1 LOGICAL CONSEQUENCE: A TURN IN STYLE KOSTA DO SEN
... successor’, ‘2 has a successor’, etc. for each natural number; as a consequence of we have the sentence ‘Every natural number has a successor’. On a rather abstract level of logic, one may envisage a deduction corresponding to the consequence relation in this example (the rule justifying this ded ...
... successor’, ‘2 has a successor’, etc. for each natural number; as a consequence of we have the sentence ‘Every natural number has a successor’. On a rather abstract level of logic, one may envisage a deduction corresponding to the consequence relation in this example (the rule justifying this ded ...
A Resolution-Based Proof Method for Temporal Logics of
... This paper presents two logics, called KLn and BLn respectively, and gives resolutionbased proof methods for both. The logic KLn is a temporal logic of knowledge. That is, in addition to the usual connectives of linear discrete temporal logic [4], KLn contains an indexed set of unary modal connectiv ...
... This paper presents two logics, called KLn and BLn respectively, and gives resolutionbased proof methods for both. The logic KLn is a temporal logic of knowledge. That is, in addition to the usual connectives of linear discrete temporal logic [4], KLn contains an indexed set of unary modal connectiv ...
Dynamic logic of propositional assignments
... interpreting abstract actions. Models are in fact made up of a single state, i.e., a valuation of classical propositional logic. Propositional assignments then update these valuations in the obvious way. Being able to do PDL-style reasoning whilst relying on such ‘degenerate’, succinctly specified m ...
... interpreting abstract actions. Models are in fact made up of a single state, i.e., a valuation of classical propositional logic. Propositional assignments then update these valuations in the obvious way. Being able to do PDL-style reasoning whilst relying on such ‘degenerate’, succinctly specified m ...
On Decidability of Intuitionistic Modal Logics
... GF containing formulas ϕ such that (i) ϕ has no more than two variables (free or bound), and (ii) all non-unary predicate letters of ϕ occur in guards. ...
... GF containing formulas ϕ such that (i) ϕ has no more than two variables (free or bound), and (ii) all non-unary predicate letters of ϕ occur in guards. ...
propositional logic extended with a pedagogically useful relevant
... not render the logic useless with respect to formalizing sentences from natural language. It hardly is any hindrance at all as I show below. So the restriction makes the logic suitable for an introductory logic course. It is hardly an exaggeration to claim that nested implications do not occur in na ...
... not render the logic useless with respect to formalizing sentences from natural language. It hardly is any hindrance at all as I show below. So the restriction makes the logic suitable for an introductory logic course. It is hardly an exaggeration to claim that nested implications do not occur in na ...
Propositional inquisitive logic: a survey
... Proposition 3. IPL ⊆ InqB ⊆ CPL Thus, from this perspective InqB is a logic stronger than intuitionistic logic, but weaker than classical logic. It is not, however, an intermediate logic in the usual sense of the term. This is because InqB is not closed under the rule of uniform substitution: in par ...
... Proposition 3. IPL ⊆ InqB ⊆ CPL Thus, from this perspective InqB is a logic stronger than intuitionistic logic, but weaker than classical logic. It is not, however, an intermediate logic in the usual sense of the term. This is because InqB is not closed under the rule of uniform substitution: in par ...
Kripke Semantics for Basic Sequent Systems
... {π0 }. In LK-legal frames, R consists of one relation Rπ0 which is the identity relation. π0 imposes a trivial condition, v(a, ψ) = v(b, ψ) whenever a = b. The basic rules of LK impose the usual truth-tables in each world, e.g. v(a, ψ ⊃ ϕ) = t iff either v(a, ψ) = f or v(a, ϕ) = t. Example 6. Assume ...
... {π0 }. In LK-legal frames, R consists of one relation Rπ0 which is the identity relation. π0 imposes a trivial condition, v(a, ψ) = v(b, ψ) whenever a = b. The basic rules of LK impose the usual truth-tables in each world, e.g. v(a, ψ ⊃ ϕ) = t iff either v(a, ψ) = f or v(a, ϕ) = t. Example 6. Assume ...
Query Answering for OWL-DL with Rules
... sort c. We translate each atomic concept into a unary predicate of sort a, each n-ary concrete domain predicate into a predicate with arguments of sort c, and each abstract (concrete) role into a binary predicate of sort a × a (a × c). The translation operator π is presented in Table 1. For rules, w ...
... sort c. We translate each atomic concept into a unary predicate of sort a, each n-ary concrete domain predicate into a predicate with arguments of sort c, and each abstract (concrete) role into a binary predicate of sort a × a (a × c). The translation operator π is presented in Table 1. For rules, w ...
