First-Order Queries over One Unary Function
... structures [FG01] (see also [FFG02]). A quasi-unary signature consists of one unary function and any number of monadic predicates. First-order logic over quasi-unary structures has been often studied and some of its aspects are quite well understood. The satisfiability problem for this kind of first ...
... structures [FG01] (see also [FFG02]). A quasi-unary signature consists of one unary function and any number of monadic predicates. First-order logic over quasi-unary structures has been often studied and some of its aspects are quite well understood. The satisfiability problem for this kind of first ...
The Logic of Provability
... Classical first-order arithmetic with induction; also called arithmetic or PA. More formally, we take the signature of PA to have ‘0’ as a constant and ‘+’, ‘·’, and ‘<’ as binary function symbols; PA is then the theory axiomatized by the following: • ∀x(sx 6= 0) • ∀x, y(sx = sy → x = y) • For every ...
... Classical first-order arithmetic with induction; also called arithmetic or PA. More formally, we take the signature of PA to have ‘0’ as a constant and ‘+’, ‘·’, and ‘<’ as binary function symbols; PA is then the theory axiomatized by the following: • ∀x(sx 6= 0) • ∀x, y(sx = sy → x = y) • For every ...
A Simple Tableau System for the Logic of Elsewhere
... that u ≡M v iff for all P ∈ φ0 , u ∈ V (P ) iff v ∈ V (P ). We write |u|V to denote the equivalence classes with respect to ≡M . Observe that for all A ∈ for(φ0 ), if |u|V = |v|V then M, u |= A iff M, v |= A. Proposition 5. Let M = (W, V ) be a model and u, u0 ∈ W such that |u|V = |u0 |V . Take any ...
... that u ≡M v iff for all P ∈ φ0 , u ∈ V (P ) iff v ∈ V (P ). We write |u|V to denote the equivalence classes with respect to ≡M . Observe that for all A ∈ for(φ0 ), if |u|V = |v|V then M, u |= A iff M, v |= A. Proposition 5. Let M = (W, V ) be a model and u, u0 ∈ W such that |u|V = |u0 |V . Take any ...
Document
... determine a set. As a result, most attempts at resolving the paradox have concentrated on various ways of restricting the principles governing set existence found within naive set theory, particularly the so-called Comprehension (or Abstraction) axiom. This axiom in effect states that any propositio ...
... determine a set. As a result, most attempts at resolving the paradox have concentrated on various ways of restricting the principles governing set existence found within naive set theory, particularly the so-called Comprehension (or Abstraction) axiom. This axiom in effect states that any propositio ...
Natural deduction for predicate logic
... This suggests that to prove a formula of the form ∀xφ, we can prove φ with some arbitrary but fresh variable x0 substituted for x. That is, we want to prove the formula φ[x0 /x]. On the previous slide, we used n as a fresh variable, but in our formal proofs, we adopt the convention of using subscri ...
... This suggests that to prove a formula of the form ∀xφ, we can prove φ with some arbitrary but fresh variable x0 substituted for x. That is, we want to prove the formula φ[x0 /x]. On the previous slide, we used n as a fresh variable, but in our formal proofs, we adopt the convention of using subscri ...
Introducing Quantified Cuts in Logic with Equality
... The method [5] for propositional logic is shown to never increase the size of proofs more than polynomially. Another approach to the compression of firstorder proofs by introduction of definitions for abbreviating terms is [19]. There is a large body of work on the generation of non-analytic formulas ...
... The method [5] for propositional logic is shown to never increase the size of proofs more than polynomially. Another approach to the compression of firstorder proofs by introduction of definitions for abbreviating terms is [19]. There is a large body of work on the generation of non-analytic formulas ...
PDF
... We can imagine a simpler principle if we work in a sub-logic in which we do not keep track of all the evidence. This kind of sub-logic is called classical logic, and we will examine it more later in the course. {Least Number Principle:} ∃x : N.A(x) ⇒ ∃y : N.(A(y)&∀z : N.z < y ⇒∼ A(z)). There are man ...
... We can imagine a simpler principle if we work in a sub-logic in which we do not keep track of all the evidence. This kind of sub-logic is called classical logic, and we will examine it more later in the course. {Least Number Principle:} ∃x : N.A(x) ⇒ ∃y : N.(A(y)&∀z : N.z < y ⇒∼ A(z)). There are man ...
The unintended interpretations of intuitionistic logic
... where x is not free in A(0) in the rule above. The system HA is strong enough to prove essentially all number theoretic results we find in any text on number theory. The main exceptions involve theorems related to incompleteness proofs of P A and statements that are true in P A only because of their ...
