Mathematical Logic
... Complexity of deciding logical consequence in Propositional Logic The truth table method is Exponential The problem of determining if a formula A containing n primitive propositions, is a logical consequence of the empty set, i.e., the problem of determining if A is valid, (|= A), takes an n-expone ...
... Complexity of deciding logical consequence in Propositional Logic The truth table method is Exponential The problem of determining if a formula A containing n primitive propositions, is a logical consequence of the empty set, i.e., the problem of determining if A is valid, (|= A), takes an n-expone ...
Predicate Calculus - National Taiwan University
... Our only alternative is proof procedures! Therefore the soundness and completeness of our proof procedures is very important! ...
... Our only alternative is proof procedures! Therefore the soundness and completeness of our proof procedures is very important! ...
pdf - Consequently.org
... On these grounds, if the background account of deducibility included the commitment that some p did not entail some q, then, relative to that background, tonk fails the demand of consistency. This is one of the tests Belnap considers in the paper. In the case of a natural deduction proof theory or a ...
... On these grounds, if the background account of deducibility included the commitment that some p did not entail some q, then, relative to that background, tonk fails the demand of consistency. This is one of the tests Belnap considers in the paper. In the case of a natural deduction proof theory or a ...
Lecture 9. Model theory. Consistency, independence, completeness
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
... If M ╞ δ for every δ ∈ ∆, then M ╞ φ. In other words, ∆ entails φ if φ is true in every model in which all the premises in ∆ are true. We write ╞ φ for ∅ ╞ φ . We say φ is valid, or logically valid, or a semantic tautology in that case. ╞ φ holds iff for every M, M ╞ φ. Validity means truth in all m ...
Monadic Second Order Logic and Automata on Infinite Words
... greater, he develops the theories in a more general (and more complicated) way than is necessary to understand Büchi’s theorem, and he only sketches the proof of Büchi’s theorem, which is given in detail here. Two theories concerned with infinite words For both of the theories considered in this r ...
... greater, he develops the theories in a more general (and more complicated) way than is necessary to understand Büchi’s theorem, and he only sketches the proof of Büchi’s theorem, which is given in detail here. Two theories concerned with infinite words For both of the theories considered in this r ...
notes
... Let P be a propositions containing the (distinct) atomic formulas A 1 , . . . , An and v1 , . . . v2n its interpretations. We denote with v P the boolean function associated with P , i.e. vP : {0, 1}n → {0, 1} is defined as follows: for each (a 1 , . . . , an ), ai ∈ {0, 1}, there exists i ∈ {1, . ...
... Let P be a propositions containing the (distinct) atomic formulas A 1 , . . . , An and v1 , . . . v2n its interpretations. We denote with v P the boolean function associated with P , i.e. vP : {0, 1}n → {0, 1} is defined as follows: for each (a 1 , . . . , an ), ai ∈ {0, 1}, there exists i ∈ {1, . ...
Completeness Theorem for Continuous Functions and Product
... theorems for many infinitary logics with generalized quantifiers [4–7, 12]. Roughly speaking, adding new quantifiers to infinitary logics is one of the most frequent and high acceptable ways to incorporate into the realm of logic those structures whose related concepts are left out of the first-order log ...
... theorems for many infinitary logics with generalized quantifiers [4–7, 12]. Roughly speaking, adding new quantifiers to infinitary logics is one of the most frequent and high acceptable ways to incorporate into the realm of logic those structures whose related concepts are left out of the first-order log ...
1 Preliminaries 2 Basic logical and mathematical definitions
... An Herbrand interpretation is then defined, according to the previous general definition, as an interpretation where the domain is the Herbrand universe, each constant in L is assigned to itself, each n-ary function symbol f ∈ L is assigned to the mapping fH : τ (Σ)n → τ (Σ) defined by fH (t1 , . . ...
... An Herbrand interpretation is then defined, according to the previous general definition, as an interpretation where the domain is the Herbrand universe, each constant in L is assigned to itself, each n-ary function symbol f ∈ L is assigned to the mapping fH : τ (Σ)n → τ (Σ) defined by fH (t1 , . . ...
Predicate Logic - Teaching-WIKI
... • Intuitionistic first-order logic uses intuitionistic rather than classical propositional calculus; for example, ¬¬φ need not be equivalent to φ; similarly, first-order fuzzy logics are first-order extensions of propositional fuzzy logics rather than classical logic • Infinitary logic allows infini ...
