Reducing Propositional Theories in Equilibrium Logic to
... programs of a special kind. Answer set semantics was already generalised and extended to arbitrary propositional theories with two negations in the system of equilibrium logic, defined in [17] and further studied in [18,19,20]. Equilibrium logic is based on a simple, minimal model construction in th ...
... programs of a special kind. Answer set semantics was already generalised and extended to arbitrary propositional theories with two negations in the system of equilibrium logic, defined in [17] and further studied in [18,19,20]. Equilibrium logic is based on a simple, minimal model construction in th ...
Herbrand Theorem, Equality, and Compactness
... terms (i.e. terms with no variables) from the underlying language. Notice that a ground instance of a ∀-sentence A is a logical consequence of A. Therefore if a set Φ0 of ground instances of A is unsatisfiable, then A is unsatisfiable. Definition: An L-truth assignment (or just truth assignment) is ...
... terms (i.e. terms with no variables) from the underlying language. Notice that a ground instance of a ∀-sentence A is a logical consequence of A. Therefore if a set Φ0 of ground instances of A is unsatisfiable, then A is unsatisfiable. Definition: An L-truth assignment (or just truth assignment) is ...
Logic and Existential Commitment
... This second way of changing a sentence’s truth value leads to what might be termed the possible meaning (PM) account of logical consequence: the conclusion of an argument is a logical consequence of its premises iff there is no possible use or meaning of its constituent nonlogical elements under whi ...
... This second way of changing a sentence’s truth value leads to what might be termed the possible meaning (PM) account of logical consequence: the conclusion of an argument is a logical consequence of its premises iff there is no possible use or meaning of its constituent nonlogical elements under whi ...
Introduction to logic
... The necessary steps to the development of logic in its modern form were taken by George Boole (1854) and Gottlob Frege (1879). Boole revolutionized logic by applying methods from the thenemerging field of symbolic algebra to logic. Where traditional (Aristotelian) logic relied on cataloging the vali ...
... The necessary steps to the development of logic in its modern form were taken by George Boole (1854) and Gottlob Frege (1879). Boole revolutionized logic by applying methods from the thenemerging field of symbolic algebra to logic. Where traditional (Aristotelian) logic relied on cataloging the vali ...
To What Type of Logic Does the "Tetralemma" Belong?
... Perhaps, as is commonly suggested, Nagarjuna was simply trying to express a mystical rejection of analytical thought itself. However, it seems worth pointing out that anhomomorphic logic opens up another interpretation, perhaps consistent with the mystical one, but not really requiring it. Namely o ...
... Perhaps, as is commonly suggested, Nagarjuna was simply trying to express a mystical rejection of analytical thought itself. However, it seems worth pointing out that anhomomorphic logic opens up another interpretation, perhaps consistent with the mystical one, but not really requiring it. Namely o ...
Chapter 5 Predicate Logic
... formula: (∀x)H(x, x). Here there is still only one quantifier and no connectives, but there is more than one quantified variable. The interpretation is that both arguments must be the same. This expression is true if H can pair all elements of D with themselves. This is true in the just preceding ca ...
... formula: (∀x)H(x, x). Here there is still only one quantifier and no connectives, but there is more than one quantified variable. The interpretation is that both arguments must be the same. This expression is true if H can pair all elements of D with themselves. This is true in the just preceding ca ...
pdf
... There has been a great deal of work on characterizing the complexity of the satisfiability and validity problem for modal logics (see [7; 9; 14; 15] for some examples). In particular, Ladner [9] showed that the validity (and satisfiability) problem for every modal logic between K and S4 is PSPACE-ha ...
... There has been a great deal of work on characterizing the complexity of the satisfiability and validity problem for modal logics (see [7; 9; 14; 15] for some examples). In particular, Ladner [9] showed that the validity (and satisfiability) problem for every modal logic between K and S4 is PSPACE-ha ...
ON PRESERVING 1. Introduction The
... This is all very well, but we haven’t really gotten to anything that would single out an inference relation from among a throng of such all of which preserve consistency. In order to do that it will be necessary to talk about a logic X preserving the consistency predicate of a logic Y , in the stron ...
... This is all very well, but we haven’t really gotten to anything that would single out an inference relation from among a throng of such all of which preserve consistency. In order to do that it will be necessary to talk about a logic X preserving the consistency predicate of a logic Y , in the stron ...
Natural deduction for predicate logic
... This suggests that to prove a formula of the form ∀xφ, we can prove φ with some arbitrary but fresh variable x0 substituted for x. That is, we want to prove the formula φ[x0 /x]. On the previous slide, we used n as a fresh variable, but in our formal proofs, we adopt the convention of using subscri ...
