3463: Mathematical Logic
... is applied to any configuration of the form αpaβ, or possibly αp if a is the blank symbol, and yields αbqβ. There are a few more cases to be considered for quintuples pabLq, but it is all quite simple. (1.7) Lemma If M is a Turing machine with initial state q0 , and x is an input string, then there ...
... is applied to any configuration of the form αpaβ, or possibly αp if a is the blank symbol, and yields αbqβ. There are a few more cases to be considered for quintuples pabLq, but it is all quite simple. (1.7) Lemma If M is a Turing machine with initial state q0 , and x is an input string, then there ...
Topological aspects of real-valued logic
... Theorem 1.0.3. Let S be a two-sorted metric signature, and let L be a countable fragment of Lω1 ,ω (S). Let T be an L-theory and let M = h M, V, . . . i be a model of T where M has density κ and V has density λ, with κ > λ ≥ ℵ0 . Then there is a model N = h N, W, . . . i ≡L M with N of density ℵ1 an ...
... Theorem 1.0.3. Let S be a two-sorted metric signature, and let L be a countable fragment of Lω1 ,ω (S). Let T be an L-theory and let M = h M, V, . . . i be a model of T where M has density κ and V has density λ, with κ > λ ≥ ℵ0 . Then there is a model N = h N, W, . . . i ≡L M with N of density ℵ1 an ...
pdf format
... Theorem 2 (Integer Induction) Let ϕ(x, y1, . . . , yk ) be a first-order formula. Let y1 , . . . , yk be fixed sets. Suppose that ϕ(0, ~y) is true and that (∀x ∈ ω)(ϕ(x, ~y) → ϕ(S(x), ~y)). Then ϕ(x, ~y) is true for all x ∈ ω . Definition A set x is transitive if every member of x is a subset of x. ...
... Theorem 2 (Integer Induction) Let ϕ(x, y1, . . . , yk ) be a first-order formula. Let y1 , . . . , yk be fixed sets. Suppose that ϕ(0, ~y) is true and that (∀x ∈ ω)(ϕ(x, ~y) → ϕ(S(x), ~y)). Then ϕ(x, ~y) is true for all x ∈ ω . Definition A set x is transitive if every member of x is a subset of x. ...
Algebraizing Hybrid Logic - Institute for Logic, Language and
... Remark 2.4.2. The systematic tableau construction is defined in [5]. Roughly speaking, this construction is needed in order to prove strong completeness. Theorem 2.4.1. ([5]) Any consistent set of formulas in countable language is satisfiable in a countable standard model. The following theorem is t ...
... Remark 2.4.2. The systematic tableau construction is defined in [5]. Roughly speaking, this construction is needed in order to prove strong completeness. Theorem 2.4.1. ([5]) Any consistent set of formulas in countable language is satisfiable in a countable standard model. The following theorem is t ...
.pdf
... if all of its finite subsets are. We gave three proofs for that: one using tableau proofs and König’s lemma, one giving a direct construction of a Hintikka set, and one using Lindenbaum’s construction, extending S to a maximally consistent set, which turned out to be a proof set. In first-order log ...
... if all of its finite subsets are. We gave three proofs for that: one using tableau proofs and König’s lemma, one giving a direct construction of a Hintikka set, and one using Lindenbaum’s construction, extending S to a maximally consistent set, which turned out to be a proof set. In first-order log ...
ppt - Purdue College of Engineering
... • A tautology is a formula that is true in every model. (also called a theorem) – for example, (A A) is a tautology – What about (AB)(AB)? – Look at tautological equivalences on pg. 8 of text ...
... • A tautology is a formula that is true in every model. (also called a theorem) – for example, (A A) is a tautology – What about (AB)(AB)? – Look at tautological equivalences on pg. 8 of text ...
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 ...
Natural Deduction Calculus for Quantified Propositional Linear
... comparing to PLTL is their ability to ”count”, for example, to express that some property occurs at every even moment of time [Wolper (1981)]. Nevertheless, each of these logics uses its own specific syntax and it makes sense to consider how easy these logics can be used in specification. We believe ...
