Systems of modal logic - Department of Computing
... Systems of modal logic can also be defined (syntactically) in other ways, usually by reference to some kind of proof system. For example: • Hilbert systems: given a set of formulas called axioms and a set of rules of proof, a formula A is a theorem of the system when it is the last formula of a sequ ...
... Systems of modal logic can also be defined (syntactically) in other ways, usually by reference to some kind of proof system. For example: • Hilbert systems: given a set of formulas called axioms and a set of rules of proof, a formula A is a theorem of the system when it is the last formula of a sequ ...
Handling Exceptions in nonmonotonic reasoning
... expansions. This is the case of the justified and constrained default logics [12,3] shown in section 5. We believe the EFP settles this issue out: only the second generalization should be applied. The situation is worse than it might seem at first glance. In example 3.3 is not the case of an isolate ...
... expansions. This is the case of the justified and constrained default logics [12,3] shown in section 5. We believe the EFP settles this issue out: only the second generalization should be applied. The situation is worse than it might seem at first glance. In example 3.3 is not the case of an isolate ...
General Dynamic Dynamic Logic
... logic [7,11,12,20]. A significant difference from the epistemic setting is the need to describe dynamic operators that change the relational structure of the underlying model, not just the size of its domain (announcement) or the propositional valuations (real-world change). For example, if one mode ...
... logic [7,11,12,20]. A significant difference from the epistemic setting is the need to describe dynamic operators that change the relational structure of the underlying model, not just the size of its domain (announcement) or the propositional valuations (real-world change). For example, if one mode ...
Propositional inquisitive logic: a survey
... out in what states of affairs the sentence is true; however, this truthconditional view does not seem suitable in the case of questions. In inquisitive logic, by contrast, the meaning of a sentence is given by laying out what information is needed in order for a sentence to be supported. Accordingly ...
... out in what states of affairs the sentence is true; however, this truthconditional view does not seem suitable in the case of questions. In inquisitive logic, by contrast, the meaning of a sentence is given by laying out what information is needed in order for a sentence to be supported. Accordingly ...
Topological Completeness of First-Order Modal Logic
... associated to the possible-world structure [23,19,4]. In this article we provide a completeness proof for first-order S4 modal logic with respect to topologicalsheaf semantics of Awodey-Kishida [3], which combines the possible-world formulation of sheaf semantics with the topos-theoretic interpretat ...
... associated to the possible-world structure [23,19,4]. In this article we provide a completeness proof for first-order S4 modal logic with respect to topologicalsheaf semantics of Awodey-Kishida [3], which combines the possible-world formulation of sheaf semantics with the topos-theoretic interpretat ...
Second-Order Logic of Paradox
... that every proposition has at least one of the two usual truth values), the classical principles of Double Negation, and the classical De Morgan equivalences and their quantificational analogues. (As a result, every valid formula of classical logic is valid in LP as well.) What is missing are some o ...
... that every proposition has at least one of the two usual truth values), the classical principles of Double Negation, and the classical De Morgan equivalences and their quantificational analogues. (As a result, every valid formula of classical logic is valid in LP as well.) What is missing are some o ...
Admissible rules in the implication-- negation fragment of intuitionistic logic
... premises form such a set, admissibility coincides with derivability. Let us fix L as a logic based on a language L containing a binary connective → for which modus ponens is derivable (ϕ, ϕ → ψ ⊢L ψ ). Generalizing Ghilardi [7,8] slightly, Γ ⊆ FmL is called L-projective if there exists an L-substitu ...
... premises form such a set, admissibility coincides with derivability. Let us fix L as a logic based on a language L containing a binary connective → for which modus ponens is derivable (ϕ, ϕ → ψ ⊢L ψ ). Generalizing Ghilardi [7,8] slightly, Γ ⊆ FmL is called L-projective if there exists an L-substitu ...
Basic Logic and Fregean Set Theory - MSCS
... of constructive logic, only failing in cases of the kind mentioned above. In areas like computer algebra constructive logic may perform relatively more prominent functions. The idea of using models of nature with a logic different from the classical one is not new. Quantum logic has been used to mo ...
... of constructive logic, only failing in cases of the kind mentioned above. In areas like computer algebra constructive logic may perform relatively more prominent functions. The idea of using models of nature with a logic different from the classical one is not new. Quantum logic has been used to mo ...
Intuitionistic Logic
... Heyting showed that this is asking too much. Consider A = “there occur twenty consecutive 7’s in the decimal expansion of π”, and B = “there occur nineteen consecutive 7’s in the decimal expansion of π”. Then ¬A ∨ B does not hold constructively, but on the interpretation following here, the implicat ...
