A BRIEF INTRODUCTION TO MODAL LOGIC Introduction Consider
... This claim has come to be seen as false. After all, if two statements are equivalent, they ought to imply each other. It seems reasonable to say that if P is the case then P must be a possible state of affairs, since what is true cannot be impossible. However, it is quite a bit less obvious to say t ...
... This claim has come to be seen as false. After all, if two statements are equivalent, they ought to imply each other. It seems reasonable to say that if P is the case then P must be a possible state of affairs, since what is true cannot be impossible. However, it is quite a bit less obvious to say t ...
Remarks on Second-Order Consequence
... only that each informally proven theorem be provable by means of the calculus (in other words, when formalizing, we do not mean to be true to proofs, but to theorems). As soon as we state this demand we see the difficulty it involves, for if the notion of an informal theorem turned out to be open-en ...
... only that each informally proven theorem be provable by means of the calculus (in other words, when formalizing, we do not mean to be true to proofs, but to theorems). As soon as we state this demand we see the difficulty it involves, for if the notion of an informal theorem turned out to be open-en ...
Elementary Logic
... A proposition is also said to be valid if it is a tautology. So, the problem of determining whether a given proposition is valid (a tautology) is also called the validity problem. Note: the notion of a tautology is restricted to propositional logic. In first-order logic, we also speak of valid formu ...
... A proposition is also said to be valid if it is a tautology. So, the problem of determining whether a given proposition is valid (a tautology) is also called the validity problem. Note: the notion of a tautology is restricted to propositional logic. In first-order logic, we also speak of valid formu ...
Document
... Show that if a is odd, then a2 is odd Then a = 2k + 1 for some integer k Then a2 = (2k + 1)(2k + 1) = 4k2 + 4k + 1 = 2(j) + 1 for some integer j and a2 is odd Therefore if a2 is even, then a is even ...
... Show that if a is odd, then a2 is odd Then a = 2k + 1 for some integer k Then a2 = (2k + 1)(2k + 1) = 4k2 + 4k + 1 = 2(j) + 1 for some integer j and a2 is odd Therefore if a2 is even, then a is even ...
Coordinate-free logic - Utrecht University Repository
... predicate ‘<’ as a collection of ordered pairs hx, yi, where x is less than y, and it interprets the predicate ‘>’ as a collection of ordered pairs hx, yi, where x is greater than y. The logic does not offer an easy way to express in a neutral way how things stand to each other. Another misleading a ...
... predicate ‘<’ as a collection of ordered pairs hx, yi, where x is less than y, and it interprets the predicate ‘>’ as a collection of ordered pairs hx, yi, where x is greater than y. The logic does not offer an easy way to express in a neutral way how things stand to each other. Another misleading a ...
Belief Revision in non
... logic axiomatisation of the semantics of the object logic L, ii) a domain-dependent notion of “acceptability” for theories of L and iii) a classical AGM belief revision operation. In general, different translation mechanisms can be defined from a given object logic to classical logic, depending on t ...
... logic axiomatisation of the semantics of the object logic L, ii) a domain-dependent notion of “acceptability” for theories of L and iii) a classical AGM belief revision operation. In general, different translation mechanisms can be defined from a given object logic to classical logic, depending on t ...
Definability properties and the congruence closure
... dividing a structure by a congruence relation [Fe, Ma]. In other words, these logics are not congruence closed in the sense of [MeSh2]. The counterexample for A-interpolation is transformed in one for Beth property using a tree technique due to Friedman [Fr], and later generalized in [MaSh1, 2]. Rec ...
... dividing a structure by a congruence relation [Fe, Ma]. In other words, these logics are not congruence closed in the sense of [MeSh2]. The counterexample for A-interpolation is transformed in one for Beth property using a tree technique due to Friedman [Fr], and later generalized in [MaSh1, 2]. Rec ...
Interpreting Lattice-Valued Set Theory in Fuzzy Set Theory
... well as its predecessor [14], builds upon results of set theory in intuitionistic logic, as given by W. C. Powell [12] and R. J. Grayson [7], which is apparent, among other things, in its spelling of axioms—in a weak setting (such as that of an intermediate logic), different but classically equivale ...
... well as its predecessor [14], builds upon results of set theory in intuitionistic logic, as given by W. C. Powell [12] and R. J. Grayson [7], which is apparent, among other things, in its spelling of axioms—in a weak setting (such as that of an intermediate logic), different but classically equivale ...
An Overview of Intuitionistic and Linear Logic
... Let N = p1 · p2 · . . . · pk + 1. Note that none of pi divide N. Now, either N is prime or it can be factored into primes. So there is at least one prime q that divides N, but q can’t be any of the pi ’s, so q is a prime number not in the list, which contradicts our assumption. ...