... comparing to PLTL is their ability to ”count”, for example, to express that some property occurs at every even moment of time [Wolper (1981)]. Nevertheless, each of these logics uses its own specific syntax and it makes sense to consider how easy these logics can be used in specification. We believe ...
on partially conservative sentences and interpretability
... every \¡i e F. In §1 this concept for T - 2°+, and Tl°+, is investigated. In §2 results from §1 are applied to interpretability in theories containing arithmetic. ...
... every \¡i e F. In §1 this concept for T - 2°+, and Tl°+, is investigated. In §2 results from §1 are applied to interpretability in theories containing arithmetic. ...
RR-01-02
... knowledge in the topic are invited to consult [1, 15, 14]. Any concept not explicitly defined in this paper refers to [1]. The research task in this paper is precisely described as follows, with some preliminaries. Definition 2.1 (Preference Logic) [20, pages 73-77] Let L be a standard logic, i.e. a ...
... knowledge in the topic are invited to consult [1, 15, 14]. Any concept not explicitly defined in this paper refers to [1]. The research task in this paper is precisely described as follows, with some preliminaries. Definition 2.1 (Preference Logic) [20, pages 73-77] Let L be a standard logic, i.e. a ...
handout
... 3 Natural Deduction (Gentzen, 1943) Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgments of the form φ1 , . . . , φn ⊢ φ, which means that φ can be derived from the assumptions φ1 , . . . , φn . If ⊢ φ without assumptions, then φ is a theorem of intu ...
... 3 Natural Deduction (Gentzen, 1943) Intuitionistic logic uses a sequent calculus to derive the truth of formulas. Assertions are judgments of the form φ1 , . . . , φn ⊢ φ, which means that φ can be derived from the assumptions φ1 , . . . , φn . If ⊢ φ without assumptions, then φ is a theorem of intu ...
A Proof of Nominalism. An Exercise in Successful
... known as type theory as a many-sorted first-order theory, each different type serving as one of the “sorts”. One can try to interpret the logics of Frege and of Russell and Whitehead in this way. The attempt fails (systematically if not historically) because there are higher-order logical principles ...
... known as type theory as a many-sorted first-order theory, each different type serving as one of the “sorts”. One can try to interpret the logics of Frege and of Russell and Whitehead in this way. The attempt fails (systematically if not historically) because there are higher-order logical principles ...
Restricted notions of provability by induction
... to develop algorithms that find proofs by induction and to implement them efficiently. This subject is characterized by a great variety of different methods (and systems implementing these methods), for example, rippling [6], theory exploration [9], integration into a superposition prover [18, 24], ...
... to develop algorithms that find proofs by induction and to implement them efficiently. This subject is characterized by a great variety of different methods (and systems implementing these methods), for example, rippling [6], theory exploration [9], integration into a superposition prover [18, 24], ...
proceedings version
... We relate the language of equilibrium logic to our bimodal language by means of a translation tr. The main clause of the translation is: tr(ϕ ⇒ ψ) = (tr(ϕ) → tr(ψ)) ∧ [T](tr(ϕ) → tr(ψ)) A first attempt to relate equilibrium logic to modal logic in the style of the present approach was presented in [ ...
... We relate the language of equilibrium logic to our bimodal language by means of a translation tr. The main clause of the translation is: tr(ϕ ⇒ ψ) = (tr(ϕ) → tr(ψ)) ∧ [T](tr(ϕ) → tr(ψ)) A first attempt to relate equilibrium logic to modal logic in the style of the present approach was presented in [ ...
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 ...
THE HITCHHIKER`S GUIDE TO THE INCOMPLETENESS
... Now we are powerful enough to deal with elementary arithmetic, but still something is missing. Let us try to write out the following theorem as a formal sentence: Theorem 2. There is an even number which is not the sum of two prime numbers. What is a prime? A number which is no divisible by any numb ...
... Now we are powerful enough to deal with elementary arithmetic, but still something is missing. Let us try to write out the following theorem as a formal sentence: Theorem 2. There is an even number which is not the sum of two prime numbers. What is a prime? A number which is no divisible by any numb ...
