Formal systems of fuzzy logic and their fragments∗
... The logic BCK plus this axiom of prelinearity will be the starting point for us in this paper—we call this logic Fuzzy BCK logic (FBCK for short). This logic is obviously complete with respect to the BCK-chains and this is the rationale for the name “Fuzzy BCK”, as the authors believe that completen ...
... The logic BCK plus this axiom of prelinearity will be the starting point for us in this paper—we call this logic Fuzzy BCK logic (FBCK for short). This logic is obviously complete with respect to the BCK-chains and this is the rationale for the name “Fuzzy BCK”, as the authors believe that completen ...
Reasoning about Programs by exploiting the environment
... would then be incomplete for this new environment. Weakening the assumptions could add feasible behaviors; the logic for the original environment would then become unsound. For example, any of the programming logics for shared-memory concurrency (e.g. [0G76]) could be used to prove that program of F ...
... would then be incomplete for this new environment. Weakening the assumptions could add feasible behaviors; the logic for the original environment would then become unsound. For example, any of the programming logics for shared-memory concurrency (e.g. [0G76]) could be used to prove that program of F ...
Strong Completeness for Iteration
... Modal logics are a much used formalism in automated verification thanks to the good balance between their expressive power and their computational properties. Recently, it has been shown that modal logics can be developed in the general framework of coalgebra [4,18], and that the expressiveness and ...
... Modal logics are a much used formalism in automated verification thanks to the good balance between their expressive power and their computational properties. Recently, it has been shown that modal logics can be developed in the general framework of coalgebra [4,18], and that the expressiveness and ...
Natural deduction
... Conditional proof and validity • At this point you might wonder. . . “yeah, I could see how the other rules were valid from the truth-tables, but this one is pretty weird! what’s the deal?” – in other words, you may not be persuaded that conditional proof preserves validity • So here is a little arg ...
... Conditional proof and validity • At this point you might wonder. . . “yeah, I could see how the other rules were valid from the truth-tables, but this one is pretty weird! what’s the deal?” – in other words, you may not be persuaded that conditional proof preserves validity • So here is a little arg ...
Propositional Logic and Methods of Inference
... The basic idea of normal form is to express wffs in a standard form that uses only the ^, v, and possibly ~ The resolution method is then applied to normal form wffs in which all other connectives and quantifiers have been eliminated Resolution is an operation on pairs of disjuncts, which produces n ...
... The basic idea of normal form is to express wffs in a standard form that uses only the ^, v, and possibly ~ The resolution method is then applied to normal form wffs in which all other connectives and quantifiers have been eliminated Resolution is an operation on pairs of disjuncts, which produces n ...
On the Finite Model Property in Order-Sorted Logic
... model-finder that deduces sort information for unsorted problems and, under certain conditions, can bound the size of domains for certain sorts and improve the performance of the instantiation procedure. Order-sorting is not used, and there are restrictions on the use of equality. Momtahan [23] defi ...
... model-finder that deduces sort information for unsorted problems and, under certain conditions, can bound the size of domains for certain sorts and improve the performance of the instantiation procedure. Order-sorting is not used, and there are restrictions on the use of equality. Momtahan [23] defi ...
First-Order Logic, Second-Order Logic, and Completeness
... case the quantifier binds a binary relation variable etc. As is well-known, standard semantics is not the only semantics available. Henkin semantics, for example, specifies a second domain of predicates and relations for the upper case constants and variables. The second-order quantifiers binding predi ...
... case the quantifier binds a binary relation variable etc. As is well-known, standard semantics is not the only semantics available. Henkin semantics, for example, specifies a second domain of predicates and relations for the upper case constants and variables. The second-order quantifiers binding predi ...
Diagrammatic Reasoning in Separation Logic
... Once the appropriate definitions are in place, the ATP would work in the following stages: 1. User provides a few example proofs for a specific instance of a theorem. 2. Program generalises from the examples, using a heuristic which suggests a schematic proof. 3. Program verifies the schematic proof ...
