Verification and Specification of Concurrent Programs
... a terminating execution is represented by an infinite behavior in which the final state is repeated.) The meaning [[P ]] of a predicate P is a Boolean-valued function on program states. For example, [[x + 1 > y]](s) equals true iff one plus the value of x in state s is greater than the value of y in st ...
... a terminating execution is represented by an infinite behavior in which the final state is repeated.) The meaning [[P ]] of a predicate P is a Boolean-valued function on program states. For example, [[x + 1 > y]](s) equals true iff one plus the value of x in state s is greater than the value of y in st ...
Non-classical metatheory for non-classical logics
... certain semantic notions such as a truth predicate or a satisfaction relation. However one might object that this is far from enough to do any serious semantics. For instance, one does not have the resources to theorise about the intended interpretation at the subsentential level: to reason about th ...
... certain semantic notions such as a truth predicate or a satisfaction relation. However one might object that this is far from enough to do any serious semantics. For instance, one does not have the resources to theorise about the intended interpretation at the subsentential level: to reason about th ...
Modal logic and the approximation induction principle
... in terms of observations. That is, a process semantics is captured by means of a sublogic of HennessyMilner logic; two states in an LTS are equivalent if and only if they make true exactly the same formulas in this sublogic. In particular, Hennessy-Milner logic itself characterizes bisimulation equi ...
... in terms of observations. That is, a process semantics is captured by means of a sublogic of HennessyMilner logic; two states in an LTS are equivalent if and only if they make true exactly the same formulas in this sublogic. In particular, Hennessy-Milner logic itself characterizes bisimulation equi ...
A game semantics for proof search: Preliminary results - LIX
... where p(t1 , . . . , tn ) is false in M then Γ cannot reduce to {}. In Figure 1, the positive and negative translations of neutral expressions into logic are provided. Notice that the only logical connectives in the range of [·]+ are positive (synchronous), while those in the range of [·]− are negat ...
... where p(t1 , . . . , tn ) is false in M then Γ cannot reduce to {}. In Figure 1, the positive and negative translations of neutral expressions into logic are provided. Notice that the only logical connectives in the range of [·]+ are positive (synchronous), while those in the range of [·]− are negat ...
Glivenko sequent classes in the light of structural proof theory
... proved that if A is a formula without implications and consists of formulas that contain disjunctions and falsity only negatively and implications only positively, then classical derivability yields minimal derivability. Instead of sequent calculus, their approach uses natural deduction that is priv ...
... proved that if A is a formula without implications and consists of formulas that contain disjunctions and falsity only negatively and implications only positively, then classical derivability yields minimal derivability. Instead of sequent calculus, their approach uses natural deduction that is priv ...
Judgment and consequence relations
... In this paper I look into the standard definitions of logical consequence and show that they can be unified under a common scheme. Standardly, consequence relations are defined via the preservation of truth. Here I propose to generalize this as follows: consequence is the preservation of a (truth re ...
... In this paper I look into the standard definitions of logical consequence and show that they can be unified under a common scheme. Standardly, consequence relations are defined via the preservation of truth. Here I propose to generalize this as follows: consequence is the preservation of a (truth re ...
Predicate Logic
... If P(x) denotes “x is an undergraduate student” and U is {Enorlled Students in COMPSCI 230}, then x P(x) is TRUE. If P(x) denotes “x > 0” and U is the integers, then x P(x) is FALSE. If P(x) denotes “x > 0” and U is the positive integers, then x P(x) is TRUE. If P(x) denotes “x is even” and U is ...
... If P(x) denotes “x is an undergraduate student” and U is {Enorlled Students in COMPSCI 230}, then x P(x) is TRUE. If P(x) denotes “x > 0” and U is the integers, then x P(x) is FALSE. If P(x) denotes “x > 0” and U is the positive integers, then x P(x) is TRUE. If P(x) denotes “x is even” and U is ...
Hoare Logic, Weakest Liberal Preconditions
... • There is only one data type: integers. They will have their mathematical meaning, that is, they are unbounded, unlike machine integers. • The relational operators return an integer: 0 meaning “false” and −1 meaning “true”. • The condition in if and while statements interprets 0 as “false” and non- ...
... • There is only one data type: integers. They will have their mathematical meaning, that is, they are unbounded, unlike machine integers. • The relational operators return an integer: 0 meaning “false” and −1 meaning “true”. • The condition in if and while statements interprets 0 as “false” and non- ...
