pdf
... It is possible to build this “Hintikka Test” into the tableau method and use it to prove that certain formulas cannot be valid. However, there are many formulas that are neither valid nor falsifiable in any finite domain. Any tableau proof attempt for these will run infinitely and at no stage of the ...
... It is possible to build this “Hintikka Test” into the tableau method and use it to prove that certain formulas cannot be valid. However, there are many formulas that are neither valid nor falsifiable in any finite domain. Any tableau proof attempt for these will run infinitely and at no stage of the ...
Polarizing Double-Negation Translations
... Gödel-Gentzen negative translation (Definition 2.3 above) removes many negations from translations and the polarization we give in Section 5 will even more. If we want to follow the pattern of Theorem 1 to show equiprovability (in the absence of cut), we can no longer systematically move formulæ fr ...
... Gödel-Gentzen negative translation (Definition 2.3 above) removes many negations from translations and the polarization we give in Section 5 will even more. If we want to follow the pattern of Theorem 1 to show equiprovability (in the absence of cut), we can no longer systematically move formulæ fr ...
Chapter 2, Logic
... What is Philosophy Chapter 2 by Richard Thompson gave rise to a good deal of debate among logicians. For sometimes we assert universal generalisations without any commitment to existence. For instance if we explained ‘unicorn’ by saying ‘Unicorn’ means ’quadruped mammal resembling a horse but with ...
... What is Philosophy Chapter 2 by Richard Thompson gave rise to a good deal of debate among logicians. For sometimes we assert universal generalisations without any commitment to existence. For instance if we explained ‘unicorn’ by saying ‘Unicorn’ means ’quadruped mammal resembling a horse but with ...
Lecture Notes 2
... mathematicians, is just as rigorous. It consists of sentences describing the situation at hand, the inferences being made, and the justification of each inference. ...
... mathematicians, is just as rigorous. It consists of sentences describing the situation at hand, the inferences being made, and the justification of each inference. ...
When is Metric Temporal Logic Expressively Complete?
... Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany ...
... Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany ...
A SHORT AND READABLE PROOF OF CUT ELIMINATION FOR
... also Latin capital letters) stand for finite sets of formulae; so do primed such letters. The expression Γ ⊢ ∆ is called a sequent and intuitively says that the set of hypotheses (formulae) in Γ proves the disjunction of the formulae in ∆. Γ is the antecedent part of the sequent, while ∆ is the succ ...
... also Latin capital letters) stand for finite sets of formulae; so do primed such letters. The expression Γ ⊢ ∆ is called a sequent and intuitively says that the set of hypotheses (formulae) in Γ proves the disjunction of the formulae in ∆. Γ is the antecedent part of the sequent, while ∆ is the succ ...
HOARE`S LOGIC AND PEANO`S ARITHMETIC
... Induction scheme : for each assertion p E L, containing free variable X,the foilowingisanaxiom[p(O)AVx l(p(x)+p(x+l))]+Vx *p(x), Thus, we may observe that equations (3)-(6) alone define N under initial algebra semantics and so we may consider (1) and (2) as additions, making a first refinement of th ...
... Induction scheme : for each assertion p E L, containing free variable X,the foilowingisanaxiom[p(O)AVx l(p(x)+p(x+l))]+Vx *p(x), Thus, we may observe that equations (3)-(6) alone define N under initial algebra semantics and so we may consider (1) and (2) as additions, making a first refinement of th ...
Logic and Resolution - Institute for Computing and Information
... developed a computer program which was capable of proving several theorems from number theory. The greatest triumph of the program was its proof that the sum of two even numbers is even. Other researchers, however, were more interested in the study of human problem solving, more in particular in heu ...
... developed a computer program which was capable of proving several theorems from number theory. The greatest triumph of the program was its proof that the sum of two even numbers is even. Other researchers, however, were more interested in the study of human problem solving, more in particular in heu ...
PHIL12A Section answers, 9 February 2011
... 2. How many different ternary sentential connectives are there? How did you arrive at this number? You should not try to list them all! We calculate the number of ternary connectives in the same way as we calculated the number of binary connectives in the last question. A truth table for a ternary ...
... 2. How many different ternary sentential connectives are there? How did you arrive at this number? You should not try to list them all! We calculate the number of ternary connectives in the same way as we calculated the number of binary connectives in the last question. A truth table for a ternary ...
A Simple Tableau System for the Logic of Elsewhere
... the size of models of the satisfiable formulae) and we show that this problem becomes linear-time when the number of propositional variables is bounded. Although E and the well-known propositional modal S5 share numerous common features we show that E is strictly more expressive than S5 (in a sense ...
... the size of models of the satisfiable formulae) and we show that this problem becomes linear-time when the number of propositional variables is bounded. Although E and the well-known propositional modal S5 share numerous common features we show that E is strictly more expressive than S5 (in a sense ...
Classical and Intuitionistic Models of Arithmetic
... m ψ(ā) ⇐⇒ m |= ψ(ā). Proof: The first claim is proved by induction on ϕ( x̄). The interesting cases are those of implication and universal quantification. So suppose first that ϕ ≡ ψ → χ. We have to show that m ∀ x̄[(ψ( x̄) → χ( x̄)) ∨ ¬(ψ( x̄) → χ( x̄))]. So let k ≥ m and c̄ ∈ Ak . If k ψ(c ...
