Knowledge Representation and Reasoning
... (and with ‘reasonable’ use of other resources such as memory). Certain classes of logical problem are not only intractable but also undecidable. This means that there is no program that, given any instance of the problem, will in finite time either: a) find a solution; or b) terminate having determi ...
... (and with ‘reasonable’ use of other resources such as memory). Certain classes of logical problem are not only intractable but also undecidable. This means that there is no program that, given any instance of the problem, will in finite time either: a) find a solution; or b) terminate having determi ...
2. First Order Logic 2.1. Expressions. Definition 2.1. A language L
... consists of prenex formulas then there is a deduction of Γ ⇒ Σ in which the quantifier rules (R∀, R∃, L∀, L∃) appear below every other inference rule. We begin with a cut-free deduction of Γ ⇒ Σ. The idea is to take the (or a, since the proof might branch) bottom-most quantifier rule in the proof an ...
... consists of prenex formulas then there is a deduction of Γ ⇒ Σ in which the quantifier rules (R∀, R∃, L∀, L∃) appear below every other inference rule. We begin with a cut-free deduction of Γ ⇒ Σ. The idea is to take the (or a, since the proof might branch) bottom-most quantifier rule in the proof an ...
Natural deduction
... Why natural deduction? • After all, we just found this nice method of truth-tables, which can be used to determine the validity or invalidity of any argument in truth-functional logic! • But there are some problems with truth-tables – They can be very inefficient. The reason is that if you have an a ...
... Why natural deduction? • After all, we just found this nice method of truth-tables, which can be used to determine the validity or invalidity of any argument in truth-functional logic! • But there are some problems with truth-tables – They can be very inefficient. The reason is that if you have an a ...
Propositional Logic: Part I - Semantics
... Then f : R → R is the signature of f meaning f takes a real argument and produce a real. Here − is a unary prefix operator meaning it takes one argument, the number immediately following − (e.g., −(5) = −5). So really −:R→R Similarly + : R2 → R + is a binary operator on R so we could treat it as a p ...
... Then f : R → R is the signature of f meaning f takes a real argument and produce a real. Here − is a unary prefix operator meaning it takes one argument, the number immediately following − (e.g., −(5) = −5). So really −:R→R Similarly + : R2 → R + is a binary operator on R so we could treat it as a p ...
Resolution Proof System for First Order Logic
... Process for removing existential quantifiers. Delete each existential quantifier, then replace the resulting free variables by terms referred to as Skolem functions. ...
... Process for removing existential quantifiers. Delete each existential quantifier, then replace the resulting free variables by terms referred to as Skolem functions. ...
THE HISTORY OF LOGIC
... satisfaction. This, however, is but a small fraction of his work, which illuminates the methodology of deductive systems, and such central notions as completeness, decidability, consistency, satisfiablility, and definability. His results are the doundation of several ongoing research programmes. Alo ...
... satisfaction. This, however, is but a small fraction of his work, which illuminates the methodology of deductive systems, and such central notions as completeness, decidability, consistency, satisfiablility, and definability. His results are the doundation of several ongoing research programmes. Alo ...
Quiz Game Midterm
... Let’s do a proof by cases on premise 3. Case 1: Assume that A. From this and premise 1, B follows by modus ponens. Case 2: Assume that C. This is equivalent to C by double negation. From this and premise 2, B follows by modus tollens. But by double negation, this is just B. So either way B follo ...
... Let’s do a proof by cases on premise 3. Case 1: Assume that A. From this and premise 1, B follows by modus ponens. Case 2: Assume that C. This is equivalent to C by double negation. From this and premise 2, B follows by modus tollens. But by double negation, this is just B. So either way B follo ...
• Above we applied the unit resolution inference rule: ℓ1 ∨ … ∨ ℓ k
... • Backward chaining is a form of goal-directed reasoning • We aim at showing the truth of a given query q by considering those implications in the knowledge base that conclude q (have it as the head) • By backward chaining one examines the whether all the premises of one of those implications can be ...
... • Backward chaining is a form of goal-directed reasoning • We aim at showing the truth of a given query q by considering those implications in the knowledge base that conclude q (have it as the head) • By backward chaining one examines the whether all the premises of one of those implications can be ...
Flowchart Thinking
... may not be the clearest way to present the steps. In such cases, it is often helpful to organize the steps in the form of a flowchart. A flowchart is a concept map that shows all the steps in a complicated procedure in their proper order. The steps in the procedure are written in boxes. Arrows conne ...
