equivalents of the compactness theorem for locally finite sets of
... As it is known (see [2]) Ff in is equivalent to some statement about propositional calculus. We consider the language {¬, ∧, ∨} and accept standard definitions of propositional formulae. A set X of propositional formulas is said to be locally satisfiable iff every finite subset X0 of X is satisfiabl ...
... As it is known (see [2]) Ff in is equivalent to some statement about propositional calculus. We consider the language {¬, ∧, ∨} and accept standard definitions of propositional formulae. A set X of propositional formulas is said to be locally satisfiable iff every finite subset X0 of X is satisfiabl ...
NONSTANDARD MODELS IN RECURSION THEORY
... It is straightforward to verify that ≤T is a transitive relation. Turing degrees are defined in the usual way. In the above definition, P is a “positive condition” of the oracle Y and N is a “negative condition”. Notice that the reduction procedure is designed to answer questions about M-finite sets ...
... It is straightforward to verify that ≤T is a transitive relation. Turing degrees are defined in the usual way. In the above definition, P is a “positive condition” of the oracle Y and N is a “negative condition”. Notice that the reduction procedure is designed to answer questions about M-finite sets ...
Syntax and Semantics of Propositional Linear Temporal Logic
... LTL is satisfiable iff it is satisfiable at a linear model in which, from a certain state on, the same finite sequence of states is repeated infinitely many times. The equivalence between satisfiability of individual formulas in general and in finite models is known as the small (finite) model prope ...
... LTL is satisfiable iff it is satisfiable at a linear model in which, from a certain state on, the same finite sequence of states is repeated infinitely many times. The equivalence between satisfiability of individual formulas in general and in finite models is known as the small (finite) model prope ...
Standardization of Formulæ
... An existential quantifier can be removed by replacing the variable it bounds by a Skolem function of the form f (x1 , ..xn ), where: f is a fresh function symbol x1 , .., xn are the variables which are universally quantified before the quantifier to be removed ∀x∃y (p(x) → ¬q(y )) ∃x∀z(q(x, z) ∨ r ( ...
... An existential quantifier can be removed by replacing the variable it bounds by a Skolem function of the form f (x1 , ..xn ), where: f is a fresh function symbol x1 , .., xn are the variables which are universally quantified before the quantifier to be removed ∀x∃y (p(x) → ¬q(y )) ∃x∀z(q(x, z) ∨ r ( ...
Nonmonotonic Logic - Default Logic
... the intended meaning is If ϕ is known, and it is consistent to assume ψ1 , . . . ψn , then assume χ. I ...
... the intended meaning is If ϕ is known, and it is consistent to assume ψ1 , . . . ψn , then assume χ. I ...
A Basis Theorem for Perfect Sets
... A.r.1. Let n be minimal so that n is greater than t, Gx (n) 6= S(r), and Gx n_ hS(n)i, the concatenation of hGx (0), . . . , Gx (n − 1)i with hS(n)i, is in T ; we replace Gx with the least Gj such that Gj extends Gx n_ hS(n)i, Gj ∈ [T ], and Gj is not eventually constant in T . A.r.2. We let s b ...
... A.r.1. Let n be minimal so that n is greater than t, Gx (n) 6= S(r), and Gx n_ hS(n)i, the concatenation of hGx (0), . . . , Gx (n − 1)i with hS(n)i, is in T ; we replace Gx with the least Gj such that Gj extends Gx n_ hS(n)i, Gj ∈ [T ], and Gj is not eventually constant in T . A.r.2. We let s b ...
1 Introduction 2 Formal logic
... After all, ϕ → ψ is just a formula whose truth value is defined in terms of the truth values of ϕ and ψ; in no sense are we deriving the truth of ψ from the truth of ϕ. Perhaps the most unexpected feature of implication is that ϕ → ψ is true if ϕ is false, no matter if ψ is true or not. Lemma 1. For ...
... After all, ϕ → ψ is just a formula whose truth value is defined in terms of the truth values of ϕ and ψ; in no sense are we deriving the truth of ψ from the truth of ϕ. Perhaps the most unexpected feature of implication is that ϕ → ψ is true if ϕ is false, no matter if ψ is true or not. Lemma 1. For ...
Logic and Categories As Tools For Building Theories
... is terminal in the category of such pairings, where the morphisms are arrows preserving the components. It is straightforward to show that if a category has a terminal object, and all binary products, then it has products of all finite families of objects. Thus these are the two cases usually consid ...
