we may assume some trivial reductions, which can later be refined to the reductions that will be necessarily involved in this generation on the concrete level of a logical calculus. ...
... step inside another tree, we do not overemphasize reduction by this: – For purely generative abstract theory expansion we may assume some trivial reductions, which can later be refined to the reductions that will be necessarily involved in this generation on the concrete level of a logical calculus. ...
Duplication of directed graphs and exponential blow up of
... these patterns appear explicitly and examples where they appear in more subtle ways. How the symmetry of a statement is related to the internal symmetry of its proofs? Given a proof, how can we reduce it to a smaller proof by exploiting the symmetricity of its subparts? These are fundamental questio ...
... these patterns appear explicitly and examples where they appear in more subtle ways. How the symmetry of a statement is related to the internal symmetry of its proofs? Given a proof, how can we reduce it to a smaller proof by exploiting the symmetricity of its subparts? These are fundamental questio ...
Programming in Logic Without Logic Programming
... 0 of the state Si to which the fluent belongs. The unstamped fluent atom p(t1, …, tn) is the same atom without this timestamp. Event predicates represent events contributing to the transition from one state to the next. The last argument of a timestamped event atom e(t1, …, tn, i) is a time paramete ...
... 0 of the state Si to which the fluent belongs. The unstamped fluent atom p(t1, …, tn) is the same atom without this timestamp. Event predicates represent events contributing to the transition from one state to the next. The last argument of a timestamped event atom e(t1, …, tn, i) is a time paramete ...
Refinement Modal Logic
... model restrictions were not sufficient to simulate informative events, and they introduced refinement trees for this purpose — a precursor of the dynamic epistemic logics developed later (for an overview, see [57]). This usage of refinement as a more general operation than model restriction is simil ...
... model restrictions were not sufficient to simulate informative events, and they introduced refinement trees for this purpose — a precursor of the dynamic epistemic logics developed later (for an overview, see [57]). This usage of refinement as a more general operation than model restriction is simil ...
slides - UCLA Computer Science
... 1987 Circa: XCON, the Digital Equipment Company expert system had reached about 10,000 rules, and was increasingly expensive to maintain. Reasons for these limits include: ...
... 1987 Circa: XCON, the Digital Equipment Company expert system had reached about 10,000 rules, and was increasingly expensive to maintain. Reasons for these limits include: ...
Default Logic (Reiter) - Department of Computing
... • α follows from (D, S W ) by ‘brave’/‘credulous’ reasoning when α in any extension of (D, W ): α ∈ ext(D, W ); • α follows from (D, T W ) by ‘cautious’/‘sceptical’ reasoning when α in all extensions of (D, W ): α ∈ ext(D, W ). ...
... • α follows from (D, S W ) by ‘brave’/‘credulous’ reasoning when α in any extension of (D, W ): α ∈ ext(D, W ); • α follows from (D, T W ) by ‘cautious’/‘sceptical’ reasoning when α in all extensions of (D, W ): α ∈ ext(D, W ). ...
Intuitionistic completeness part I
... formula A ⇒ B, then we add to the context of the evidence structure the assumption that x : A and continue by analyzing b(x) after normalizing it by symbolic computation. This computation reveals the operations that must be performed on the context to expose more of the evidence term b(x). For examp ...
... formula A ⇒ B, then we add to the context of the evidence structure the assumption that x : A and continue by analyzing b(x) after normalizing it by symbolic computation. This computation reveals the operations that must be performed on the context to expose more of the evidence term b(x). For examp ...
Language, Proof and Logic
... object in the domain, so our conclusion, ∀x Small(x), follows by universal generalization. Any proof using general conditional proof can be converted into a proof using universal generalization, together with the method of conditional proof. Suppose we have managed to prove ∀x [P(x) → Q(x)] using ge ...