First-order possibility models and finitary
... A propositional modal logic has a finitary completeness proof if it has a canonical model all of whose possibilities are finitely specified in this sense. This was one of Humberstone’s original motivations for considering possibility models. For many normal modal logics extending K with standard axi ...
... A propositional modal logic has a finitary completeness proof if it has a canonical model all of whose possibilities are finitely specified in this sense. This was one of Humberstone’s original motivations for considering possibility models. For many normal modal logics extending K with standard axi ...
Document
... An argument has the form ∀x (P(x ) → Q(x )), where 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) ...
... An argument has the form ∀x (P(x ) → Q(x )), where 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) ...
Propositional Logic: Normal Forms
... P1 ∧ . . . ∧ Pki → false of φ such that all Pi s are marked. If φ is satisfiable, by Corollary 1, this means (true → false) = false whenever φ is true. This is impossible, so φ is unsatisfiable. Reductio ad absurdum. SAT: if false is NOT marked, let ν be an interpretation that assign true to all mar ...
... P1 ∧ . . . ∧ Pki → false of φ such that all Pi s are marked. If φ is satisfiable, by Corollary 1, this means (true → false) = false whenever φ is true. This is impossible, so φ is unsatisfiable. Reductio ad absurdum. SAT: if false is NOT marked, let ν be an interpretation that assign true to all mar ...
Predicate logic
... Proving universal statements Claim: For any integers a and b, if a and b are odd, then ab is also odd. Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m ...
... Proving universal statements Claim: For any integers a and b, if a and b are odd, then ab is also odd. Definition: integer a is odd iff a = 2m + 1 for some integer m Let a, b ∈ Z s.t. a and b are odd. Then by definition of odd a = 2m + 1.m ∈ Z and b = 2n + 1.n ∈ Z So ab = (2m + 1)(2n + 1) = 4mn + 2m ...
Clausal Logic and Logic Programming in Algebraic Domains*
... the distinction, at least temporarily, between syntax and semantics. It is customary to describe a pta by a string of literals: variables or their complements. For example, abcd represents the pta which maps a and b to 1, and c and d to 0 (with all other variables mapping to ⊥). The set of all ptas ...
... the distinction, at least temporarily, between syntax and semantics. It is customary to describe a pta by a string of literals: variables or their complements. For example, abcd represents the pta which maps a and b to 1, and c and d to 0 (with all other variables mapping to ⊥). The set of all ptas ...
First-Order Logic, Second-Order Logic, and Completeness
... that a feature of SOL gets lost when a Henkin semantics is adopted; the very feature that attracts many of those who are interested in SOL to it.13 SOL with standard semantics allows for categorical axiomatizations of certain mathematical theories, such as arithmetic or real analysis. A mathematical ...
... that a feature of SOL gets lost when a Henkin semantics is adopted; the very feature that attracts many of those who are interested in SOL to it.13 SOL with standard semantics allows for categorical axiomatizations of certain mathematical theories, such as arithmetic or real analysis. A mathematical ...
this PDF file
... situations (cf. Remark 4 below). Remark 1: 1. The axiom m.p is sometimes labelled “pseudo-modus/ponens” in order to distinguish it from the rule modus ponens. 2. The variable-sharing property (vsp) is the following: A logic S has the vsp if in any theorem of S of the form A → B, A and B share at lea ...
... situations (cf. Remark 4 below). Remark 1: 1. The axiom m.p is sometimes labelled “pseudo-modus/ponens” in order to distinguish it from the rule modus ponens. 2. The variable-sharing property (vsp) is the following: A logic S has the vsp if in any theorem of S of the form A → B, A and B share at lea ...
What is "formal logic"?
... we can present a proof system where axioms and rules are schemes, then the substitution theorem appears rather as a axiom, expressing the formal character of logic. In fact in the 1950s, the substitution theorem was explicitly stated as an axiom in the abstract definition of logic by Los and Suszko ...
... we can present a proof system where axioms and rules are schemes, then the substitution theorem appears rather as a axiom, expressing the formal character of logic. In fact in the 1950s, the substitution theorem was explicitly stated as an axiom in the abstract definition of logic by Los and Suszko ...
On Herbrand`s Theorem for Intuitionistic Logic
... to Sig denoting the constructed extension by eSig. For example, 1 ∨, 3 ⊃, and 5 ∀ are symbols of the extended signature. These left upper indices are used to distinguish connectives in different copies of the same formula, stemming from multiplicities, to encode impermutabilities. The notions of ter ...
