The Foundations
... (or fact) that the proposition is intended to represent occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now(the current situation), the statement It_is_raining is false (in the current situation). But if it were raining now, ...
... (or fact) that the proposition is intended to represent occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now(the current situation), the statement It_is_raining is false (in the current situation). But if it were raining now, ...
A pragmatic dialogic interpretation of bi
... identify, among the mathematical models of bi-intuitionism, those which may be regarded as its intended interpretations. The quest for an intended interpretation of a formal system often arises when several mathematical structures have been proposed to characterise an informal, perhaps vague notion ...
... identify, among the mathematical models of bi-intuitionism, those which may be regarded as its intended interpretations. The quest for an intended interpretation of a formal system often arises when several mathematical structures have been proposed to characterise an informal, perhaps vague notion ...
The Foundations
... (or fact) that the proposition is intended to represent occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now(the current situation), the statement It_is_raining is false (in the current situation). But if it were raining now, ...
... (or fact) that the proposition is intended to represent occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now(the current situation), the statement It_is_raining is false (in the current situation). But if it were raining now, ...
Constraint Logic Programming with Hereditary Harrop Formula
... of σ to every constraint of the set Γ. In the sequel, the notation F σ will also be used for other formulas F , not necessarily constraints. Note that the three conditions i), ii), iii) are meant as minimal requirements. In particular, the availability of the equality symbol ≈ is granted in any cons ...
... of σ to every constraint of the set Γ. In the sequel, the notation F σ will also be used for other formulas F , not necessarily constraints. Note that the three conditions i), ii), iii) are meant as minimal requirements. In particular, the availability of the equality symbol ≈ is granted in any cons ...
The Foundations
... (or fact) that the proposition is intended to represent occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now(the current situation), the statement It_is_raining is false (in the current situation). But if it were raining now, ...
... (or fact) that the proposition is intended to represent occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now(the current situation), the statement It_is_raining is false (in the current situation). But if it were raining now, ...
LINEAR LOGIC AS A FRAMEWORK FOR SPECIFYING SEQUENT
... In this paper, we make use of linear logic as a meta-logic and find that we can specify a variety of proof systems for object-level systems. By making use of classical linear logic, we are able to capture not only natural deduction proof systems but also many sequent calculus proof systems. We will ...
... In this paper, we make use of linear logic as a meta-logic and find that we can specify a variety of proof systems for object-level systems. By making use of classical linear logic, we are able to capture not only natural deduction proof systems but also many sequent calculus proof systems. We will ...
First-Order Theorem Proving and VAMPIRE
... simplifying ones. This distinction will be made more clear when we later discuss saturation and redundancy elimination in Section 4. Though the most complex part of proof search is the use of the resolution and superposition inference system, preprocessing is also very important, especially when the ...
... simplifying ones. This distinction will be made more clear when we later discuss saturation and redundancy elimination in Section 4. Though the most complex part of proof search is the use of the resolution and superposition inference system, preprocessing is also very important, especially when the ...
logic for the mathematical
... not true; but the conclusion does not follow from them anyway. Actually, in that argument, the word “should” is probably better left out. Mostly, we want to deal with statements which simply state some kind of claimed fact, statements which are clearly either true or false, though which of the two m ...
... not true; but the conclusion does not follow from them anyway. Actually, in that argument, the word “should” is probably better left out. Mostly, we want to deal with statements which simply state some kind of claimed fact, statements which are clearly either true or false, though which of the two m ...
An argumentation framework in default logic
... view, defeasible rules express approximations to reality, which might need to be corrected in specific circumstances: people reason with defaults as if they were true, until they give rise to an inconsistency. The attractiveness of this idea is that if nonmonotonic reasoning is regarded as a kind of ...
... view, defeasible rules express approximations to reality, which might need to be corrected in specific circumstances: people reason with defaults as if they were true, until they give rise to an inconsistency. The attractiveness of this idea is that if nonmonotonic reasoning is regarded as a kind of ...