... where x is not free in A(0) in the rule above. The system HA is strong enough to prove essentially all number theoretic results we find in any text on number theory. The main exceptions involve theorems related to incompleteness proofs of P A and statements that are true in P A only because of their ...
Cylindric Modal Logic - Homepages of UvA/FNWI staff
... motivation for adopting this particular restriction will be given below. For α < ω, we get a logic with finitely many variables. Such logics have been studied in the literature, for purely logical reasons (Henkin [15], Henkin, Monk & Tarski [16], Tarski-Givant [41], Sain [37], Monk [24]) or because ...
... motivation for adopting this particular restriction will be given below. For α < ω, we get a logic with finitely many variables. Such logics have been studied in the literature, for purely logical reasons (Henkin [15], Henkin, Monk & Tarski [16], Tarski-Givant [41], Sain [37], Monk [24]) or because ...
A modal perspective on monadic second
... (found in [18]) is of no particular importance for the present paper, as we give a virtually self-contained exposition of all our results. As a by-product of our investigations we obtain a simple, effective procedure (inspired by the approach of ten Cate [8]) that translates MSOsentences to equivale ...
... (found in [18]) is of no particular importance for the present paper, as we give a virtually self-contained exposition of all our results. As a by-product of our investigations we obtain a simple, effective procedure (inspired by the approach of ten Cate [8]) that translates MSOsentences to equivale ...
Reaching transparent truth
... Truth is a generalization device insofar as it allows us to report that the conjunction of a set of sentences, or their disjunction, holds, without having to enumerate all sentences in the set, and even without having to know what sentences are in the set. For instance, if I accept the sentence (1) ...
... Truth is a generalization device insofar as it allows us to report that the conjunction of a set of sentences, or their disjunction, holds, without having to enumerate all sentences in the set, and even without having to know what sentences are in the set. For instance, if I accept the sentence (1) ...
2. First Order Logic 2.1. Expressions. Definition 2.1. A language L
... consists of prenex formulas then there is a deduction of Γ ⇒ Σ in which the quantifier rules (R∀, R∃, L∀, L∃) appear below every other inference rule. We begin with a cut-free deduction of Γ ⇒ Σ. The idea is to take the (or a, since the proof might branch) bottom-most quantifier rule in the proof an ...
... consists of prenex formulas then there is a deduction of Γ ⇒ Σ in which the quantifier rules (R∀, R∃, L∀, L∃) appear below every other inference rule. We begin with a cut-free deduction of Γ ⇒ Σ. The idea is to take the (or a, since the proof might branch) bottom-most quantifier rule in the proof an ...
Gödel on Conceptual Realism and Mathematical Intuition
... themselves on us allows us to deduce that they must exist. The system of basic laws that need to be conserved, which makes the expansion a forced one, is unique. We have a strong requirement that the concept of the number of a set should be independent of the properties of its elements. When we have ...
... themselves on us allows us to deduce that they must exist. The system of basic laws that need to be conserved, which makes the expansion a forced one, is unique. We have a strong requirement that the concept of the number of a set should be independent of the properties of its elements. When we have ...
Intuitionistic Logic
... that some proposition has as yet no proof, but it is not excluded that eventually a proof may be found. In formal logic there is a similar distinction: 6` A and ` ¬A. The Brouwerian counter examples are similar to the first case, strong counterexamples cannot always be expected. For example, althoug ...
... that some proposition has as yet no proof, but it is not excluded that eventually a proof may be found. In formal logic there is a similar distinction: 6` A and ` ¬A. The Brouwerian counter examples are similar to the first case, strong counterexamples cannot always be expected. For example, althoug ...
PROPERTIES PRESERVED UNDER ALGEBRAIC
... theory of fields that holds for the one-element "field" is preserved under homomorphism ; a moment's reflection shows that this jibes with Theorem H, using the appropriate interpretation of "equivalence" of sentences. Tarski ordinarily considers more general systems that possess relations other than ...
... theory of fields that holds for the one-element "field" is preserved under homomorphism ; a moment's reflection shows that this jibes with Theorem H, using the appropriate interpretation of "equivalence" of sentences. Tarski ordinarily considers more general systems that possess relations other than ...
AppA - txstateprojects
... abstract machines and problems which they are able to solve. It is closely related to formal language theory as the automata are often classified by the class of formal languages they are able to recognize. – An abstract machine, also called an abstract computer, is a theoretical model of a computer ...