... This suggests that to prove a formula of the form ∀xφ, we can prove φ with some arbitrary but fresh variable x0 substituted for x. That is, we want to prove the formula φ[x0 /x]. On the previous slide, we used n as a fresh variable, but in our formal proofs, we adopt the convention of using subscri ...
When is Metric Temporal Logic Expressively Complete?
... timing devices which are initially synchronized but have independent unit time length. In this paper we consider arbitrary subsets of R for K; however we observe that with simple arithmetic any integer linear combination of elements in K can be derived as a unary function. Thus we restrict our atten ...
... timing devices which are initially synchronized but have independent unit time length. In this paper we consider arbitrary subsets of R for K; however we observe that with simple arithmetic any integer linear combination of elements in K can be derived as a unary function. Thus we restrict our atten ...
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 ...
Version 1.5 - Trent University
... interpretation of statements, truth, and reasoning. This volume develops the basics of two kinds of formal logical systems, propositional logic and first-order logic. Propositional logic attempts to make precise the relationships that certain connectives like not, and , or , and if . . . then are us ...
... interpretation of statements, truth, and reasoning. This volume develops the basics of two kinds of formal logical systems, propositional logic and first-order logic. Propositional logic attempts to make precise the relationships that certain connectives like not, and , or , and if . . . then are us ...
(pdf)
... One important distinction to make is that fuzzy logic is NOT probability. Although both employ values between 0 and 1 that represent something about the symbol or event, it is the meaning of this number that differs. In probability, the number represents the likelihood of an event’s occurrence. In f ...
... One important distinction to make is that fuzzy logic is NOT probability. Although both employ values between 0 and 1 that represent something about the symbol or event, it is the meaning of this number that differs. In probability, the number represents the likelihood of an event’s occurrence. In f ...
characterization of classes of frames in modal language
... If a logic consists of K, φ → φ, φ → φ, grz, then it is characterized by the class of reflexive, transitive and antisymmetric Kripke frames which do not contain any infinite ascending chains of distinct points. S4 is valid in frames defined by grz. S4 laws in K ∪ grz were proved around 1979 by W. J ...
... If a logic consists of K, φ → φ, φ → φ, grz, then it is characterized by the class of reflexive, transitive and antisymmetric Kripke frames which do not contain any infinite ascending chains of distinct points. S4 is valid in frames defined by grz. S4 laws in K ∪ grz were proved around 1979 by W. J ...
1 The calculus of “predicates”
... adds to the language of the propositional calculus: names of individuals belonging to some domain or universe of discourse; variables standing for these names (ranging over the domain), predicate symbols, and quantifiers. In first-order logic there are also function symbols, but we concentrate for p ...
... adds to the language of the propositional calculus: names of individuals belonging to some domain or universe of discourse; variables standing for these names (ranging over the domain), predicate symbols, and quantifiers. In first-order logic there are also function symbols, but we concentrate for p ...
Guarded fragments with constants - Institute for Logic, Language
... than zero. A first-order formula φ of such a language is called guarded if it is built up from atomic formulas using the Boolean connectives and guarded quantifiers of the form ∃x1 . . . xn .(π∧ψ) or ∀x1 . . . xn .(π → ψ), where π is an atomic formula and the free variables of ψ all occur in π. A f ...
... than zero. A first-order formula φ of such a language is called guarded if it is built up from atomic formulas using the Boolean connectives and guarded quantifiers of the form ∃x1 . . . xn .(π∧ψ) or ∀x1 . . . xn .(π → ψ), where π is an atomic formula and the free variables of ψ all occur in π. A f ...
Extending modal logic
... Theorem: for frame classes K defined by universal Horn conditions x1...xn(φ1 ... φk → ψ) the following are equivalent: 1. K is modally definable 2. K is closed under bounded morphic images and disjoint unions 3. The Horn conditions can be written so that their left hand sides are tree-shaped. 4. K i ...
... Theorem: for frame classes K defined by universal Horn conditions x1...xn(φ1 ... φk → ψ) the following are equivalent: 1. K is modally definable 2. K is closed under bounded morphic images and disjoint unions 3. The Horn conditions can be written so that their left hand sides are tree-shaped. 4. K i ...
Formal logic
... But how and why can we conclude that this last sentence follows from the previous two premises? Or, more generally, how can we determine whether a formula ϕ is a valid consequence of a set of formulas {ϕ1 , . . . , ϕn }? Modern logic offers two possible ways, that used to be fused in the time of syl ...