Chapter 2 Propositional Logic
... this “circularity” is benign, because the definition is recursive. A recursive (or “inductive”) definition of a concept F contains a circular-seeming clause, often called the “inductive” clause, which specifies that if such-and-such objects are F , then so-and-so objects are also F . But a recursive ...
... this “circularity” is benign, because the definition is recursive. A recursive (or “inductive”) definition of a concept F contains a circular-seeming clause, often called the “inductive” clause, which specifies that if such-and-such objects are F , then so-and-so objects are also F . But a recursive ...
Predicate logic
... 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 + 2n + 1 = 2(2mn + m + n) + 1 and since m, n ∈ Z it holds that (2mn + m + n) ∈ Z, so ab = 2k + 1 for some k ∈ Z. Thus ab is odd by definition of odd. QED ...
... 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 + 2n + 1 = 2(2mn + m + n) + 1 and since m, n ∈ Z it holds that (2mn + m + n) ∈ Z, so ab = 2k + 1 for some k ∈ Z. Thus ab is odd by definition of odd. QED ...
preliminary version
... In classical logic, the situation is different. Here the notion of truth that is absolute, in the sense that it is independent of whether this truth can be observed. In classical logic, A ∨ ¬A is a tautology because every proposition is either true or false. Interpretation. There is an intuitive sem ...
... In classical logic, the situation is different. Here the notion of truth that is absolute, in the sense that it is independent of whether this truth can be observed. In classical logic, A ∨ ¬A is a tautology because every proposition is either true or false. Interpretation. There is an intuitive sem ...
John Nolt – Logics, chp 11-12
... quantifiers too—only they range, not over worlds, but over moments of time. Time and possibility: an odd juxtaposition, yet illuminating, for there are rich analogies here. For one thing, just as there is a specific temporal moment, the present, which is in a sense uniquely real (for the past exists ...
... quantifiers too—only they range, not over worlds, but over moments of time. Time and possibility: an odd juxtaposition, yet illuminating, for there are rich analogies here. For one thing, just as there is a specific temporal moment, the present, which is in a sense uniquely real (for the past exists ...
article in press - School of Computer Science
... 4. Intuitionistic modal logics One of the most promising applications of the result above is propositional intuitionistic modal logic. Intuitionistic modal logic is simply a modal logic with intuitionistic, rather than classical, base. The work on intuitionistic modal logic has several motivations: ...
... 4. Intuitionistic modal logics One of the most promising applications of the result above is propositional intuitionistic modal logic. Intuitionistic modal logic is simply a modal logic with intuitionistic, rather than classical, base. The work on intuitionistic modal logic has several motivations: ...
Introduction to Linear Logic
... Implication is then defined as A ⇒ B = ¬A ∨ B. We would then have a perfectly symmetric system. However, we have chosen the system of Appendix A.1 with the aim of making clear the connection to Intuitionistic Logic. One of the most important properties of the proof-rules for Classical Logic is that ...
... Implication is then defined as A ⇒ B = ¬A ∨ B. We would then have a perfectly symmetric system. However, we have chosen the system of Appendix A.1 with the aim of making clear the connection to Intuitionistic Logic. One of the most important properties of the proof-rules for Classical Logic is that ...
A Unified View of Induction Reasoning for First-Order Logic
... and implicit induction principles, [16, 22, 27, 30, 40, 56] being among the most notable. Other studies have been conducted to reduce the gap between them. Protzen [42] proposed a proof strategy to perform lazy induction on particular explicit induction proofs. Kapur and Subramaniam [29] devised a m ...
... and implicit induction principles, [16, 22, 27, 30, 40, 56] being among the most notable. Other studies have been conducted to reduce the gap between them. Protzen [42] proposed a proof strategy to perform lazy induction on particular explicit induction proofs. Kapur and Subramaniam [29] devised a m ...
Argumentations and logic
... chain of reasoning that deduces the hypothesis from the premises. This problem is, of course, not always solvable. In the ideal case, application of the hypothetico-deductive method begins after four things are available: the hypothesis itself, a set of premises known to be true, a proposition known ...
