Modal Logic for Artificial Intelligence
... is valid, regardless of the sentences we use in the place of A and B. The only items that need to be fixed are ‘or’ and ‘not’ in this case. If we would replace ‘not’ by ‘maybe’, then the argument would not be valid anymore. We call ‘or’ and ‘not’ logical constants. Together with ‘and’, ‘if . . . the ...
... is valid, regardless of the sentences we use in the place of A and B. The only items that need to be fixed are ‘or’ and ‘not’ in this case. If we would replace ‘not’ by ‘maybe’, then the argument would not be valid anymore. We call ‘or’ and ‘not’ logical constants. Together with ‘and’, ‘if . . . the ...
The Development of Mathematical Logic from Russell to Tarski
... to a specific result in mathematical logic, that is Padoa’s method for proving indefinability (see below). The result was stated in the talks Padoa gave in 1900 at the two meetings mentioned at the outset ( Padoa 1901 , 1902). We will follow the “Essai d’une théorie algébrique des nombre entiers, ...
... to a specific result in mathematical logic, that is Padoa’s method for proving indefinability (see below). The result was stated in the talks Padoa gave in 1900 at the two meetings mentioned at the outset ( Padoa 1901 , 1902). We will follow the “Essai d’une théorie algébrique des nombre entiers, ...
071 Embeddings
... For technical explanation of this term see Monk [1976] chapters 13 to 16, and Tarski, Mostowski and Robinson ...
... For technical explanation of this term see Monk [1976] chapters 13 to 16, and Tarski, Mostowski and Robinson ...
Interpreting and Applying Proof Theories for Modal Logic
... Y holds here. Provided that here is arbitrary, this is exactly the same fact about frames, described in two different ways. Despite these pleasing features, display logic has not been widely used.6 Part of this may be explained in terms of the unique features of display calculi: systems for modal lo ...
... Y holds here. Provided that here is arbitrary, this is exactly the same fact about frames, described in two different ways. Despite these pleasing features, display logic has not been widely used.6 Part of this may be explained in terms of the unique features of display calculi: systems for modal lo ...
Propositional Discourse Logic
... and f all statements can even be assigned classical truth-values. At f 0 , however, it dissolves again. At this point it exemplifies all phenomena we will address, so there is no need to extend it. Open-endedness means, in particular, that many statements are undetermined at the moment they are made ...
... and f all statements can even be assigned classical truth-values. At f 0 , however, it dissolves again. At this point it exemplifies all phenomena we will address, so there is no need to extend it. Open-endedness means, in particular, that many statements are undetermined at the moment they are made ...
x - Homepages | The University of Aberdeen
... Applications of Predicate Logic It is one of the most-used formal notations for writing mathematical definitions, axioms, and theorems. For example, in linear algebra, a partial order is introduced saying that a relation R is reflexive and transitive – and these notions are defined using predicate l ...
... Applications of Predicate Logic It is one of the most-used formal notations for writing mathematical definitions, axioms, and theorems. For example, in linear algebra, a partial order is introduced saying that a relation R is reflexive and transitive – and these notions are defined using predicate l ...
Formal systems of fuzzy logic and their fragments∗
... and especially papers [8, 4] where mathematical, philosophical, and methodological reasons for this belief are presented). The expansions of BCK to richer propositional languages was thoroughly studied in the famous paper by Ono and Komori [46] (see also older papers by Idziak [30, 31]). They constr ...
... and especially papers [8, 4] where mathematical, philosophical, and methodological reasons for this belief are presented). The expansions of BCK to richer propositional languages was thoroughly studied in the famous paper by Ono and Komori [46] (see also older papers by Idziak [30, 31]). They constr ...
FC §1.1, §1.2 - Mypage at Indiana University
... with which we began this chapter. Logical deduction will be a major topic of this chapter; under the name of proof , it will be the last major topic of this chapter, and a major tool for the rest of this book. ...
... with which we began this chapter. Logical deduction will be a major topic of this chapter; under the name of proof , it will be the last major topic of this chapter, and a major tool for the rest of this book. ...
The Herbrand Manifesto
... This allows us to give complete definitions to things that cannot be completely defined with Tarskian Semantics. We have already seen Peano arithmetic. It turns out that, under Herbrand semantics, we can also define some other useful concepts that are not definable with Tarskian semantics, and we ca ...
... This allows us to give complete definitions to things that cannot be completely defined with Tarskian Semantics. We have already seen Peano arithmetic. It turns out that, under Herbrand semantics, we can also define some other useful concepts that are not definable with Tarskian semantics, and we ca ...