... Once the appropriate definitions are in place, the ATP would work in the following stages: 1. User provides a few example proofs for a specific instance of a theorem. 2. Program generalises from the examples, using a heuristic which suggests a schematic proof. 3. Program verifies the schematic proof ...
Truth in the limit
... Kronecker postulates that natural numbers are based on counting procedure. So in every moment only finitely many of them are generated. Of course mathematics deals with what can happen further. Hilbert – evidently influenced by Kronecker – recalled the Aristotelian notions of actual and potential in ...
... Kronecker postulates that natural numbers are based on counting procedure. So in every moment only finitely many of them are generated. Of course mathematics deals with what can happen further. Hilbert – evidently influenced by Kronecker – recalled the Aristotelian notions of actual and potential in ...
Introduction to Logic
... • Formal logic replaces the ordinary language of argument with a symbolic language. • This language is meant to be free of all ambiguity and vagueness. • The language is meant to wear its logical structure on its face. • Our formal languages: SL and QL. ...
... • Formal logic replaces the ordinary language of argument with a symbolic language. • This language is meant to be free of all ambiguity and vagueness. • The language is meant to wear its logical structure on its face. • Our formal languages: SL and QL. ...
Interpolation for McCain
... a question can be regarded as denoting its set of possible answers (out of which an appropriate answer selects one). For example, in Harrah’s system our @P would be called the “assertive core” of the question, whereas his indicated replies would, in our system, be combinations of rule bodies φ, . . ...
... a question can be regarded as denoting its set of possible answers (out of which an appropriate answer selects one). For example, in Harrah’s system our @P would be called the “assertive core” of the question, whereas his indicated replies would, in our system, be combinations of rule bodies φ, . . ...
Logic 3
... • Thus, if we can use different propositions and logical equivalences to show two statements are tautologies, we can do proofs. • Proofs are conditional and biconditional statements that are tautologies • Notation: p and q are atomic statements, while A and B are statements of all types, including a ...
... • Thus, if we can use different propositions and logical equivalences to show two statements are tautologies, we can do proofs. • Proofs are conditional and biconditional statements that are tautologies • Notation: p and q are atomic statements, while A and B are statements of all types, including a ...
Lecture 2
... A Boolean variable that can denote a proposition is sometimes called a propositional variable. The operator greater than (<) is a predicate (covered in Later). Predicates are not directly representable as propositions. Greater than can be regarded as a function of two numerical arguments and a Boole ...
... A Boolean variable that can denote a proposition is sometimes called a propositional variable. The operator greater than (<) is a predicate (covered in Later). Predicates are not directly representable as propositions. Greater than can be regarded as a function of two numerical arguments and a Boole ...
9. “… if and only if …”
... In English, it appears that there are several phrases that usually have the same meaning as the biconditional. Each of the following sentences would be translated as “(P↔Q)”. P if and only if Q. P just in case Q. P is necessary and sufficient for Q. P is equivalent to Q. ...
... In English, it appears that there are several phrases that usually have the same meaning as the biconditional. Each of the following sentences would be translated as “(P↔Q)”. P if and only if Q. P just in case Q. P is necessary and sufficient for Q. P is equivalent to Q. ...
Introduction to proposition
... Solution: There are many ways to translate this sentence into a logical expression. Although it is possible to represent the sentence by a single propositional variable, such as p, this would not be useful when analyzing its meaning or reasoning with it. Instead, we will use propositional variables ...
... Solution: There are many ways to translate this sentence into a logical expression. Although it is possible to represent the sentence by a single propositional variable, such as p, this would not be useful when analyzing its meaning or reasoning with it. Instead, we will use propositional variables ...
Formal Logic, Models, Reality
... world is a consequence of an ontic interpretation of the indeterminacy relations. If the logical law (5-4) is applied to premises which are true in a local quantum reality, this can lead to false conclusions like for instance Bell's inequality. Therefore classical formal logic is not sound when it i ...
