CHAPTER 1 The main subject of Mathematical Logic is
... In a given context we shall adopt the following convention. Once a formula has been introduced as A(x), i.e., A with a designated variable x, we write A(r) for A[x := r], and similarly with more variables. 1.1.3. Subformulas. Unless stated otherwise, the notion of subformula will be that defined by ...
... In a given context we shall adopt the following convention. Once a formula has been introduced as A(x), i.e., A with a designated variable x, we write A(r) for A[x := r], and similarly with more variables. 1.1.3. Subformulas. Unless stated otherwise, the notion of subformula will be that defined by ...
Query Answering for OWL-DL with Rules
... only provides insight into the causes for the undecidability of the full combination, but also enables a more detailed complexity analysis and, ultimately, the design of “specialized” decision procedures. Applications that do not require the expressive power of the full combination can use such proc ...
... only provides insight into the causes for the undecidability of the full combination, but also enables a more detailed complexity analysis and, ultimately, the design of “specialized” decision procedures. Applications that do not require the expressive power of the full combination can use such proc ...
Truth-Functional Propositional Logic
... why the rules work or not. The uniformity, simplicity, and regularity of these arithmetical rules, and their applicability with minimal understanding, is shown by the existence of extremely simple artificial devices for effective arithmetical calculation such as the ancient abacus. Before any system ...
... why the rules work or not. The uniformity, simplicity, and regularity of these arithmetical rules, and their applicability with minimal understanding, is shown by the existence of extremely simple artificial devices for effective arithmetical calculation such as the ancient abacus. Before any system ...
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 ...
Classical First-Order Logic Introduction
... First-order logic (FOL) is a richer language than propositional logic. Its lexicon contains not only the symbols ∧, ∨, ¬, and → (and parentheses) from propositional logic, but also the symbols ∃ and ∀ for “there exists” and “for all”, along with various symbols to represent variables, constants, fun ...
... First-order logic (FOL) is a richer language than propositional logic. Its lexicon contains not only the symbols ∧, ∨, ¬, and → (and parentheses) from propositional logic, but also the symbols ∃ and ∀ for “there exists” and “for all”, along with various symbols to represent variables, constants, fun ...
A Proof Theory for Generic Judgments: An extended abstract
... say t is a Σ-term (of type γ), and, if γ is o, t is a Σ-formula. In the displayed sequent above, n ≥ 0 and B0 , B1 , . . . , Bn are Σ-formulas. Informally, the “extensional” reading of this sequent would be that for every substitution θ that maps a variable x : γ ∈ Σ to a term of type γ, if Bi θ hol ...
... say t is a Σ-term (of type γ), and, if γ is o, t is a Σ-formula. In the displayed sequent above, n ≥ 0 and B0 , B1 , . . . , Bn are Σ-formulas. Informally, the “extensional” reading of this sequent would be that for every substitution θ that maps a variable x : γ ∈ Σ to a term of type γ, if Bi θ hol ...
predicate
... program exists which, given any , can determine in a finite amount of time if ⊨ • Proof reduce to Post Correspondence problem. I.E. show that if the decision problem is solvable, we could solve the Post Correspondence problem. This is a contradiction. ...
... program exists which, given any , can determine in a finite amount of time if ⊨ • Proof reduce to Post Correspondence problem. I.E. show that if the decision problem is solvable, we could solve the Post Correspondence problem. This is a contradiction. ...