Predicate Logic
... We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat is ...
... We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat is ...
Predicate logic
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
INTRODUCTION TO LOGIC Lecture 6 Natural Deduction Proofs in
... The occurrence of a sentence φ with no sentence above it is an assumption. An assumption of φ is a proof of φ. This may seem odd. Suppose I assume, the following: ∃x∃y(Rxy ∨ P ) By the rule, this counts as a proof of ∃x∃y(Rxy ∨ P ) But it is not an outright proof of ∃x∃y(Rxy ∨ P ) This proof does no ...
... The occurrence of a sentence φ with no sentence above it is an assumption. An assumption of φ is a proof of φ. This may seem odd. Suppose I assume, the following: ∃x∃y(Rxy ∨ P ) By the rule, this counts as a proof of ∃x∃y(Rxy ∨ P ) But it is not an outright proof of ∃x∃y(Rxy ∨ P ) This proof does no ...
The Diagonal Lemma Fails in Aristotelian Logic
... Gödel's theorem does not need an introduction. But it is generally not acknowledged that all instances of Gödel's sentence are vacuous (see below). On the contrary Aristotelian logic permits only sentences that are not vacuous. The consequences of these observations are surprising. What does the art ...
... Gödel's theorem does not need an introduction. But it is generally not acknowledged that all instances of Gödel's sentence are vacuous (see below). On the contrary Aristotelian logic permits only sentences that are not vacuous. The consequences of these observations are surprising. What does the art ...
03_Artificial_Intelligence-PredicateLogic
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
Tools-Slides-3 - Michael Johnson`s Homepage
... For any two sets A and B, A = B iff (for all x)(x ∈ A iff x ∈ B) Therefore, {1, 1} = {1} iff (for all x)(x ∈ {1, 1} iff x ∈ {1}) ...
... For any two sets A and B, A = B iff (for all x)(x ∈ A iff x ∈ B) Therefore, {1, 1} = {1} iff (for all x)(x ∈ {1, 1} iff x ∈ {1}) ...
Predicate logic - Teaching-WIKI
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
Predicate logic
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
... • We'd like to conclude that Jan will get wet. But each of these sentences would just be a represented by some proposition, say P, Q and R. What relationship is there between these propositions? We can say P /\ Q → R Then, given P /\ Q, we could indeed conclude R. But now, suppose we were told Pat i ...
Philosophy 240: Symbolic Logic
... P “It seems to me obvious that the only rational approach to [questions about the correct notion of truth] would be the following: We should reconcile ourselves with the fact that we are confronted, not with one concept, but with several different concepts which are denoted by one word; we should tr ...
... P “It seems to me obvious that the only rational approach to [questions about the correct notion of truth] would be the following: We should reconcile ourselves with the fact that we are confronted, not with one concept, but with several different concepts which are denoted by one word; we should tr ...
Knowledge representation 1
... Some well-established rules of inference (i.e. ways of proving an argument is sound). For instance, if you know that CD and you know that D isn’t true, then you know that ¬C must be true - a rule of inference known as modus tolens. ...
... Some well-established rules of inference (i.e. ways of proving an argument is sound). For instance, if you know that CD and you know that D isn’t true, then you know that ¬C must be true - a rule of inference known as modus tolens. ...
on fuzzy intuitionistic logic
... The value T(a) € L is the degree of truth of a in interpretation T. If T(a) = 1 we say that T is a model of a. If T(a) = 0, then a is false in interpretation T. By semantics Sx with respect to some set of formulae X, we understand the set of all models of X. Similarly, as in classical logic, it is r ...
... The value T(a) € L is the degree of truth of a in interpretation T. If T(a) = 1 we say that T is a model of a. If T(a) = 0, then a is false in interpretation T. By semantics Sx with respect to some set of formulae X, we understand the set of all models of X. Similarly, as in classical logic, it is r ...
Lesson 1
... (depending on the expressive power of the logical system). Logical connectives (‘and’, ‘or’, ‘if …then …’) and quantifiers (‘all’, ‘some’, ‘every’, …) have a fixed interpretation; we interpret elementary propositions and/or their parts. In our example, if “this apple” and “agarics” were interpreted ...
... (depending on the expressive power of the logical system). Logical connectives (‘and’, ‘or’, ‘if …then …’) and quantifiers (‘all’, ‘some’, ‘every’, …) have a fixed interpretation; we interpret elementary propositions and/or their parts. In our example, if “this apple” and “agarics” were interpreted ...
2/TRUTH-FUNCTIONS
... `Or' implies alternative in which 1. only one alternative is possible both conceptually and factually; 2. that both alternatives are in fact possible. The exclusion of joined truth gives the idea of a strict disjunction. Its symbolic formulation: [(pvq).-(p.q)] In logic, as in mathematics, the `or’ ...
... `Or' implies alternative in which 1. only one alternative is possible both conceptually and factually; 2. that both alternatives are in fact possible. The exclusion of joined truth gives the idea of a strict disjunction. Its symbolic formulation: [(pvq).-(p.q)] In logic, as in mathematics, the `or’ ...
Notes5
... In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial intelligence. We start with propositional logic, using symbols to stand for things that can be eith ...