... chain of reasoning that deduces the hypothesis from the premises. This problem is, of course, not always solvable. In the ideal case, application of the hypothetico-deductive method begins after four things are available: the hypothesis itself, a set of premises known to be true, a proposition known ...
Refinement Modal Logic
... model restrictions were not sufficient to simulate informative events, and they introduced refinement trees for this purpose — a precursor of the dynamic epistemic logics developed later (for an overview, see [57]). This usage of refinement as a more general operation than model restriction is simil ...
... model restrictions were not sufficient to simulate informative events, and they introduced refinement trees for this purpose — a precursor of the dynamic epistemic logics developed later (for an overview, see [57]). This usage of refinement as a more general operation than model restriction is simil ...
Harmony, Normality and Stability
... Dummett singles out two features of the use of expressions that are of central importance for specifying their meanings. The two features are intended to apply very generally to all kinds of expressions, but I’m only concerned with the logical constants. ‘The first category [of principles governing ...
... Dummett singles out two features of the use of expressions that are of central importance for specifying their meanings. The two features are intended to apply very generally to all kinds of expressions, but I’m only concerned with the logical constants. ‘The first category [of principles governing ...
Programming with Classical Proofs
... corresponds to a proof of totality of a recursive function. This leads to the area of classical program extraction. There have been several approaches to extracting the computational content of these classical proofs. It was discovered by Griffin in 1989 [20] that inference by contradiction correspo ...
... corresponds to a proof of totality of a recursive function. This leads to the area of classical program extraction. There have been several approaches to extracting the computational content of these classical proofs. It was discovered by Griffin in 1989 [20] that inference by contradiction correspo ...
CUED PhD and MPhil Thesis Classes
... Substructural logics are logics which omit some structural rules, e.g. contraction, weakening, commutativity. Nonassociative Lambek calulus (NL) introduced by Lambek is a propositional logic omitting all structural rules, which can be treated as a basic core of substructural logics. NL can be enrich ...
... Substructural logics are logics which omit some structural rules, e.g. contraction, weakening, commutativity. Nonassociative Lambek calulus (NL) introduced by Lambek is a propositional logic omitting all structural rules, which can be treated as a basic core of substructural logics. NL can be enrich ...
Sound and Complete Inference Rules in FOL Example
... For fill-in-the-blank questions, we can have: • Termination with a clause which is a single answer literal Ans(c1 , . . . , cn ). In this case, the constants c1 , . . . , cn gives us an answer to the query. There might be more answers depending on whether there are more resolution refutations of Ans ...
... For fill-in-the-blank questions, we can have: • Termination with a clause which is a single answer literal Ans(c1 , . . . , cn ). In this case, the constants c1 , . . . , cn gives us an answer to the query. There might be more answers depending on whether there are more resolution refutations of Ans ...
An Overview of Intuitionistic and Linear Logic
... the natural numbers are “God-given”, the rest have to be explained in terms of natural numbers. There are several branches in constructivism, each with a varying degree of preference towards constructive concepts and methods. The (perhaps) most well-known view is the idea of intuitionism, pioneered ...
... the natural numbers are “God-given”, the rest have to be explained in terms of natural numbers. There are several branches in constructivism, each with a varying degree of preference towards constructive concepts and methods. The (perhaps) most well-known view is the idea of intuitionism, pioneered ...
Reductio ad Absurdum Argumentation in Normal Logic
... define, in several ways, the meaning, the semantics of a Logic Program. Several semantics were defined, some 2-valued, some 3-valued, and even multi-valued semantics. The current standard 2-valued semantics for Normal Logic Programs— the Stable Models Semantics [11] — has been around for almost 20 y ...
... define, in several ways, the meaning, the semantics of a Logic Program. Several semantics were defined, some 2-valued, some 3-valued, and even multi-valued semantics. The current standard 2-valued semantics for Normal Logic Programs— the Stable Models Semantics [11] — has been around for almost 20 y ...