... But how and why can we conclude that this last sentence follows from the previous two premises? Or, more generally, how can we determine whether a formula ϕ is a valid consequence of a set of formulas {ϕ1 , . . . , ϕn }? Modern logic offers two possible ways, that used to be fused in the time of syl ...
Analysis of the paraconsistency in some logics
... 1. We will say that a theory Γ is contradictory, with respect to ¬, if there exists a formula A such that Γ ` A y Γ ` ¬A; 2. We say that a theory Γ is trivial if ∀A : Γ ` A; 3. We say that a theory is explosive if, when adding to it any couple of contradictory formulas, the theory becomes trivial; 4 ...
... 1. We will say that a theory Γ is contradictory, with respect to ¬, if there exists a formula A such that Γ ` A y Γ ` ¬A; 2. We say that a theory Γ is trivial if ∀A : Γ ` A; 3. We say that a theory is explosive if, when adding to it any couple of contradictory formulas, the theory becomes trivial; 4 ...
Speaking Logic - SRI International
... meanings of other symbols, e.g., variables, functions, and predicates. An assertion is valid if it holds in all interpretations. Checking validity through interpretations is not always efficient and often, not even possible, so proofs in the form axioms and inference rules are used to demonstrate th ...
... meanings of other symbols, e.g., variables, functions, and predicates. An assertion is valid if it holds in all interpretations. Checking validity through interpretations is not always efficient and often, not even possible, so proofs in the form axioms and inference rules are used to demonstrate th ...
8 predicate logic
... invoke simplification to prove the validity of the argument (x)(Ax · Bx) / (x)Ax. But many of the rules of inference of propositional logic (such as simplification) may be applied only to whole lines in a proof. Thus, we need rules for dropping initial quantifiers from quantified propositions. If we ...
... invoke simplification to prove the validity of the argument (x)(Ax · Bx) / (x)Ax. But many of the rules of inference of propositional logic (such as simplification) may be applied only to whole lines in a proof. Thus, we need rules for dropping initial quantifiers from quantified propositions. If we ...
AI Principles, Semester 2, Week 2, Lecture 5 Propositional Logic
... A PROPOSITION LETTER is any symbol from following list: A, ...Z, A0...Z0, A1...Z1... The PROPOSITIONAL CONNECTIVES are ¬, ∨, ∧, →, ↔ An EXPRESSION of propositional logic is any sequence of sentence letters, propositional connectives, or left and right parentheses. METAVARIABLES such as Φ and Ψ are n ...
... A PROPOSITION LETTER is any symbol from following list: A, ...Z, A0...Z0, A1...Z1... The PROPOSITIONAL CONNECTIVES are ¬, ∨, ∧, →, ↔ An EXPRESSION of propositional logic is any sequence of sentence letters, propositional connectives, or left and right parentheses. METAVARIABLES such as Φ and Ψ are n ...
Bilattices In Logic Programming
... The domain of the intended model for L(B) will be the Herbrand universe (of closed terms), familiar from Prolog semantics (see [17]). We greatly broaden the notion of formula, however. To this end we assume we have formal symbols, ∧, ∨, ⊗, ⊕ and ¬, corresponding to the various operations on the bila ...
... The domain of the intended model for L(B) will be the Herbrand universe (of closed terms), familiar from Prolog semantics (see [17]). We greatly broaden the notion of formula, however. To this end we assume we have formal symbols, ∧, ∨, ⊗, ⊕ and ¬, corresponding to the various operations on the bila ...
G - Courses
... can be extended to arbitrary FO-sentences by forming structures that are obtained from Herbrand structures via taking the equivalence classes of terms according to the equalities between them in some structure satisfying the FO-sentence at hand. Here, we used the resolution procedure only for form ...
... can be extended to arbitrary FO-sentences by forming structures that are obtained from Herbrand structures via taking the equivalence classes of terms according to the equalities between them in some structure satisfying the FO-sentence at hand. Here, we used the resolution procedure only for form ...
Logic: Introduction - Department of information engineering and
... • Design Validation and verification: to verify the correctness of a design with a certainty beyond that of conventional testing. It uses temporal logic . • AI: mechanized reasoning and expert systems. • Security: With increasing use of network, security has become a big issue. Hence, the concept o ...
... • Design Validation and verification: to verify the correctness of a design with a certainty beyond that of conventional testing. It uses temporal logic . • AI: mechanized reasoning and expert systems. • Security: With increasing use of network, security has become a big issue. Hence, the concept o ...