... • Intuitionistic first-order logic uses intuitionistic rather than classical propositional calculus; for example, ¬¬φ need not be equivalent to φ; similarly, first-order fuzzy logics are first-order extensions of propositional fuzzy logics rather than classical logic • Infinitary logic allows infini ...
Infinitistic Rules of Proof and Their Semantics
... (every non-empty analytical family of unary functions has an analytical element} holds, which is known to be independent from the axioms of set theory. 4. Searching a satisfactory syntactical ,8-rule. It seems that the question raised by Mostowski in [4] about the existence of a syntactical ,8-rule ...
... (every non-empty analytical family of unary functions has an analytical element} holds, which is known to be independent from the axioms of set theory. 4. Searching a satisfactory syntactical ,8-rule. It seems that the question raised by Mostowski in [4] about the existence of a syntactical ,8-rule ...
Notes and exercises on First Order Logic
... Exercise 9 Repeat the above exercise, but replace the formula P by x2 + y 2 = z 2 and add the squaring function to the structure N. From the exercises above (and our intuition) we suspect that the truth value of φ depends only on the values substituted for the free variables. The next theorem shows ...
... Exercise 9 Repeat the above exercise, but replace the formula P by x2 + y 2 = z 2 and add the squaring function to the structure N. From the exercises above (and our intuition) we suspect that the truth value of φ depends only on the values substituted for the free variables. The next theorem shows ...
Introduction to Theoretical Computer Science, lesson 3
... Formula B logically follows from A1, …, An, denoted A1,…,An |= B, iff B is true in every model of {A1,…,An}. Thus for every interpretation I in which the formulas A1, …, An are true it holds that the formula B is true as well: A1,…,An |= B: If |=I A1,…, |=I An then |=I B, for all I. Note that the “c ...
... Formula B logically follows from A1, …, An, denoted A1,…,An |= B, iff B is true in every model of {A1,…,An}. Thus for every interpretation I in which the formulas A1, …, An are true it holds that the formula B is true as well: A1,…,An |= B: If |=I A1,…, |=I An then |=I B, for all I. Note that the “c ...
Second-Order Logic and Fagin`s Theorem
... Let n = ||A||, so A is a boolean formula with at most n variables and n clauses. The construction of f (A) is shown in Figure 7.19. Notice the triangle, with vertices labeled T, F, R. Any three-coloring of the graph must color these three vertices distinct colors. We may assume without loss of gener ...
... Let n = ||A||, so A is a boolean formula with at most n variables and n clauses. The construction of f (A) is shown in Figure 7.19. Notice the triangle, with vertices labeled T, F, R. Any three-coloring of the graph must color these three vertices distinct colors. We may assume without loss of gener ...
←→ ↓ ↓ ←→ ←→ ←→ ←→ −→ −→ → The diagonal lemma as
... Notation Our formal language is that of first-order arithmetic; F m is the set of formulas with at most one free variable; g is any one of the standard Gödel numberings; N denotes the set of Gödel numbers of formulas in F m; Q stands for Robinson arithmetic. The closed terms corresponding to natur ...
... Notation Our formal language is that of first-order arithmetic; F m is the set of formulas with at most one free variable; g is any one of the standard Gödel numberings; N denotes the set of Gödel numbers of formulas in F m; Q stands for Robinson arithmetic. The closed terms corresponding to natur ...
pdf
... Then I maps all the predicate symbols in S to relations over U . What remains to be shown is ∀Y ∈ S. U,I|=Y. We prove this by structural induction on formulas, keeping in mind that the cases for γ and δ are straightforward generalizations of those for α and β. base case: If Y is an atomic formula th ...
... Then I maps all the predicate symbols in S to relations over U . What remains to be shown is ∀Y ∈ S. U,I|=Y. We prove this by structural induction on formulas, keeping in mind that the cases for γ and δ are straightforward generalizations of those for α and β. base case: If Y is an atomic formula th ...
Computer Science 202a Homework #2, due in class
... 1. (15 points) For integers a and b, we say that a divides b if and only if there exists an integer d such that b = d · a. Express each of the following false statements about the domain of integers as a closed predicate logic formula, and give a counterexample to show that it is false. Use the bina ...
... 1. (15 points) For integers a and b, we say that a divides b if and only if there exists an integer d such that b = d · a. Express each of the following false statements about the domain of integers as a closed predicate logic formula, and give a counterexample to show that it is false. Use the bina ...