... Let N = p1 · p2 · . . . · pk + 1. Note that none of pi divide N. Now, either N is prime or it can be factored into primes. So there is at least one prime q that divides N, but q can’t be any of the pi ’s, so q is a prime number not in the list, which contradicts our assumption. ...
Document
... Let P be a stratified and allowed program and Q an allowed query, such that P is strict wrt. Q If comp(P) |= Qq for some q such that Qq is ground, then there is a successful SLDNF-derivation of P {Q} with CAS q. ...
... Let P be a stratified and allowed program and Q an allowed query, such that P is strict wrt. Q If comp(P) |= Qq for some q such that Qq is ground, then there is a successful SLDNF-derivation of P {Q} with CAS q. ...
Classical BI - UCL Computer Science
... “vending machine” model for BI given by Pym, O’Hearn and Yang [27], which itself was inspired by Girard’s well-known “Marlboro and Camel” illustration of linear logic [18]. Let hZ, +, 0, −i be the Abelian group of integers under addition with identity 0, where − is the usual unary minus. This group ...
... “vending machine” model for BI given by Pym, O’Hearn and Yang [27], which itself was inspired by Girard’s well-known “Marlboro and Camel” illustration of linear logic [18]. Let hZ, +, 0, −i be the Abelian group of integers under addition with identity 0, where − is the usual unary minus. This group ...
A Concurrent Logical Framework: The Propositional Fragment Kevin Watkins , Iliano Cervesato
... as a formal meta-logic or type theory together with a representation methodology. A single implementation of a logical framework can then be used to study a variety of deductive systems, thereby factoring the effort that would be required to implement each deductive system separately. Applications o ...
... as a formal meta-logic or type theory together with a representation methodology. A single implementation of a logical framework can then be used to study a variety of deductive systems, thereby factoring the effort that would be required to implement each deductive system separately. Applications o ...
Boolean Connectives and Formal Proofs - FB3
... We could also introduce Till Mossakowski Logic rules justified by the meanings of ot ...
... We could also introduce Till Mossakowski Logic rules justified by the meanings of ot ...
Seventy-five problems for testing automatic
... ATPers in mind that the following list is offered. None of these problems will be the sort whose solution is, of itself, of any mathematical or logical interest. Such ‘open problems’ are regularly published in the Newsletter of the Association for Automated Reasoning. Most (but not all) of my proble ...
... ATPers in mind that the following list is offered. None of these problems will be the sort whose solution is, of itself, of any mathematical or logical interest. Such ‘open problems’ are regularly published in the Newsletter of the Association for Automated Reasoning. Most (but not all) of my proble ...
Logic Programming, Functional Programming, and Inductive
... rules in logic programs. The rules in an inductive definition contain no variables. A schematic rule (like the rule for cousin) abbreviates an infinite set of rules: all ground instances under the Herbrand universe. In the semantics of logic programming, such theory has long been used as a technical ...
... rules in logic programs. The rules in an inductive definition contain no variables. A schematic rule (like the rule for cousin) abbreviates an infinite set of rules: all ground instances under the Herbrand universe. In the semantics of logic programming, such theory has long been used as a technical ...
Normal modal logics (Syntactic characterisations)
... (i) Trivial. We need to show that L contains PL and is closed under MP and US. This is trivial, because L is the set of all formulas. (ii) Easy. PL is a subset of every Σi , so also a subset of the intersection. To show the intersection is closed under MP: suppose A and A → B are formulas in the int ...
... (i) Trivial. We need to show that L contains PL and is closed under MP and US. This is trivial, because L is the set of all formulas. (ii) Easy. PL is a subset of every Σi , so also a subset of the intersection. To show the intersection is closed under MP: suppose A and A → B are formulas in the int ...
Can Modalities Save Naive Set Theory?
... A third reason for restricting set comprehension as in (Comp2) is that this restriction fits certain views in the philosophy of mathematics and logic, on suitable ways of understanding the qualification “in a special way”. One example is fictionalism, which will be discussed below. For another examp ...
... A third reason for restricting set comprehension as in (Comp2) is that this restriction fits certain views in the philosophy of mathematics and logic, on suitable ways of understanding the qualification “in a special way”. One example is fictionalism, which will be discussed below. For another examp ...
Discrete Mathematics
... A propositional variable (lowercase letters p, q, r) is a proposition. These variables model true/false statements. The negation of a proposition P, written ¬ P, is a proposition. The conjunction (and) of two propositions, written P ∧ Q, is a proposition. The disjunction (or) of two propositions, wr ...