... to Sig denoting the constructed extension by eSig. For example, 1 ∨, 3 ⊃, and 5 ∀ are symbols of the extended signature. These left upper indices are used to distinguish connectives in different copies of the same formula, stemming from multiplicities, to encode impermutabilities. The notions of ter ...
F - Teaching-WIKI
... • One of the simplest and most common logic – The core of (almost) all other logics ...
... • One of the simplest and most common logic – The core of (almost) all other logics ...
CHAPTER 1 The main subject of Mathematical Logic is
... In a given context we shall adopt the following convention. Once a formula has been introduced as A(x), i.e., A with a designated variable x, we write A(r) for A[x := r], and similarly with more variables. 1.1.3. Subformulas. Unless stated otherwise, the notion of subformula will be that defined by ...
... In a given context we shall adopt the following convention. Once a formula has been introduced as A(x), i.e., A with a designated variable x, we write A(r) for A[x := r], and similarly with more variables. 1.1.3. Subformulas. Unless stated otherwise, the notion of subformula will be that defined by ...
A Recursively Axiomatizable Subsystem of Levesque`s Logic of Only
... It is obvious that subjective sentences do not depend on the w in question, and objective sentences do not depend on the W chosen. So we can write W j= and w j= in these cases. (W; w) j= i (W; w) j= for all 2 . Some comments on the semantics de nition are in order. Levesque's models di er i ...
... It is obvious that subjective sentences do not depend on the w in question, and objective sentences do not depend on the W chosen. So we can write W j= and w j= in these cases. (W; w) j= i (W; w) j= for all 2 . Some comments on the semantics de nition are in order. Levesque's models di er i ...
Least and greatest fixed points in linear logic
... Exponentials As shown above, µMALL= can be encoded using exponentials and second-order quantifiers. But at first-order, exponentials and fixed points are incomparable. We could add exponentials in further work, but conjecture that the essential observations done in this work would stay the same. Non ...
... Exponentials As shown above, µMALL= can be encoded using exponentials and second-order quantifiers. But at first-order, exponentials and fixed points are incomparable. We could add exponentials in further work, but conjecture that the essential observations done in this work would stay the same. Non ...
INTRODUCTION TO LOGIC Natural Deduction
... becomes available: An argument is valid iff the conclusion can be derived from the premisses using the specified rules. The notion of proof can be precisely defined. In cases of disagreement, one can always break down an argument into elementary steps that are covered by these rules. The point is th ...
... becomes available: An argument is valid iff the conclusion can be derived from the premisses using the specified rules. The notion of proof can be precisely defined. In cases of disagreement, one can always break down an argument into elementary steps that are covered by these rules. The point is th ...
Answer Sets for Propositional Theories
... (i) Γ1 is strongly equivalent to Γ2 , (ii) Γ1 is equivalent to Γ2 in the logic of here-and-there, and (iii) for each set X of atoms, Γ1X is equivalent to Γ2X in classical logic. The equivalence between (i) and (ii) is a generalization of the main result of [Lifschitz et al., 2001], and it is an imme ...
... (i) Γ1 is strongly equivalent to Γ2 , (ii) Γ1 is equivalent to Γ2 in the logic of here-and-there, and (iii) for each set X of atoms, Γ1X is equivalent to Γ2X in classical logic. The equivalence between (i) and (ii) is a generalization of the main result of [Lifschitz et al., 2001], and it is an imme ...
Systems of modal logic - Department of Computing
... Systems of modal logic In common with most modern approaches, we will define systems of modal logic (‘modal logics’ or just ‘logics’ for short) in rather abstract terms — a system of modal logic is just a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ ...
... Systems of modal logic In common with most modern approaches, we will define systems of modal logic (‘modal logics’ or just ‘logics’ for short) in rather abstract terms — a system of modal logic is just a set of formulas satisfying certain closure conditions. A formula A is a theorem of the system Σ ...
Concept Hierarchies from a Logical Point of View
... logic is commonly seen as the most basic sort of logic – witness any textbook on logic. Conceptually, however, it seems rather awkward to regard attributes as propositions. If attributes are formalized within a logical language at all then the most natural way to do so is to represent them as monadi ...
... logic is commonly seen as the most basic sort of logic – witness any textbook on logic. Conceptually, however, it seems rather awkward to regard attributes as propositions. If attributes are formalized within a logical language at all then the most natural way to do so is to represent them as monadi ...