... abstract machines and problems which they are able to solve. It is closely related to formal language theory as the automata are often classified by the class of formal languages they are able to recognize. – An abstract machine, also called an abstract computer, is a theoretical model of a computer ...
Points, lines and diamonds: a two-sorted modal logic for projective
... Compared to temporal logics, modal logics of space have received very little attention. I can see two reasons for this. First, temporal logic has its roots in the semantics of natural language; here, the notion of tense naturally leads to an extension of classical logics with temporal modal operator ...
... Compared to temporal logics, modal logics of space have received very little attention. I can see two reasons for this. First, temporal logic has its roots in the semantics of natural language; here, the notion of tense naturally leads to an extension of classical logics with temporal modal operator ...
Intuitionistic modal logic made explicit
... We present the intuitionistic modal logic iS4. We will start with introducing the language LI of iS4. For our purpose, we will only consider the -modality but not the ♦-modality. Definition 2.1 (Intuitionistic modal language). We assume a countable set Prop of atomic propositions. The set of formul ...
... We present the intuitionistic modal logic iS4. We will start with introducing the language LI of iS4. For our purpose, we will only consider the -modality but not the ♦-modality. Definition 2.1 (Intuitionistic modal language). We assume a countable set Prop of atomic propositions. The set of formul ...
slides
... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional and (mostly) syntax-directed axioms and inference rules ...
... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional and (mostly) syntax-directed axioms and inference rules ...
Notions of locality and their logical characterizations over nite
... Let be an isomorphism type of a structure in the language 1 ( extended with one constant). A point a in a structure A d-realizes , written as d (A; a) = , if NdA (a) is of isomorphism type . By #d [A; ] we denote the number of elements of A which d-realize , that is, the cardinality o ...
... Let be an isomorphism type of a structure in the language 1 ( extended with one constant). A point a in a structure A d-realizes , written as d (A; a) = , if NdA (a) is of isomorphism type . By #d [A; ] we denote the number of elements of A which d-realize , that is, the cardinality o ...
A Proof of Cut-Elimination Theorem for U Logic.
... Basic Propositional Logic, BPL, was invented by Albert Visser in 1981 [5]. He wanted to interpret implication as formal provability. To protect his system against the liar paradox, modus ponens is weakened. His axiomatization of BPL uses natural deduction[3, p. 8]. The first sequent calculus for BPL ...
... Basic Propositional Logic, BPL, was invented by Albert Visser in 1981 [5]. He wanted to interpret implication as formal provability. To protect his system against the liar paradox, modus ponens is weakened. His axiomatization of BPL uses natural deduction[3, p. 8]. The first sequent calculus for BPL ...
CHAPTER 1 The main subject of Mathematical Logic is
... reasons for this choice are twofold. First, as the name says this is a natural notion of formal proof, which means that the way proofs are represented corresponds very much to the way a careful mathematician writing out all details of an argument would proceed anyway. Second, formal proofs in natura ...
... reasons for this choice are twofold. First, as the name says this is a natural notion of formal proof, which means that the way proofs are represented corresponds very much to the way a careful mathematician writing out all details of an argument would proceed anyway. Second, formal proofs in natura ...
Logic, deontic. The study of principles of reasoning pertaining to
... Post formulated a technically advantageous system in which ºukasiewicz's negation is replaced by a "cyclic" negation--the truth values are 0,1,..,m and the truth value of ¬A is 0 if the truth value of A is m and it is 1+the truth value of A otherwise. Post's negation and disjunction are truth functi ...
... Post formulated a technically advantageous system in which ºukasiewicz's negation is replaced by a "cyclic" negation--the truth values are 0,1,..,m and the truth value of ¬A is 0 if the truth value of A is m and it is 1+the truth value of A otherwise. Post's negation and disjunction are truth functi ...
Distributed Knowledge
... structure, in the sense that their generated submodels are isomorphic (in particular, this means that the same sentences of LD are true in the two models). The de nition 1.4 of distributed knowledge, then, only makes sense if we take a view on Kripke models in which the di erence between isomorphic ...
... structure, in the sense that their generated submodels are isomorphic (in particular, this means that the same sentences of LD are true in the two models). The de nition 1.4 of distributed knowledge, then, only makes sense if we take a view on Kripke models in which the di erence between isomorphic ...
Ordered Groups: A Case Study In Reverse Mathematics 1 Introduction
... gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. ...
... gives the following list of areas which can be analyzed by reverse mathematics: number theory, geometry, calculus, differential equations, real and complex analysis, combinatorics, countable algebra, separable Banach spaces, computability theory, and the topology of complete separable metric spaces. ...