PROPOSITIONAL LOGIC 1 Propositional Logic - Glasnost!
... x could represent the class of all sheep, and y could represent the class of all white objects. They xy represents the class of all white sheep. In particular, x2 = xx represents the class of all sheep which are sheep, which is clearly just the class of all sheep! Thus in all cases this statement si ...
... x could represent the class of all sheep, and y could represent the class of all white objects. They xy represents the class of all white sheep. In particular, x2 = xx represents the class of all sheep which are sheep, which is clearly just the class of all sheep! Thus in all cases this statement si ...
Answer Sets for Propositional Theories
... Equilibrium logic, introduced by Pearce [1997, 1999], extends the concept of an answer set [Gelfond and Lifschitz, 1988, 1991] from logic programs to arbitrary sets of formulas. Logic programs correspond to the special case in which every formula is a “rule” — an implication that has no implications ...
... Equilibrium logic, introduced by Pearce [1997, 1999], extends the concept of an answer set [Gelfond and Lifschitz, 1988, 1991] from logic programs to arbitrary sets of formulas. Logic programs correspond to the special case in which every formula is a “rule” — an implication that has no implications ...
Logic in Proofs (Valid arguments) A theorem is a hypothetical
... A theorem is a hypothetical statement of the form H 6 C, where H is a (compound) statement which is taken as being true, and C is a statement which follows from H by logical reasoning. Example: [(p 6 q) v (q 6 r) v (¬ r)] 6 (¬ p) An argument in logic is a way to reach a conclusion based on prior sta ...
... A theorem is a hypothetical statement of the form H 6 C, where H is a (compound) statement which is taken as being true, and C is a statement which follows from H by logical reasoning. Example: [(p 6 q) v (q 6 r) v (¬ r)] 6 (¬ p) An argument in logic is a way to reach a conclusion based on prior sta ...
equivalents of the compactness theorem for locally finite sets of
... greater than twice the number of those a ∈ A for which ¬(π(x)Ra). Thus R is dense. Since every R–consistent choice on A is also an R∗ –consistent choice on A∗ , we get an R∗ –consistent choice S on the family A∗ . Then we easily see that {π(x) : x ∈ S} is an R–consistent choice on A. 2 As it is know ...
... greater than twice the number of those a ∈ A for which ¬(π(x)Ra). Thus R is dense. Since every R–consistent choice on A is also an R∗ –consistent choice on A∗ , we get an R∗ –consistent choice S on the family A∗ . Then we easily see that {π(x) : x ∈ S} is an R–consistent choice on A. 2 As it is know ...
Logic
... but that C is true if all the premises are true. That is, there are NO counterexamples. P1: Bertil is a professional musician. P2: All professional musicians have pony-tail. Therefore: Bertil has pony-tail. ...
... but that C is true if all the premises are true. That is, there are NO counterexamples. P1: Bertil is a professional musician. P2: All professional musicians have pony-tail. Therefore: Bertil has pony-tail. ...
PDF
... In this entry, we will prove the substitution theorem for propositional logic based on the axiom system found here. Besides the deduction theorem, below are some additional results we will need to prove the theorem: 1. If ∆ ` A → B and Γ ` B → C, then ∆, Γ ` A → C. 2. ∆ ` A and ∆ ` B iff ∆ ` A ∧ B. ...
... In this entry, we will prove the substitution theorem for propositional logic based on the axiom system found here. Besides the deduction theorem, below are some additional results we will need to prove the theorem: 1. If ∆ ` A → B and Γ ` B → C, then ∆, Γ ` A → C. 2. ∆ ` A and ∆ ` B iff ∆ ` A ∧ B. ...
Cylindric Modal Logic - Homepages of UvA/FNWI staff
... semantics and axiomatics. The motivation for introducing this formalism is twofold: first, it forms an interesting bridge over the gap between propositional formalisms and first-order logic. And second, the modal tools developed in studying cylindric modal logic will be applied to analyze some probl ...
... semantics and axiomatics. The motivation for introducing this formalism is twofold: first, it forms an interesting bridge over the gap between propositional formalisms and first-order logic. And second, the modal tools developed in studying cylindric modal logic will be applied to analyze some probl ...
Predicate Calculus - National Taiwan University
... But if the domain is infinite? Intuitively, this is why a computer cannot be programmed to determine if an arbitrary formula in predicate logic is a tautology (for all tautologies). Our only alternative is proof procedures! Therefore the soundness and completeness of our proof procedures is very imp ...
... But if the domain is infinite? Intuitively, this is why a computer cannot be programmed to determine if an arbitrary formula in predicate logic is a tautology (for all tautologies). Our only alternative is proof procedures! Therefore the soundness and completeness of our proof procedures is very imp ...
compactness slides
... The language of sentential logic, that is, the set of all wffs, corresponds to C ∗ , the intersection of all inductive sets w.r.t. B and F. By the unique readability theorem C ∗ is freely generated from the set of sentence symbols by the functions in F. This guarantees the uniqueness of the extensi ...
... The language of sentential logic, that is, the set of all wffs, corresponds to C ∗ , the intersection of all inductive sets w.r.t. B and F. By the unique readability theorem C ∗ is freely generated from the set of sentence symbols by the functions in F. This guarantees the uniqueness of the extensi ...
Pseudo-finite model theory
... Cherlin, Harrington and Lachlan [3] proved that all models of totally categorical ...
... Cherlin, Harrington and Lachlan [3] proved that all models of totally categorical ...
Implementable Set Theory and Consistency of ZFC
... looks like, at first sight. Especially with complicated sets, where the elements themselves in turn are sets of sets (towers of sets), the look and feel of a set can become quite messy. One way to get rid of that messiness, at least in part, is to adopt the following more detailed rules: - if x and ...
... looks like, at first sight. Especially with complicated sets, where the elements themselves in turn are sets of sets (towers of sets), the look and feel of a set can become quite messy. One way to get rid of that messiness, at least in part, is to adopt the following more detailed rules: - if x and ...
Annals of Pure and Applied Logic Ordinal machines and admissible
... enumerable. So α -recursion theory is closely connected to set theory, in particular to constructibility theory. Its methods involve set-theoretic definability arguments up to the beginnings of Jensen’s fine structure theory of the constructible hierarchy. Further information on the connection to se ...
... enumerable. So α -recursion theory is closely connected to set theory, in particular to constructibility theory. Its methods involve set-theoretic definability arguments up to the beginnings of Jensen’s fine structure theory of the constructible hierarchy. Further information on the connection to se ...
Slides from 10/20/14
... --if there is a tilde, the quantifier applies to the tilde and to whatever the tilde applies to --if there is a parenthesis (or bracket), the quantifier applies to everything in that pair of parentheses (or brackets) A variable is bound if and only if it is within the scope of a quantifier that con ...
... --if there is a tilde, the quantifier applies to the tilde and to whatever the tilde applies to --if there is a parenthesis (or bracket), the quantifier applies to everything in that pair of parentheses (or brackets) A variable is bound if and only if it is within the scope of a quantifier that con ...
COMPLETENESS OF THE RANDOM GRAPH
... Note that we do not need to include the ∀ quantifier because, following the rules of predicate logic, every statement of the form (∀y)θ is equivalent to a statement of the form (¬∃y)(φ). Also, we call repeated but finite applications of step (ii) to formulas a boolean combination of formulas. In a f ...
... Note that we do not need to include the ∀ quantifier because, following the rules of predicate logic, every statement of the form (∀y)θ is equivalent to a statement of the form (¬∃y)(φ). Also, we call repeated but finite applications of step (ii) to formulas a boolean combination of formulas. In a f ...