... is terminal in the category of such pairings, where the morphisms are arrows preserving the components. It is straightforward to show that if a category has a terminal object, and all binary products, then it has products of all finite families of objects. Thus these are the two cases usually consid ...
Backwards and Forwards - Cornell Math
... Elementary Equiv. and Substructures: Examples In the language of rings, consider the field of real numbers R = hR; +, ×, −, 0, 1i. If we let A be the set of all real algebraic numbers (i.e. roots of integer polynomials), and restrict our interpretations to A, we obtain an elementary substructure A = ...
... Elementary Equiv. and Substructures: Examples In the language of rings, consider the field of real numbers R = hR; +, ×, −, 0, 1i. If we let A be the set of all real algebraic numbers (i.e. roots of integer polynomials), and restrict our interpretations to A, we obtain an elementary substructure A = ...
Mathematicians
... word which means 'witch'. The equation for this bell-shaped curve was given the name 'witch of Agnesi' and it stuck and can be found in some textbooks today. 1738 she published Propositiones Philosophicae a series of essays on philosophy and natural science. First woman to be appointed as professor ...
... word which means 'witch'. The equation for this bell-shaped curve was given the name 'witch of Agnesi' and it stuck and can be found in some textbooks today. 1738 she published Propositiones Philosophicae a series of essays on philosophy and natural science. First woman to be appointed as professor ...
Classicality as a Property of Predicate Symbols
... lattice. Intuitionistic logic is the lower bound of the lattice and classical logic is the upper bound. All the extensions are consistent. It will be shown later that there is a straightforward translation between the extensions for ordered pairs of sets of classical predicate symbols. These extensi ...
... lattice. Intuitionistic logic is the lower bound of the lattice and classical logic is the upper bound. All the extensions are consistent. It will be shown later that there is a straightforward translation between the extensions for ordered pairs of sets of classical predicate symbols. These extensi ...
Is the Liar Sentence Both True and False? - NYU Philosophy
... with acceptance. To a Þrst approximation anyway, accepting A is having a high degree of belief in it; say a degree of belief over a certain threshold T , which may depend on context but must be greater than 12 . (Degrees of belief are assumed to be real numbers in the interval [0, 1].) To the same ...
... with acceptance. To a Þrst approximation anyway, accepting A is having a high degree of belief in it; say a degree of belief over a certain threshold T , which may depend on context but must be greater than 12 . (Degrees of belief are assumed to be real numbers in the interval [0, 1].) To the same ...
slides1
... A proof of A ⇒ B is a function f that maps each proof p of A to the proof f (p) of B. ¬A is treated as A ⇒ ⊥ where ⊥ is a sentence without proof. A proof of ∀ξ.A is a function f that maps each point a in the domain of definition to a proof f (a) of A[a/ξ]. A proof of ∃ξ.A is a pair (a, p) where a is ...
... A proof of A ⇒ B is a function f that maps each proof p of A to the proof f (p) of B. ¬A is treated as A ⇒ ⊥ where ⊥ is a sentence without proof. A proof of ∀ξ.A is a function f that maps each point a in the domain of definition to a proof f (a) of A[a/ξ]. A proof of ∃ξ.A is a pair (a, p) where a is ...
Mathematical Logic Fall 2004 Professor R. Moosa Contents
... the field itself, and this is the case with logic. We often discover connections to core areas of math itself (number theory, geometry, analysis, and algebra). There is a dichotomy in logic. Given a statement (theorem/axiom/whatever), there is the syntax of the statement (what is written down on the ...
... the field itself, and this is the case with logic. We often discover connections to core areas of math itself (number theory, geometry, analysis, and algebra). There is a dichotomy in logic. Given a statement (theorem/axiom/whatever), there is the syntax of the statement (what is written down on the ...
The complexity of the dependence operator
... is, transitive model of Kripke-Platek set theory) beyond ω1ck . Thus the quantification is really (but implicitly) a bounded universal quantification. (The reason for this pleasantly bounded state of affairs is the Kleene Basis Theorem (see, eg., again Rogers [4], Theorem XLII), which in our contex ...
... is, transitive model of Kripke-Platek set theory) beyond ω1ck . Thus the quantification is really (but implicitly) a bounded universal quantification. (The reason for this pleasantly bounded state of affairs is the Kleene Basis Theorem (see, eg., again Rogers [4], Theorem XLII), which in our contex ...