ND for predicate logic ∀-elimination, first attempt Variable capture
... The introduction rules for the logical connectives are called “logical rules”. Besides those and the axiom rule, there is another essential set of rules: the structural rules. ...
... The introduction rules for the logical connectives are called “logical rules”. Besides those and the axiom rule, there is another essential set of rules: the structural rules. ...
pdf file
... E1 = Th(C ∧ A, B → F, C → B} and E2 = Th(C ∧ A, B → F, A → ¬F} E1 is the extension vindicated by common sense while E2 is an anomalous extension. The principle that states that the derivation of an exception has priority over the derivation of the default to which it is an exception, the situation o ...
... E1 = Th(C ∧ A, B → F, C → B} and E2 = Th(C ∧ A, B → F, A → ¬F} E1 is the extension vindicated by common sense while E2 is an anomalous extension. The principle that states that the derivation of an exception has priority over the derivation of the default to which it is an exception, the situation o ...
Complete Sequent Calculi for Induction and Infinite Descent
... Any LKID proof can be transformed into a CLKIDω proof. (Proof: We show how to derive any induction rule in CLKIDω .) ...
... Any LKID proof can be transformed into a CLKIDω proof. (Proof: We show how to derive any induction rule in CLKIDω .) ...
Strong Logics of First and Second Order
... 2 we shall start by investigating two traditional strong logics (ω-logic and β-logic) that share many of these features of absoluteness, only now absoluteness is secured relative to ZFC. These logics will serve as our guide in setting up stronger logics that are absolute relative to stronger backgro ...
... 2 we shall start by investigating two traditional strong logics (ω-logic and β-logic) that share many of these features of absoluteness, only now absoluteness is secured relative to ZFC. These logics will serve as our guide in setting up stronger logics that are absolute relative to stronger backgro ...
485-291 - Wseas.us
... The first order theory of can be axiomatized by a very concrete set of axioms, such a is explicitly given in [2], Example 3.2.11. Here we don't recall the concrete form of these axioms because we need only some properties of them. Namely, forms a complete theory and, in addition, the asympto ...
... The first order theory of can be axiomatized by a very concrete set of axioms, such a is explicitly given in [2], Example 3.2.11. Here we don't recall the concrete form of these axioms because we need only some properties of them. Namely, forms a complete theory and, in addition, the asympto ...
Clausal Connection-Based Theorem Proving in
... Automated reasoning in intuitionistic first-order logic is an important task within the formal approach of constructing verifiable correct software. Interactive proof assistants, like NuPRL [5] and Coq [2], use constructive type theory to formalise the notion of computation and would greatly benefit ...
... Automated reasoning in intuitionistic first-order logic is an important task within the formal approach of constructing verifiable correct software. Interactive proof assistants, like NuPRL [5] and Coq [2], use constructive type theory to formalise the notion of computation and would greatly benefit ...
Subset Types and Partial Functions
... This paper develops a unified approach to partial functions and subset types, which does not suffer from this anomalous behavior. We begin with a higherorder logic that allows functions to be undefined on some arguments. We extend this logic’s type system to include subset types, but we retain deci ...
... This paper develops a unified approach to partial functions and subset types, which does not suffer from this anomalous behavior. We begin with a higherorder logic that allows functions to be undefined on some arguments. We extend this logic’s type system to include subset types, but we retain deci ...
A Cut-Free Calculus for Second
... Definition 2.10. A (standard) Gödel set is a bounded complete linearly ordered set V = hV, ≤i. We denote by 0V and 1V the maximal and minimal elements (respectively) of V with respect to ≤. The operations minV , maxV , inf V and supV are defined as usual (where minV ∅ = 1V and maxV ∅ = 0V ). For ev ...
... Definition 2.10. A (standard) Gödel set is a bounded complete linearly ordered set V = hV, ≤i. We denote by 0V and 1V the maximal and minimal elements (respectively) of V with respect to ≤. The operations minV , maxV , inf V and supV are defined as usual (where minV ∅ = 1V and maxV ∅ = 0V ). For ev ...
On Elkan`s theorems: Clarifying their meaning
... omitted from the first version of Elkan’s theorem. As to the rest of the assumptions, both t~A ∧ B! ⫽ min$t~A!, t~B!% and t~¬A! ⫽ 1 ⫺ t~A! are quite reasonable and, in fact, are often used in applications of fuzzy logic. Let us now concentrate on the last assumption, that is, on t~A! ⫽ t~B! if A and ...