... In this part of the course we consider logic. Logic is used in many places in computer science including digital circuit design, relational databases, automata theory and computability, and artificial intelligence. We start with propositional logic, using symbols to stand for things that can be eith ...
T - RTU
... An inference rule is sound, if the conclusion is true in all cases where the premises are true. To prove the soundness, the truth table must be constructed with one line for each possible model of the proposition symbols in the premises. In all models where the premise is true, the conclusion must b ...
... An inference rule is sound, if the conclusion is true in all cases where the premises are true. To prove the soundness, the truth table must be constructed with one line for each possible model of the proposition symbols in the premises. In all models where the premise is true, the conclusion must b ...
First-Order Predicate Logic (2) - Department of Computer Science
... F |= G versus X |= G • Note that F |= G or F |= ¬G, for every sentence G. Thus, we have complete information about the domain of discourse. There are many examples where X 6|= G and X 6|= ¬G. We have incomplete information. • F |= G means that G is true in the structure F . Checking whether this is ...
... F |= G versus X |= G • Note that F |= G or F |= ¬G, for every sentence G. Thus, we have complete information about the domain of discourse. There are many examples where X 6|= G and X 6|= ¬G. We have incomplete information. • F |= G means that G is true in the structure F . Checking whether this is ...
Logical nihilism - University of Notre Dame
... I shall not argue for the thesis that there are no correct logics. Although I do find attempts from our history to paint a convincing picture of a relation of logical consequence that attains among propositions (or sentences, or whatever) dubious, I should not know how to cast general doubt on the v ...
... I shall not argue for the thesis that there are no correct logics. Although I do find attempts from our history to paint a convincing picture of a relation of logical consequence that attains among propositions (or sentences, or whatever) dubious, I should not know how to cast general doubt on the v ...
slides
... Want a way to prove partial correctness statements valid... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional ...
... Want a way to prove partial correctness statements valid... ... without having to consider explicitly every store and interpretation! Idea: develop a proof system in which every theorem is a valid partial correctness statement Judgements of the form ⊢ {P} c {Q} De ned inductively using compositional ...
Plural Quantifiers
... cannot be given a first-order formulation. The proof uses some concepts from metalogic, so don’t worry if you can’t understand it. For those who are interested, though, here’s the basic idea: 1. If there were a first-order formula that captured the meaning of (B), it would be possible to give first- ...
... cannot be given a first-order formulation. The proof uses some concepts from metalogic, so don’t worry if you can’t understand it. For those who are interested, though, here’s the basic idea: 1. If there were a first-order formula that captured the meaning of (B), it would be possible to give first- ...
Notes
... Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponents were Kronecker and Brouwer around the beginning of the last cent ...
... Intuitionistic logic is the basis of constructive mathematics. Constructive mathematics takes a much more conservative view of truth than classical mathematics. It is concerned less with truth than with provability. Its main proponents were Kronecker and Brouwer around the beginning of the last cent ...
Propositional Logic Proof
... propositional logic statement, (2) each statement is a premise or follows unequivocally by a previously established rule of inference from the truth of previous statements, and (3) the last statement is the conclusion. A very constrained form of proof, but a good starting point. Interesting proofs w ...
... propositional logic statement, (2) each statement is a premise or follows unequivocally by a previously established rule of inference from the truth of previous statements, and (3) the last statement is the conclusion. A very constrained form of proof, but a good starting point. Interesting proofs w ...
Lesson 2
... • The simplest logical system. It analyzes a way of composing a complex sentence (proposition) from elementary propositions by means of logical connectives. • What is a proposition? A proposition (sentence) is a statement that can be said to be true or false. • The Two-Value Principle – tercium non ...
... • The simplest logical system. It analyzes a way of composing a complex sentence (proposition) from elementary propositions by means of logical connectives. • What is a proposition? A proposition (sentence) is a statement that can be said to be true or false. • The Two-Value Principle – tercium non ...
Assumption Sets for Extended Logic Programs
... For any models M = hH, T i, M0 = hH 0 , T 0 i, we set M ≤ M0 iff T = T 0 and H ⊆ H 0 . A model M of a program Π is said to be a minimal model of Π, if it is minimal under the ≤-ordering among all models of Π. Definition 3 An N 2-model hH, T i of Π is said to be an equilibrium model of Π iff it is mi ...
... For any models M = hH, T i, M0 = hH 0 , T 0 i, we set M ≤ M0 iff T = T 0 and H ⊆ H 0 . A model M of a program Π is said to be a minimal model of Π, if it is minimal under the ≤-ordering among all models of Π. Definition 3 An N 2-model hH, T i of Π is said to be an equilibrium model of Π iff it is mi ...
Definition - Rogelio Davila
... Def. An inference rule is a machinery for producing new valid statements from previously obtained ones. Def. An axiomatic system consists of a set of axioms (or axioms schemata) and a set of inference rules. Def. In an axiomatic system, valid statements produced by the system are called theorems. ...
... Def. An inference rule is a machinery for producing new valid statements from previously obtained ones. Def. An axiomatic system consists of a set of axioms (or axioms schemata) and a set of inference rules. Def. In an axiomatic system, valid statements produced by the system are called theorems. ...
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.