... Heyting showed that this is asking too much. Consider A = “there occur twenty consecutive 7’s in the decimal expansion of π”, and B = “there occur nineteen consecutive 7’s in the decimal expansion of π”. Then ¬A ∨ B does not hold constructively, but on the interpretation following here, the implicat ...
On Elkan`s theorems: Clarifying their meaning
... omitted from the first version of Elkan’s theorem. As to the rest of the assumptions, both t~A ∧ B! ⫽ min$t~A!, t~B!% and t~¬A! ⫽ 1 ⫺ t~A! are quite reasonable and, in fact, are often used in applications of fuzzy logic. Let us now concentrate on the last assumption, that is, on t~A! ⫽ t~B! if A and ...
... omitted from the first version of Elkan’s theorem. As to the rest of the assumptions, both t~A ∧ B! ⫽ min$t~A!, t~B!% and t~¬A! ⫽ 1 ⫺ t~A! are quite reasonable and, in fact, are often used in applications of fuzzy logic. Let us now concentrate on the last assumption, that is, on t~A! ⫽ t~B! if A and ...
LOGIC MAY BE SIMPLE Logic, Congruence - Jean
... notion of “monotony” has nothing to do with the use of the word in the context of non-monotonic logics. If we define algebraizable logics as Fregean logics, thus the above theorem gives a sufficient condition to algebraize logics. It is interesting to compare this result with the more recent work of [B ...
... notion of “monotony” has nothing to do with the use of the word in the context of non-monotonic logics. If we define algebraizable logics as Fregean logics, thus the above theorem gives a sufficient condition to algebraize logics. It is interesting to compare this result with the more recent work of [B ...
Action Logic and Pure Induction
... no finite list of equations of REG from which the rest of REG may be inferred. But in addition to this syntactic problem, REG has a semantic problem. It is not strong enough to constrain a∗ to be the reflexive transitive closure of a. We shall call a reflexive when 1 ≤ a and transitive when aa ≤ a, ...
... no finite list of equations of REG from which the rest of REG may be inferred. But in addition to this syntactic problem, REG has a semantic problem. It is not strong enough to constrain a∗ to be the reflexive transitive closure of a. We shall call a reflexive when 1 ≤ a and transitive when aa ≤ a, ...
Classical Logic and the Curry–Howard Correspondence
... proceed line by line, with each line derived from those preceding it by means of some inference rule. Nowadays such logics are known as ‘Hilbert systems’. This format can be somewhat cumbersome and inelegant, both because it does not follow the reasoning-patterns of ordinary mathematics and because ...
... proceed line by line, with each line derived from those preceding it by means of some inference rule. Nowadays such logics are known as ‘Hilbert systems’. This format can be somewhat cumbersome and inelegant, both because it does not follow the reasoning-patterns of ordinary mathematics and because ...
KnotandTonk 1 Preliminaries
... for Tonk fail even to define a meaningful connective, on the grounds that Tonk cannot be given semantic conditions. By exactly the same token, inferentialists might allege that the semantic conditions for Knot fail even to define a meaningful connective, on the grounds that Knot cannot be given natu ...
... for Tonk fail even to define a meaningful connective, on the grounds that Tonk cannot be given semantic conditions. By exactly the same token, inferentialists might allege that the semantic conditions for Knot fail even to define a meaningful connective, on the grounds that Knot cannot be given natu ...
Quadripartitaratio - Revistas Científicas de la Universidad de
... Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, contextualism, tolerance, and pluralism. Accordingly, logic teaching in this century can hasten the decline or at least slow the growth of the recurring spirit of subjectivity, intolerance, ...
... Logic teaching in this century can exploit the new spirit of objectivity, humility, clarity, observationalism, contextualism, tolerance, and pluralism. Accordingly, logic teaching in this century can hasten the decline or at least slow the growth of the recurring spirit of subjectivity, intolerance, ...
Tactics for Separation Logic Abstract Andrew W. Appel INRIA Rocquencourt & Princeton University
... portions are disjoint, and their union forms the entire heap reasoned about by P ∗ Q. This is therefore a linear logic, in that in general P ∗ Q does not entail P ∗ P ∗ Q or vice versa. When doing machine-checked proofs of imperative programs, one faces a choice: one could implement Hoare logic (or ...