Argument construction and reinstatement in logics for
... In recent years, researchers in nonmonotonic logic have turned increasing attention to formal systems in which nonmonotonic reasoning is analyzed through the study of interactions among competing defeasible arguments; a survey appears in Prakken and Vreewsijk (forthcoming). These argument systems ar ...
... In recent years, researchers in nonmonotonic logic have turned increasing attention to formal systems in which nonmonotonic reasoning is analyzed through the study of interactions among competing defeasible arguments; a survey appears in Prakken and Vreewsijk (forthcoming). These argument systems ar ...
Introduction to Linear Logic
... The main concern of this report is to give an introduction to Linear Logic. For pedagogical purposes we shall also have a look at Classical Logic as well as Intuitionistic Logic. Linear Logic was introduced by J.-Y. Girard in 1987 and it has attracted much attention from computer scientists, as it i ...
... The main concern of this report is to give an introduction to Linear Logic. For pedagogical purposes we shall also have a look at Classical Logic as well as Intuitionistic Logic. Linear Logic was introduced by J.-Y. Girard in 1987 and it has attracted much attention from computer scientists, as it i ...
Essentials Of Symbolic Logic
... special logical notation is not peculiar to modern logic. Aristotle, the ancient founder of the subject, used variables to facilitate his own work. Although the difference in this respect between modern and classical logic is not one of kind but of degree, the difference in degree is tremendous. The ...
... special logical notation is not peculiar to modern logic. Aristotle, the ancient founder of the subject, used variables to facilitate his own work. Although the difference in this respect between modern and classical logic is not one of kind but of degree, the difference in degree is tremendous. The ...
On the Notion of Coherence in Fuzzy Answer Set Semantics
... Notice that the condition “(∗, ←) is an adjoint pair ” is not a restriction since for all t-norm ∗ we can define an implication operator ←R such that (∗, ←R ) forms an adjoint pair (see [5]). The statement in Proposition cannot be extended for α 6= 0: consider an L-interpretation I which assigns to ...
... Notice that the condition “(∗, ←) is an adjoint pair ” is not a restriction since for all t-norm ∗ we can define an implication operator ←R such that (∗, ←R ) forms an adjoint pair (see [5]). The statement in Proposition cannot be extended for α 6= 0: consider an L-interpretation I which assigns to ...
X - UOW
... In constructing a truth table for a compound statement comprised of n statements, there will be 2n combinations of truth values. This method can be long for large numbers of statements. We will consider a quicker method for determining if a compound statement is a tautology. However, truth tables ar ...
... In constructing a truth table for a compound statement comprised of n statements, there will be 2n combinations of truth values. This method can be long for large numbers of statements. We will consider a quicker method for determining if a compound statement is a tautology. However, truth tables ar ...
Proofs in Propositional Logic
... The tactic unfold not allows to expand the constant not in a goal, but is seldom used. The introduction tactic for ∼A is the introduction tactic for A→False, i.e. intro H where H is a fresh name. This tactic pushes the hypothesis H : A into the context and leaves False as the proposition to prove. ...
... The tactic unfold not allows to expand the constant not in a goal, but is seldom used. The introduction tactic for ∼A is the introduction tactic for A→False, i.e. intro H where H is a fresh name. This tactic pushes the hypothesis H : A into the context and leaves False as the proposition to prove. ...
Proofs in Propositional Logic
... The tactic unfold not allows to expand the constant not in a goal, but is seldom used. The introduction tactic for ∼A is the introduction tactic for A→False, i.e. intro H where H is a fresh name. This tactic pushes the hypothesis H : A into the context and leaves False as the proposition to prove. ...
... The tactic unfold not allows to expand the constant not in a goal, but is seldom used. The introduction tactic for ∼A is the introduction tactic for A→False, i.e. intro H where H is a fresh name. This tactic pushes the hypothesis H : A into the context and leaves False as the proposition to prove. ...
A Mathematical Introduction to Modal Logic
... Nevertheless, we will try our best not to lose our basic intuition by making occasional remarks to the philosophical and applied considerations. The intended course is a short one, and these notes will cover only the basics. For this reason, based on a subjective judgement, I left out several import ...
... Nevertheless, we will try our best not to lose our basic intuition by making occasional remarks to the philosophical and applied considerations. The intended course is a short one, and these notes will cover only the basics. For this reason, based on a subjective judgement, I left out several import ...
Introduction to first order logic for knowledge representation
... In describing a phenomena or a portion of the world, we adopt a language. The phrases of this language are used to describe objects of the real worlds, their properties, and facts that holds. This language can be informal (natural language, graphical language, icons, etc...) or a formal (logical lan ...