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 ...
A constructive approach to nonstandard analysis*
... shown that it has a full transfer principle relative to the standard model. It remains to see whether it holds also for higher type arithmetic. To give a quick but incomplete picture of the basic idea in Schmieden and Laugwitz’ paper, we could say that they work with the reduced power of the reals, ...
... shown that it has a full transfer principle relative to the standard model. It remains to see whether it holds also for higher type arithmetic. To give a quick but incomplete picture of the basic idea in Schmieden and Laugwitz’ paper, we could say that they work with the reduced power of the reals, ...
Redundancies in the Hilbert-Bernays derivability conditions for
... The elimination of the first derivability condition allows the application of the Consistency Theorem to cut-free logics which cannot prove that they are closed under cut. It is Theorem 1 which will probably have primary interest for readers who are not concerned with technical proof theory or with ...
... The elimination of the first derivability condition allows the application of the Consistency Theorem to cut-free logics which cannot prove that they are closed under cut. It is Theorem 1 which will probably have primary interest for readers who are not concerned with technical proof theory or with ...
Computational foundations of basic recursive function theory
... such concepts in this framework. We represent in the theory those mathematical objects that algorithms compute, numbers andfunctions; iffand g compute functions then asfunctions they are equal precisely iff(a) = g(u) for all a in the domain offand g. This is thus an extensional notion of equality. F ...
... such concepts in this framework. We represent in the theory those mathematical objects that algorithms compute, numbers andfunctions; iffand g compute functions then asfunctions they are equal precisely iff(a) = g(u) for all a in the domain offand g. This is thus an extensional notion of equality. F ...
Factoring out the impossibility of logical aggregation
... either or ¬ belongs to B. This maximal consistency property implies the weaker one that B is deductively closed in the same relative sense, i.e., for all ∈ ∗ , if B, then ∈ B. It follows in particular that ∈ B ⇔ ∈ B when ↔ , and that ∈ B when . Deductive closure and its consequ ...
... either or ¬ belongs to B. This maximal consistency property implies the weaker one that B is deductively closed in the same relative sense, i.e., for all ∈ ∗ , if B, then ∈ B. It follows in particular that ∈ B ⇔ ∈ B when ↔ , and that ∈ B when . Deductive closure and its consequ ...
Predicate Logic - Teaching-WIKI
... functions, variables, constants) onto objects, relations, and functions of the “world” (formally: Domain, relational Structure, or Universe) • Valuation – Assigns domain objects to variables – The Valuation function can be used for describing value assignments and constraints in case of nested quant ...
... functions, variables, constants) onto objects, relations, and functions of the “world” (formally: Domain, relational Structure, or Universe) • Valuation – Assigns domain objects to variables – The Valuation function can be used for describing value assignments and constraints in case of nested quant ...
Document
... the domain of discourse is D To show that this implication is not true in the domain D, it must be shown that there exists some x in D such that (P(x ) → Q(x )) is not true This means that there exists some x in D such that P(x) is true but Q(x) is not true. Such an x is called a counterexample of ...
... the domain of discourse is D To show that this implication is not true in the domain D, it must be shown that there exists some x in D such that (P(x ) → Q(x )) is not true This means that there exists some x in D such that P(x) is true but Q(x) is not true. Such an x is called a counterexample of ...
characterization of classes of frames in modal language
... – the model of S4.Grz. The same is true in the case of Kt ∪ {H(H(φ → Hφ) → φ) → φ}. Now, as in the case of the former system, it is complete with respect to the class that instead of well-foundedness is conversely wellfounded. 7 Moreover, in [Anselm, Saint Archbishop of Canterbury, 1929, Book II, ch ...
... – the model of S4.Grz. The same is true in the case of Kt ∪ {H(H(φ → Hφ) → φ) → φ}. Now, as in the case of the former system, it is complete with respect to the class that instead of well-foundedness is conversely wellfounded. 7 Moreover, in [Anselm, Saint Archbishop of Canterbury, 1929, Book II, ch ...
Semi-constr. theories - Stanford Mathematics
... be definite, but not quantification applied to variables for sets or functions of natural numbers. Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical logic or entirely on intuitionistic logic, while almost all axiomatizations of ...
... be definite, but not quantification applied to variables for sets or functions of natural numbers. Most axiomatizations of set theory that have been treated metamathematically have been based either entirely on classical logic or entirely on intuitionistic logic, while almost all axiomatizations of ...