... m ψ(ā) ⇐⇒ m |= ψ(ā). Proof: The first claim is proved by induction on ϕ( x̄). The interesting cases are those of implication and universal quantification. So suppose first that ϕ ≡ ψ → χ. We have to show that m ∀ x̄[(ψ( x̄) → χ( x̄)) ∨ ¬(ψ( x̄) → χ( x̄))]. So let k ≥ m and c̄ ∈ Ak . If k ψ(c ...
Logic as a Tool 3mm Chapter 2: Deductive Reasoning in
... The underlying idea of Propositional Resolution: in order to prove the validity of a logical consequence A1 , . . . , An |= B, show that there is no truth assignment which falsifies it, i.e., show that the formulae A1 , . . . , An and ¬B cannot be satisfied simultaneously. That is done by transformi ...
... The underlying idea of Propositional Resolution: in order to prove the validity of a logical consequence A1 , . . . , An |= B, show that there is no truth assignment which falsifies it, i.e., show that the formulae A1 , . . . , An and ¬B cannot be satisfied simultaneously. That is done by transformi ...
An Introduction to SOFL
... The use of parenthesis An expression is interpreted by applying the operator priority order unless parenthesis is used. For example: the expression not p and q or r <=> p => q and r is equivalent to the expression: (((not p) and q) or r) <=> (p => (q and r)) Parenthesis can be used to change the pr ...
... The use of parenthesis An expression is interpreted by applying the operator priority order unless parenthesis is used. For example: the expression not p and q or r <=> p => q and r is equivalent to the expression: (((not p) and q) or r) <=> (p => (q and r)) Parenthesis can be used to change the pr ...
8.1 Symbols and Translation
... If Rx means “x is a rabbit,” and Sx means “x is a snake,” then the premise translates as, “If everything in the universe is a rabbit, everything in the universe is snake.” ◦ The statement is true because the antecedent is false: not everything in the universe is a rabbit. However, the conclusion is ...
... If Rx means “x is a rabbit,” and Sx means “x is a snake,” then the premise translates as, “If everything in the universe is a rabbit, everything in the universe is snake.” ◦ The statement is true because the antecedent is false: not everything in the universe is a rabbit. However, the conclusion is ...
Completeness and Decidability of a Fragment of Duration Calculus
... logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a real-time hybrid system from its specification in DC. In that paper, we intro ...
... logic to specify the requirements for real-time systems. DC has been used successfully in many case studies, see e.g. [ZZ94,YWZP94,HZ94,DW94,BHCZ94,XH95], [Dan98,ED99]. In [DW94], we have developed a method for designing a real-time hybrid system from its specification in DC. In that paper, we intro ...
Ambient Logic II.fm
... The π-calculus notion of name restriction [12], initially intended to represent hidden communication channels, has been used also to represent hidden encryption keys [2] and as the basis for definitions of secrecy [2, 4]. In the context of the ambient calculus [6], name restriction can be used to re ...
... The π-calculus notion of name restriction [12], initially intended to represent hidden communication channels, has been used also to represent hidden encryption keys [2] and as the basis for definitions of secrecy [2, 4]. In the context of the ambient calculus [6], name restriction can be used to re ...
Lecture - 04 (Logic Knowledge Base)
... Argument and Proof in Propositional Logic • An argument is a relationship between a set of propositions called premises and another proposition called the conclusion. • Proof is intended to show deductively that an argument is sound (or valid). – An argument is sound iff it cannot be the case that ...
... Argument and Proof in Propositional Logic • An argument is a relationship between a set of propositions called premises and another proposition called the conclusion. • Proof is intended to show deductively that an argument is sound (or valid). – An argument is sound iff it cannot be the case that ...
ND for predicate logic ∀-elimination, first attempt Variable capture
... In the conclusion of each rule, the formula not in the context is called the main formula or principal formula. In the rule Ax , both occurrences of A are principal. ...
... In the conclusion of each rule, the formula not in the context is called the main formula or principal formula. In the rule Ax , both occurrences of A are principal. ...
Introducing Quantified Cuts in Logic with Equality
... of first-order logic. One interesting property of non-analytic proofs is their considerably smaller length. The exact difference depends on the logic (or theory) under consideration, but it is typically enormous. In (classical and intuitionistic) first-order logic there are proofs with cut of length n ...
... of first-order logic. One interesting property of non-analytic proofs is their considerably smaller length. The exact difference depends on the logic (or theory) under consideration, but it is typically enormous. In (classical and intuitionistic) first-order logic there are proofs with cut of length n ...
A game semantics for proof search: Preliminary results - LIX
... The proviso † requires that t and s are unifiable and θ is their most general unifier (∆θ is the multiset resulting from applying θ to all formulas in ∆). The proviso ‡ requires that t and s are not unifiable. The free variables of a sequent are also called eigenvariables. Notice that the equality r ...
... The proviso † requires that t and s are unifiable and θ is their most general unifier (∆θ is the multiset resulting from applying θ to all formulas in ∆). The proviso ‡ requires that t and s are not unifiable. The free variables of a sequent are also called eigenvariables. Notice that the equality r ...