... object in the domain, so our conclusion, ∀x Small(x), follows by universal generalization. Any proof using general conditional proof can be converted into a proof using universal generalization, together with the method of conditional proof. Suppose we have managed to prove ∀x [P(x) → Q(x)] using ge ...
ppt - UBC Computer Science
... Without loss of generality, let x be any element of D (or an equivalent expression like those shown on next page) Verify that the predicate P holds for this x. o Note: the only assumption we can make about x is the fact that it belongs to D. So we can only use properties common to all elements o ...
... Without loss of generality, let x be any element of D (or an equivalent expression like those shown on next page) Verify that the predicate P holds for this x. o Note: the only assumption we can make about x is the fact that it belongs to D. So we can only use properties common to all elements o ...
Specifying and Verifying Fault-Tolerant Systems
... SpecParams. The module is the basic unit of a TLA+ specification. It is a collection of declarations, definitions, assumptions, and theorems. The import statement imports the contents of the modules FiniteSets and Reals. This statement has almost the same effect as inserting the text of these modules i ...
... SpecParams. The module is the basic unit of a TLA+ specification. It is a collection of declarations, definitions, assumptions, and theorems. The import statement imports the contents of the modules FiniteSets and Reals. This statement has almost the same effect as inserting the text of these modules i ...
relevance logic - Consequently.org
... We should add a word about the delimitation of our topic. There are by now a host of formal systems that can be said with some justification to be ‘relevance logics’. Some of these antedate the Anderson–Belnap approach, some are more recent. Some have been studied somewhat extensively, whereas other ...
... We should add a word about the delimitation of our topic. There are by now a host of formal systems that can be said with some justification to be ‘relevance logics’. Some of these antedate the Anderson–Belnap approach, some are more recent. Some have been studied somewhat extensively, whereas other ...
A Computationally-Discovered Simplification of the Ontological
... us that if there is a unique object in the domain satisfying φ, then the definite description, ıxφ, is well-defined (i.e., has a denotation). A typical instance of Lemma 1 might be y = ıxF x → F y, which asserts that if the object y is identical to the x that is F , then y is F . Lemma 1 can then be ...
... us that if there is a unique object in the domain satisfying φ, then the definite description, ıxφ, is well-defined (i.e., has a denotation). A typical instance of Lemma 1 might be y = ıxF x → F y, which asserts that if the object y is identical to the x that is F , then y is F . Lemma 1 can then be ...
A Computationally-Discovered Simplification of the Ontological
... us that if there is a unique object in the domain satisfying φ, then the definite description, ıxφ, is well-defined (i.e., has a denotation). A typical instance of Lemma 1 might be y = ıxF x → F y, which asserts that if the object y is identical to the x that is F , then y is F . Lemma 1 can then be ...
... us that if there is a unique object in the domain satisfying φ, then the definite description, ıxφ, is well-defined (i.e., has a denotation). A typical instance of Lemma 1 might be y = ıxF x → F y, which asserts that if the object y is identical to the x that is F , then y is F . Lemma 1 can then be ...
Lecture Slides
... Assume x is any rational number, y is any irrational number and that x+y is a rational number. Then x+y = a / b for some aZ and some bZ+ Since x is rational, x = c /d for some cZ and some dZ+ Then (c /d ) + y = a / b and y = (a / b) - (c /d ) = (ab – bc) / bd Since ab – bc and bd are ...
... Assume x is any rational number, y is any irrational number and that x+y is a rational number. Then x+y = a / b for some aZ and some bZ+ Since x is rational, x = c /d for some cZ and some dZ+ Then (c /d ) + y = a / b and y = (a / b) - (c /d ) = (ab – bc) / bd Since ab – bc and bd are ...
A Logical Expression of Reasoning
... in law, economics, medicine and so on, is a complex, plural, multifaceted matter. It is in such a broad spectrum that it is considered in the present paper. Once reasoning is considered in those general terms its relation to logic becomes much more problematic and much less strict and clear. Questio ...