Hilbert Calculus
... Assume S ∪ {F } ⊢ G. Proof by induction on the derivation (length): Axiom/Hypothesis: G is instance of an axiom or G ∈ S ∪ {F }. If F = G use example of derivation to prove S ⊢ F → F . Otherwise S ⊢ G and by Axiom (1) S ⊢ G → (F → G). By Modus Ponens we get S ⊢ F → G. Modus Ponens: Then S ∪ {F } ⊢ G ...
... Assume S ∪ {F } ⊢ G. Proof by induction on the derivation (length): Axiom/Hypothesis: G is instance of an axiom or G ∈ S ∪ {F }. If F = G use example of derivation to prove S ⊢ F → F . Otherwise S ⊢ G and by Axiom (1) S ⊢ G → (F → G). By Modus Ponens we get S ⊢ F → G. Modus Ponens: Then S ∪ {F } ⊢ G ...
Classical and Intuitionistic Models of Arithmetic
... 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(c̄) → χ(c̄), we have nothi ...
... 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(c̄) → χ(c̄), we have nothi ...
Gödel incompleteness theorems and the limits of their applicability. I
... the representatives of Hilbert’s school as the central problem of mathematical logic. However, it follows from Gödel’s second theorem that it is impossible to formalize the ‘finitary tools’ that are able to establish the consistency of mathematics even in the framework of a very strong system P .2 ...
... the representatives of Hilbert’s school as the central problem of mathematical logic. However, it follows from Gödel’s second theorem that it is impossible to formalize the ‘finitary tools’ that are able to establish the consistency of mathematics even in the framework of a very strong system P .2 ...
Theories of arithmetics in finite models
... formulas in this vocabulary. Similarly, if X is a set of predicates and functions (of known arities) we write FX to denote the set of first order formulas with predicates and functions from X. E.g. F{+} is the set of formulas with addition. Moreover, we always assume to have equality in our language ...
... formulas in this vocabulary. Similarly, if X is a set of predicates and functions (of known arities) we write FX to denote the set of first order formulas with predicates and functions from X. E.g. F{+} is the set of formulas with addition. Moreover, we always assume to have equality in our language ...
na.
... Proof: Let .A be the set of all D's tor which there is a 9 -consistent and complete theory T such that D:: Cl (m). Notice that here we consider Cl (tJ) not a -(a.). One may see that ..A •s . is a pseudo-model for m, where s and e are defined below and e· is the retiexlve and transitive closure of B ...
... Proof: Let .A be the set of all D's tor which there is a 9 -consistent and complete theory T such that D:: Cl (m). Notice that here we consider Cl (tJ) not a -(a.). One may see that ..A •s . is a pseudo-model for m, where s and e are defined below and e· is the retiexlve and transitive closure of B ...
in every real in a class of reals is - Math Berkeley
... merely a version of …rst order logic gotten by adding on predicates for N and M , this logic can be turned into a much stronger one (N -logic) by requiring that all models have their …rst sort (with some functions and relations on it as given in the structure) isomorphic to some given countable …rst ...
... merely a version of …rst order logic gotten by adding on predicates for N and M , this logic can be turned into a much stronger one (N -logic) by requiring that all models have their …rst sort (with some functions and relations on it as given in the structure) isomorphic to some given countable …rst ...
Propositional Dynamic Logic of Regular Programs*+
... “contentless” assertions since they do not depend on the meanings of the basic assertions or the basic programs. In this section, we show that the complexity of the validity problem for PDL is in co-NTIME(cn) f or some c, where 71is the size of the formula being tested. (That is, the complement of t ...
... “contentless” assertions since they do not depend on the meanings of the basic assertions or the basic programs. In this section, we show that the complexity of the validity problem for PDL is in co-NTIME(cn) f or some c, where 71is the size of the formula being tested. (That is, the complement of t ...
connections to higher type Recursion Theory, Proof-Theory
... As Mathematics is relevant when it is both beautiful and applicable, I think that the founders of λcalculus and related systems should be happy with all of this. As it should be clear by now, the focus of this lecture will be more on the interface of λ-calculus with other theories than on its "pure ...
... As Mathematics is relevant when it is both beautiful and applicable, I think that the founders of λcalculus and related systems should be happy with all of this. As it should be clear by now, the focus of this lecture will be more on the interface of λ-calculus with other theories than on its "pure ...