... omitted from the first version of Elkan’s theorem. As to the rest of the assumptions, both t~A ∧ B! ⫽ min$t~A!, t~B!% and t~¬A! ⫽ 1 ⫺ t~A! are quite reasonable and, in fact, are often used in applications of fuzzy logic. Let us now concentrate on the last assumption, that is, on t~A! ⫽ t~B! if A and ...
Bounded Proofs and Step Frames - Università degli Studi di Milano
... using formulae of modal complexity at most n. The bounded proof property is a kind of an analytic subformula property limiting the proof search space. This property holds for proof systems enjoying the subformula property (the latter is a property that usually follows from cut elimination). The bou ...
... using formulae of modal complexity at most n. The bounded proof property is a kind of an analytic subformula property limiting the proof search space. This property holds for proof systems enjoying the subformula property (the latter is a property that usually follows from cut elimination). The bou ...
sentential logic
... In addition to deductive valid , we will be interested in some other logical concepts. Truth Values: True or false said to be truth values of sentence. We define sentences as things that could be true or false. We could have said instead that sentences are thing that can have truth values. Logical t ...
... In addition to deductive valid , we will be interested in some other logical concepts. Truth Values: True or false said to be truth values of sentence. We define sentences as things that could be true or false. We could have said instead that sentences are thing that can have truth values. Logical t ...
Definability in Boolean bunched logic
... Proof. In each case we build models M and M 0 such that there is a bounded morphism from M to M 0 , but M has the property ...
... Proof. In each case we build models M and M 0 such that there is a bounded morphism from M to M 0 , but M has the property ...
Logic Programming, Functional Programming, and Inductive
... If all rules in Φ have a finite number of premises then φω = I(Φ) and φω is the least fixed point of φ. The full theory of inductive definitions is complicated, but much of it need not concern us. A rule p ← P could have an infinite number of premises, unlike rules in logic programs. The rules in an ...
... If all rules in Φ have a finite number of premises then φω = I(Φ) and φω is the least fixed point of φ. The full theory of inductive definitions is complicated, but much of it need not concern us. A rule p ← P could have an infinite number of premises, unlike rules in logic programs. The rules in an ...
Document
... • P(x) is true for every x in the universe of discourse. • Notation: universal quantifier ∀xP(x) • ‘For all x, P(x)’, ‘For every x, P(x)’ • The variable x is bound by the universal quantifier producing a proposition. • An element for which P(x) is false is called a counterexample of ∀xP(x). • Exampl ...
... • P(x) is true for every x in the universe of discourse. • Notation: universal quantifier ∀xP(x) • ‘For all x, P(x)’, ‘For every x, P(x)’ • The variable x is bound by the universal quantifier producing a proposition. • An element for which P(x) is false is called a counterexample of ∀xP(x). • Exampl ...
The Semantic Complexity of some Fragments of English
... or as $% , where is a quantifier-free forit can be written either as $%$ mula involving only unary predicates (and no functions or constants). It is well known that the satisfiability problem for a set of standard two-variable formulas is NEXPTIME-hard (Börger et al. 1997, pp. 253 ff). Any su ...
... or as $% , where is a quantifier-free forit can be written either as $%$ mula involving only unary predicates (and no functions or constants). It is well known that the satisfiability problem for a set of standard two-variable formulas is NEXPTIME-hard (Börger et al. 1997, pp. 253 ff). Any su ...
Reaching transparent truth
... they assign to the variable bound by the quantifier [Kleene, 1952]. We can define disjunction ∨, material conditional ⊃, material biconditional ≡ and an existential quantifier ∃ as usual. We also include constants > and ⊥, which are required on every model to take values 1 and 0 respectively. Theori ...
... they assign to the variable bound by the quantifier [Kleene, 1952]. We can define disjunction ∨, material conditional ⊃, material biconditional ≡ and an existential quantifier ∃ as usual. We also include constants > and ⊥, which are required on every model to take values 1 and 0 respectively. Theori ...
Automata vs. Logics on Data Words
... into equivalent automaton-based specifications, easing, in this way, the various reasoning tasks. Different models of automata which process words over infinite alphabets have been proposed and studied in the literature (see, for instance, the surveys [6, 7]). Pebble automata [8] use special markers ...
... into equivalent automaton-based specifications, easing, in this way, the various reasoning tasks. Different models of automata which process words over infinite alphabets have been proposed and studied in the literature (see, for instance, the surveys [6, 7]). Pebble automata [8] use special markers ...