... A propositional variable (lowercase letters p, q, r) is a proposition. These variables model true/false statements. The negation of a proposition P, written ¬ P, is a proposition. The conjunction (and) of two propositions, written P ∧ Q, is a proposition. The disjunction (or) of two propositions, wr ...
pdf file
... theories, since it requires the order ‹ to be strict. Definition 5. Let be an acyclic default
theory. A set S ⊆ D is a generating set of an
extension complying with the exceptions-first
principle iff S is a maximal set of defaults such
that
1. W ∪ CONS(S) is consistent;
2. If β ∝ γ ∈ S , then ...
... theories, since it requires the order ‹ to be strict. Definition 5. Let
Propositional Logic - Department of Computer Science
... show the following. Let P be a propositional formula. • Termination: The algorithm terminates: there is no infinite tableau path S0 , S1 , . . . starting with {P }. • Soundness: If there exists a complete tableau path S0 , S1 , . . . , Sn with {P } = S0 and without clash, then P is satisfiable. • Co ...
... show the following. Let P be a propositional formula. • Termination: The algorithm terminates: there is no infinite tableau path S0 , S1 , . . . starting with {P }. • Soundness: If there exists a complete tableau path S0 , S1 , . . . , Sn with {P } = S0 and without clash, then P is satisfiable. • Co ...
On the specification of sequent systems
... Other variations on the cut rule appear in the literature and many of these can be encoded by changing one or both of the −◦ to ⇒. Since the formula Cut entails these other variations, so we shall not consider them further. The Init and Cut clauses together proves that ⌊·⌋ and ⌈·⌉ are duals of each ...
... Other variations on the cut rule appear in the literature and many of these can be encoded by changing one or both of the −◦ to ⇒. Since the formula Cut entails these other variations, so we shall not consider them further. The Init and Cut clauses together proves that ⌊·⌋ and ⌈·⌉ are duals of each ...
Many-Valued Models
... finite algebras) is arguably older than many-valued logics themselves, and has provided much information not only about non-classical systems as relevant logics, linear logic, intuitionistic logics and paraconsistent logics, but also about fragments of classical logic. The problem of enumerating suc ...
... finite algebras) is arguably older than many-valued logics themselves, and has provided much information not only about non-classical systems as relevant logics, linear logic, intuitionistic logics and paraconsistent logics, but also about fragments of classical logic. The problem of enumerating suc ...
A Well-Founded Semantics for Logic Programs with Abstract
... While ASP assumes that solutions are given by answer sets, well-founded models (Van Gelder, Ross, and Schlipf 1991) have been found to be very useful as well. First, computing the well-founded model of a normal logic program is tractable. This compares to the NP-completeness of computing an answer s ...
... While ASP assumes that solutions are given by answer sets, well-founded models (Van Gelder, Ross, and Schlipf 1991) have been found to be very useful as well. First, computing the well-founded model of a normal logic program is tractable. This compares to the NP-completeness of computing an answer s ...
Willard Van Orman Quine
Willard Van Orman Quine (/kwaɪn/; June 25, 1908 – December 25, 2000) (known to intimates as ""Van"") was an American philosopher and logician in the analytic tradition, recognized as ""one of the most influential philosophers of the twentieth century."" From 1930 until his death 70 years later, Quine was continually affiliated with Harvard University in one way or another, first as a student, then as a professor of philosophy and a teacher of logic and set theory, and finally as a professor emeritus who published or revised several books in retirement. He filled the Edgar Pierce Chair of Philosophy at Harvard from 1956 to 1978. A recent poll conducted among analytic philosophers named Quine as the fifth most important philosopher of the past two centuries. He won the first Schock Prize in Logic and Philosophy in 1993 for ""his systematical and penetrating discussions of how learning of language and communication are based on socially available evidence and of the consequences of this for theories on knowledge and linguistic meaning."" In 1996 he was awarded the Kyoto Prize in Arts and Philosophy for his ""outstanding contributions to the progress of philosophy in the 20th century by proposing numerous theories based on keen insights in logic, epistemology, philosophy of science and philosophy of language.""Quine falls squarely into the analytic philosophy tradition while also being the main proponent of the view that philosophy is not conceptual analysis but the abstract branch of the empirical sciences. His major writings include ""Two Dogmas of Empiricism"" (1951), which attacked the distinction between analytic and synthetic propositions and advocated a form of semantic holism, and Word and Object (1960), which further developed these positions and introduced Quine's famous indeterminacy of translation thesis, advocating a behaviorist theory of meaning. He also developed an influential naturalized epistemology that tried to provide ""an improved scientific explanation of how we have developed elaborate scientific theories on the basis of meager sensory input."" He is also important in philosophy of science for his ""systematic attempt to understand science from within the resources of science itself"" and for his conception of philosophy as continuous with science. This led to his famous quip that ""philosophy of science is philosophy enough."" In philosophy of mathematics, he and his Harvard colleague Hilary Putnam developed the ""Quine–Putnam indispensability thesis,"" an argument for the reality of mathematical entities.