... In describing a phenomena or a portion of the world, we adopt a language. The phrases of this language are used to describe objects of the real worlds, their properties, and facts that holds. This language can be informal (natural language, graphical language, icons, etc...) or a formal (logical lan ...
On Decidability of Intuitionistic Modal Logics
... τx (ϕ ⇒ ψ) := ∀y(R(x, y) → (¬τy (ϕ) ∨ τy (ψ))) τx (2ϕ) := ∀y(R(x, y) → ∀x(R2 (y, x) → τx (ϕ))) τx (3ϕ) := ∀y(R(x, y) → ∃x(R3 (y, x) ∧ τx (ϕ))) τy is defined analogously, switching the roles of x and y. This translation assumes modal truth clauses (22 ) and (32 ). Clauses for (21 ) and (31 ) are even ...
... τx (ϕ ⇒ ψ) := ∀y(R(x, y) → (¬τy (ϕ) ∨ τy (ψ))) τx (2ϕ) := ∀y(R(x, y) → ∀x(R2 (y, x) → τx (ϕ))) τx (3ϕ) := ∀y(R(x, y) → ∃x(R3 (y, x) ∧ τx (ϕ))) τy is defined analogously, switching the roles of x and y. This translation assumes modal truth clauses (22 ) and (32 ). Clauses for (21 ) and (31 ) are even ...
A Proof Theory for Generic Judgments
... There are, of course, at least a few ways to prove the universally quantified formula ∀τ x.B. The extensional approach attempts to prove B[t/x] for all (closed) terms t of type τ . This rule might involve an infinite number of premises if the domain of the type τ is infinite. If the type τ is define ...
... There are, of course, at least a few ways to prove the universally quantified formula ∀τ x.B. The extensional approach attempts to prove B[t/x] for all (closed) terms t of type τ . This rule might involve an infinite number of premises if the domain of the type τ is infinite. If the type τ is define ...
propositional logic extended with a pedagogically useful relevant
... nonsense to say that the implication is inappropriate because the child knows that the consequent is true. Similarly, “It is not so that it rains if I want it to rain” and, more idiomatically, “It does not rain if I want it to” are decent English sentences. Their logical form is obviously not W → ¬R ...
... nonsense to say that the implication is inappropriate because the child knows that the consequent is true. Similarly, “It is not so that it rains if I want it to rain” and, more idiomatically, “It does not rain if I want it to” are decent English sentences. Their logical form is obviously not W → ¬R ...
Teach Yourself Logic 2017: A Study Guide
... for further work – the core chapters of these cover the so-called ‘baby logic’ that it would be ideal for a non-mathematician to have under his or her belt: 1. My Introduction to Formal Logic* (CUP 2003: a second edition is in preparation): for more details see the IFL pages, where there are also an ...
... for further work – the core chapters of these cover the so-called ‘baby logic’ that it would be ideal for a non-mathematician to have under his or her belt: 1. My Introduction to Formal Logic* (CUP 2003: a second edition is in preparation): for more details see the IFL pages, where there are also an ...
Teach Yourself Logic 2016: A Study Guide
... for further work – the core chapters of these cover the so-called ‘baby logic’ that it would be ideal for a non-mathematician to have under his or her belt: 1. My Introduction to Formal Logic* (CUP 2003, corrected reprint 2013): for more details see the IFL pages, where there are also answers to the ...
... for further work – the core chapters of these cover the so-called ‘baby logic’ that it would be ideal for a non-mathematician to have under his or her belt: 1. My Introduction to Formal Logic* (CUP 2003, corrected reprint 2013): for more details see the IFL pages, where there are also answers to the ...
Conjunctive normal form - Computer Science and Engineering
... boolean formula expressed in Conjunctive Normal Form, such that the formula is true. The k-SAT problem is the problem of finding a satisfying assignment to a boolean formula expressed in CNF in which each disjunction contains at most k variables. 3-SAT is NP-complete (like any other k-SAT problem wi ...
... boolean formula expressed in Conjunctive Normal Form, such that the formula is true. The k-SAT problem is the problem of finding a satisfying assignment to a boolean formula expressed in CNF in which each disjunction contains at most k variables. 3-SAT is NP-complete (like any other k-SAT problem wi ...
higher-order logic - University of Amsterdam
... In addition to its primitives all and some, a first-order predicate language with identity can also express such quantifiers as precisely one, all but two, at most three, etcetera, referring to specific finite quantities. What is lacking, however, is the general mathematical concept of finiteness. E ...
... In addition to its primitives all and some, a first-order predicate language with identity can also express such quantifiers as precisely one, all but two, at most three, etcetera, referring to specific finite quantities. What is lacking, however, is the general mathematical concept of finiteness. E ...
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.