... portions are disjoint, and their union forms the entire heap reasoned about by P ∗ Q. This is therefore a linear logic, in that in general P ∗ Q does not entail P ∗ P ∗ Q or vice versa. When doing machine-checked proofs of imperative programs, one faces a choice: one could implement Hoare logic (or ...
Clausal Logic and Logic Programming in Algebraic Domains*
... theory to the semantics of logic programming [Apt90], in particular disjunctive logic programming [LMR92]. Most of this work focuses on first-order logic. Extensions to higher types have been made — for example, λ-Prolog [NM98] — but domain theory has not played as much of a role here as it has for ...
... theory to the semantics of logic programming [Apt90], in particular disjunctive logic programming [LMR92]. Most of this work focuses on first-order logic. Extensions to higher types have been made — for example, λ-Prolog [NM98] — but domain theory has not played as much of a role here as it has for ...
Nonmonotonic Reasoning - Computer Science Department
... in 1928. More recently, there has been much activity in formalizing the logics of ”I know ...”, and ”I believe... P”. But it is only since 1980 or so, under the influence of John McCarthy, that non-monotonic reasoning as such has been systematically formalized. McCarthy, building on the efforts of ear ...
... in 1928. More recently, there has been much activity in formalizing the logics of ”I know ...”, and ”I believe... P”. But it is only since 1980 or so, under the influence of John McCarthy, that non-monotonic reasoning as such has been systematically formalized. McCarthy, building on the efforts of ear ...
Between Truth and Falsity
... to be a tautology. Modus ponens will be a valid inference rule in K3 iff it is unexceptionable, or never false. This, of course, means that whenever ((A →B) & A) is assigned true in the truth table, B is also T. The reliability of MP as an inference rule (put differently, ...
... to be a tautology. Modus ponens will be a valid inference rule in K3 iff it is unexceptionable, or never false. This, of course, means that whenever ((A →B) & A) is assigned true in the truth table, B is also T. The reliability of MP as an inference rule (put differently, ...
The unintended interpretations of intuitionistic logic
... development of intuitionistic logic: it was not until 1923 that Brouwer discovered the equivalence in intuitionistic mathematics of triple negation and single negation [Brouwer 1925]. While Brouwer may have been uncompromising with respect to his philosophy, his mathematical and philosophical talent ...
... development of intuitionistic logic: it was not until 1923 that Brouwer discovered the equivalence in intuitionistic mathematics of triple negation and single negation [Brouwer 1925]. While Brouwer may have been uncompromising with respect to his philosophy, his mathematical and philosophical talent ...
Propositional Logic
... • Sometimes exponential in time. Relatively spaceefficient. • Somewhat mysterious to non-technical users • Algorithmically simple but more complex than perfect induction. • Not considered appropriate for general problem solving. ...
... • Sometimes exponential in time. Relatively spaceefficient. • Somewhat mysterious to non-technical users • Algorithmically simple but more complex than perfect induction. • Not considered appropriate for general problem solving. ...
Slide 1
... contradictions, equivalence, and logical proofs is no different for fuzzy sets; the results, however, can differ considerably from those in classical logic. If the truth values for the simple propositions of a fuzzy logic compound proposition are strictly true (1) or false (0), the results follow id ...
... contradictions, equivalence, and logical proofs is no different for fuzzy sets; the results, however, can differ considerably from those in classical logic. If the truth values for the simple propositions of a fuzzy logic compound proposition are strictly true (1) or false (0), the results follow id ...
SORT LOGIC AND FOUNDATIONS OF MATHEMATICS 1
... In a sense ordinary second order logic also “looks outside” the model as well as one can think of the bound second order variables as first order variables ranging over the domain of all subsets and relations on the original domain. In fact, one of the best ways to understand second order logic is t ...
... In a sense ordinary second order logic also “looks outside” the model as well as one can think of the bound second order variables as first order variables ranging over the domain of all subsets and relations on the original domain. In fact, one of the best ways to understand second order logic is t ...
POSSIBLE WORLDS SEMANTICS AND THE LIAR Reflections on a
... Now, Kaplan’s argument shows that the principle of plenitude is incompatible with assumptions commonly made in possible worlds semantics. Here is how the argument goes: (i) There is a set W of possible worlds and a set P rop of propositions. (ii) There is, for every subset X of W , a corresponding p ...
... Now, Kaplan’s argument shows that the principle of plenitude is incompatible with assumptions commonly made in possible worlds semantics. Here is how the argument goes: (i) There is a set W of possible worlds and a set P rop of propositions. (ii) There is, for every subset X of W , a corresponding p ...
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.