... in law, economics, medicine and so on, is a complex, plural, multifaceted matter. It is in such a broad spectrum that it is considered in the present paper. Once reasoning is considered in those general terms its relation to logic becomes much more problematic and much less strict and clear. Questio ...
Consequence relations and admissible rules
... and considered as one. For example, substitutions in propositional logic are often considered to be maps on formulas commuting with the connectives. In this paper we will do so too where possible. So when talking about propositional or predicate logic, Lm = Lo and the atoms, respectively the atomic ...
... and considered as one. For example, substitutions in propositional logic are often considered to be maps on formulas commuting with the connectives. In this paper we will do so too where possible. So when talking about propositional or predicate logic, Lm = Lo and the atoms, respectively the atomic ...
... Another type of inference that has been considered is what we shall call truth inference (with respect to Y ) ,and has usually been called logical implication in the literature. We write o FI cp if, for all structures S E Y and all substitutions 7, if S z[o] then S z[q]. An axiom can be viewed as a ...
The Foundations
... Propositional(Boolean; logical) operators operate on logical (instead of numerical) operands. Transparency No. 1-10 ...
... Propositional(Boolean; logical) operators operate on logical (instead of numerical) operands. Transparency No. 1-10 ...
The Development of Mathematical Logic from Russell to Tarski
... strive for. Let us consider first Pieri’s description of his work on the axiomatization of geometry, which had been carried out independently of Hilbert’s famous Foundations of Geometry ( 1899). In his presentation to the International Congress of Philosophy in 1900, Pieri emphasized that the study ...
... strive for. Let us consider first Pieri’s description of his work on the axiomatization of geometry, which had been carried out independently of Hilbert’s famous Foundations of Geometry ( 1899). In his presentation to the International Congress of Philosophy in 1900, Pieri emphasized that the study ...
Proof Theory for Propositional Logic
... particular the fact that a conditional is counted as true whenever the antecedent (the first term, above) is false. Again, let’s just get comfortable doing the proofs for now. When we do truth tables we will discuss why this is the case for propositional logic. In both cases, the problem reveals f ...
... particular the fact that a conditional is counted as true whenever the antecedent (the first term, above) is false. Again, let’s just get comfortable doing the proofs for now. When we do truth tables we will discuss why this is the case for propositional logic. In both cases, the problem reveals f ...
Non-Classical Logic
... ¬ ¬ ¬ p2 ”, etc. (Some other books may reverse the convendecision procedures for determining logical validity; each tions for upper vs. lowercase for these purposes.) row represents a different possible interpretation. ...
... ¬ ¬ ¬ p2 ”, etc. (Some other books may reverse the convendecision procedures for determining logical validity; each tions for upper vs. lowercase for these purposes.) row represents a different possible interpretation. ...
Sound and Complete Inference Rules in FOL Example
... For fill-in-the-blank questions, we can have: • Termination with a clause which is a single answer literal Ans(c1 , . . . , cn ). In this case, the constants c1 , . . . , cn gives us an answer to the query. There might be more answers depending on whether there are more resolution refutations of Ans ...
... For fill-in-the-blank questions, we can have: • Termination with a clause which is a single answer literal Ans(c1 , . . . , cn ). In this case, the constants c1 , . . . , cn gives us an answer to the query. There might be more answers depending on whether there are more resolution refutations of Ans ...
MODAL LANGUAGES AND BOUNDED FRAGMENTS OF
... Ehrenfeucht Game with restricted choices of objects in each move – which has a natural generalization to the case with whole families of n-ary accessibility relations.) In the above theorem, the first-order formula may contain any other relation symbols, or equality, too. A formula φ with one free v ...
... Ehrenfeucht Game with restricted choices of objects in each move – which has a natural generalization to the case with whole families of n-ary accessibility relations.) In the above theorem, the first-order formula may contain any other relation symbols, or equality, too. A formula φ with one free v ...