... world is a consequence of an ontic interpretation of the indeterminacy relations. If the logical law (5-4) is applied to premises which are true in a local quantum reality, this can lead to false conclusions like for instance Bell's inequality. Therefore classical formal logic is not sound when it i ...
Logic, Sets, and Proofs
... In the discussion that follows, this fixed set will be denoted U . A variable such as x represents some unspecified element from the fixed set U . Example: If Z is the fixed set, then “x is even” is a statement that involves the variable x, and “x > y” involves x and y. When a logical statement cont ...
... In the discussion that follows, this fixed set will be denoted U . A variable such as x represents some unspecified element from the fixed set U . Example: If Z is the fixed set, then “x is even” is a statement that involves the variable x, and “x > y” involves x and y. When a logical statement cont ...
The Foundations: Logic and Proofs
... Show that for all real numbers a, b, c (a @b) @ c = a @ (b @ c) (This means the operation @ is associative.) Proof: Let a, b, and c be arbitrary real numbers. Then one of the following 6 cases must hold. 1. a ≥ b ≥ c 2. a ≥ c ≥ b 3. b ≥ a ≥c 4. b ≥ c ≥a 5. c ≥ a ≥ b 6. c ≥ b ≥ a ...
... Show that for all real numbers a, b, c (a @b) @ c = a @ (b @ c) (This means the operation @ is associative.) Proof: Let a, b, and c be arbitrary real numbers. Then one of the following 6 cases must hold. 1. a ≥ b ≥ c 2. a ≥ c ≥ b 3. b ≥ a ≥c 4. b ≥ c ≥a 5. c ≥ a ≥ b 6. c ≥ b ≥ a ...
From Answer Set Logic Programming to Circumscription via Logic of
... Answer Set Programming (ASP) is a new paradigm of constraint-based programming based on logic programming with answer set semantics 17,9,13]. It started out with normal logic programs, which are programs that can have negation but not disjunction. Driven by the need of applications, various extensi ...
... Answer Set Programming (ASP) is a new paradigm of constraint-based programming based on logic programming with answer set semantics 17,9,13]. It started out with normal logic programs, which are programs that can have negation but not disjunction. Driven by the need of applications, various extensi ...
Completeness of Propositional Logic Truth Assignments and Truth
... Completeness of Propositional Logic Truth Assignments and Truth Tables Let us define a truth assignment for a first-order language to be any function h from the set of all atomic sentences of that language into the set {TRUE, FALSE}. That is, for each atomic sentence A of the language, h gives us a ...
... Completeness of Propositional Logic Truth Assignments and Truth Tables Let us define a truth assignment for a first-order language to be any function h from the set of all atomic sentences of that language into the set {TRUE, FALSE}. That is, for each atomic sentence A of the language, h gives us a ...
Introduction to Proofs, Rules of Equivalence, Rules of
... true claims, and reason according to legitimate rules, you MUST end up with true claims. • A proof demonstrates validity by giving a set of instructions on how to get from the premises to the conclusion. ...
... true claims, and reason according to legitimate rules, you MUST end up with true claims. • A proof demonstrates validity by giving a set of instructions on how to get from the premises to the conclusion. ...
Boolean Connectives and Formal Proofs - FB3
... Substitution of equivalents: If P and Q are logically equivalent: P ⇔ Q then the results of substituting one for the other in the context of a larger sentence are also logically equivalent: S(P) ⇔ S(Q) A sentence is in negation normal form (NNF) if all occurrences of ¬ apply directly to atomic sente ...
... Substitution of equivalents: If P and Q are logically equivalent: P ⇔ Q then the results of substituting one for the other in the context of a larger sentence are also logically equivalent: S(P) ⇔ S(Q) A sentence is in negation normal form (NNF) if all occurrences of ¬ apply directly to atomic sente ...