The Decision Problem for Standard Classes
... We say that a class K of formulas is decidable if both satisfiability and finite satisfiability (that is, satisfiability in a finite model) are decidable for formulas in K. K is conservative [8] if there exists an algorithm a. '> a' which associates a formula a' E K with each formula a in such a way ...
... We say that a class K of formulas is decidable if both satisfiability and finite satisfiability (that is, satisfiability in a finite model) are decidable for formulas in K. K is conservative [8] if there exists an algorithm a. '> a' which associates a formula a' E K with each formula a in such a way ...
CS3234 Logic and Formal Systems
... 6 D The problem of finding a validity preserving translation from propositional logic formulas to predicate logic sentences is undecidable. ...
... 6 D The problem of finding a validity preserving translation from propositional logic formulas to predicate logic sentences is undecidable. ...
Completeness through Flatness in Two
... In section 5 we pay special attention to the well-ordered flows of time and in particular, to the flow of time ω of the natural numbers. There are two reasons to do so: first of all, for these structures we can prove a completeness result for flat validity of a system without any non-orthodox deriva ...
... In section 5 we pay special attention to the well-ordered flows of time and in particular, to the flow of time ω of the natural numbers. There are two reasons to do so: first of all, for these structures we can prove a completeness result for flat validity of a system without any non-orthodox deriva ...
(draft)
... In 1968, Mathematician William Howard, building on work by Haskel Curry, identified a one-to-one relationship between propositional formulas and logical proofs to types and programs respectively. More genreally, it was noticed that logical ideas have computational significance. This idea became known, ...
... In 1968, Mathematician William Howard, building on work by Haskel Curry, identified a one-to-one relationship between propositional formulas and logical proofs to types and programs respectively. More genreally, it was noticed that logical ideas have computational significance. This idea became known, ...
Predicate Languages - Computer Science, Stony Brook University
... Observe, that once the set of propositional connectives is fixed, the predicate language is determined by the sets P, F and C. We use the notation L(P, F, C) for the predicate language L determined by P, F and C. If there is no danger of confusion, we may abbreviate L(P, F, C) to just L. If for som ...
... Observe, that once the set of propositional connectives is fixed, the predicate language is determined by the sets P, F and C. We use the notation L(P, F, C) for the predicate language L determined by P, F and C. If there is no danger of confusion, we may abbreviate L(P, F, C) to just L. If for som ...
Lesson 2
... • The simplest logical system. It analyzes a way of composing a complex sentence (proposition) from elementary propositions by means of logical connectives. • What is a proposition? A proposition (sentence) is a statement that can be said to be true or false. • The Two-Value Principle – tercium non ...
... • The simplest logical system. It analyzes a way of composing a complex sentence (proposition) from elementary propositions by means of logical connectives. • What is a proposition? A proposition (sentence) is a statement that can be said to be true or false. • The Two-Value Principle – tercium non ...
Automata theory
... As usual, variables within the scope of an existential quantifier are bounded, and otherwise free. A formula without free variables is a sentence. Sentences of FO(Σ) are interpreted on words over Σ. For instance, ∀x Qa (x) is true for the word aa, but false for word ab. Formulas with free variables ...
... As usual, variables within the scope of an existential quantifier are bounded, and otherwise free. A formula without free variables is a sentence. Sentences of FO(Σ) are interpreted on words over Σ. For instance, ∀x Qa (x) is true for the word aa, but false for word ab. Formulas with free variables ...
Logic, deontic. The study of principles of reasoning pertaining to
... interpretation according to which ~A is true at w exactly when A is true in all worlds "deontically accessible" from w, i.e., all worlds in which the all obligations of w are fulfilled. Much of the contemporary work in deontic logic has been inspired by the deontic paradoxes, a collection of puzzle ...
... interpretation according to which ~A is true at w exactly when A is true in all worlds "deontically accessible" from w, i.e., all worlds in which the all obligations of w are fulfilled. Much of the contemporary work in deontic logic has been inspired by the deontic paradoxes, a collection of puzzle ...
A Proof of Nominalism. An Exercise in Successful
... faith among logicians. It was what prevented Tarski from formulating a truth definition for a first-order language in the same language, as is shown in Hintikka and Sandu (1999). It might also be at the bottom of Zermelo’s unfortunate construal of the axiom of choice as a non-logical, mathematical a ...
... faith among logicians. It was what prevented Tarski from formulating a truth definition for a first-order language in the same language, as is shown in Hintikka and Sandu (1999). It might also be at the bottom of Zermelo’s unfortunate construal of the axiom of choice as a non-logical, mathematical a ...