... may not be the clearest way to present the steps. In such cases, it is often helpful to organize the steps in the form of a flowchart. A flowchart is a concept map that shows all the steps in a complicated procedure in their proper order. The steps in the procedure are written in boxes. Arrows conne ...
2/TRUTH-FUNCTIONS
... 2 refers to the number of truth values and n refers to the number of distinct statement variables. s9. Truth-Functions: Any expression whose truth value is defined in all cases by its logical operator. 9a. Negation: Not (-) curl -> -p Expressions: it is not true that/ it is false that/ it is not the ...
... 2 refers to the number of truth values and n refers to the number of distinct statement variables. s9. Truth-Functions: Any expression whose truth value is defined in all cases by its logical operator. 9a. Negation: Not (-) curl -> -p Expressions: it is not true that/ it is false that/ it is not the ...
CS389L: Automated Logical Reasoning Lecture 1
... Semantic argument method is essentially a proof by contradiction, and is also applicable for theories with non-finite domain. ...
... Semantic argument method is essentially a proof by contradiction, and is also applicable for theories with non-finite domain. ...
The Taming of the (X)OR
... by the shorter clauses towards the longer clauses. For a non-triangular form, the system is under-determined and more than one model exists. Unfortunately, unrestricted application of the Gauss− and Gauss+ rules to a set of xor-clauses may be a non-terminating process. The system might cycle among a ...
... by the shorter clauses towards the longer clauses. For a non-triangular form, the system is under-determined and more than one model exists. Unfortunately, unrestricted application of the Gauss− and Gauss+ rules to a set of xor-clauses may be a non-terminating process. The system might cycle among a ...
STEPS for INDIRECT PROOF - Fairfield Public Schools
... 1) Assume the negation of the “PROVE” is TRUE. (like above when we said ‘if it WAS equilateral’) 2) Use some of the “GIVENS” and other geometry truths to show your assumption from step 1 can’t be true, either because it CONTRADICTS one of these facts, or it leads to a statement that is ABSURD! (like ...
... 1) Assume the negation of the “PROVE” is TRUE. (like above when we said ‘if it WAS equilateral’) 2) Use some of the “GIVENS” and other geometry truths to show your assumption from step 1 can’t be true, either because it CONTRADICTS one of these facts, or it leads to a statement that is ABSURD! (like ...
PPTX
... • By the end of this module, you should be able to • Determine whether or not a propositional logic proof is valid, and explain why it is valid or invalid. • Explore the consequences of a set of propositional logic statements by application of equivalence and inference rules, especially in order to ...
... • By the end of this module, you should be able to • Determine whether or not a propositional logic proof is valid, and explain why it is valid or invalid. • Explore the consequences of a set of propositional logic statements by application of equivalence and inference rules, especially in order to ...
Module 4: Propositional Logic Proofs
... Translating English into propositional logic expressions • Premise 1: If women are too close to femininity to portray women then men must be too close to masculinity to play men, and vice versa. • Premise 2: And yet, if the onnagata are correct, women are too close to femininity to portray wom ...
... Translating English into propositional logic expressions • Premise 1: If women are too close to femininity to portray women then men must be too close to masculinity to play men, and vice versa. • Premise 2: And yet, if the onnagata are correct, women are too close to femininity to portray wom ...
1. Kripke`s semantics for modal logic
... Kripke’s defense of quantified modal logic was basically to embrace the “Aristotelian Essentialism” that Quine assumed was beyond the pale: Some philosophers have distinguished between essentialism, the belief in modality de re, and a mere advocacy of necessity, the belief in modality de dicto. Now, ...
... Kripke’s defense of quantified modal logic was basically to embrace the “Aristotelian Essentialism” that Quine assumed was beyond the pale: Some philosophers have distinguished between essentialism, the belief in modality de re, and a mere advocacy of necessity, the belief in modality de dicto. Now, ...
A Brief Introduction to Propositional Logic
... • If the truth value of a sentence is 1, the truth value of its negation is 0. If the truth value of a sentence is 0, the truth value of its negation is 1. • The truth value of a conjunction is true if and only if the truth value of its conjuncts are both true; otherwise, the truth value is false. • ...
... • If the truth value of a sentence is 1, the truth value of its negation is 0. If the truth value of a sentence is 0, the truth value of its negation is 1. • The truth value of a conjunction is true if and only if the truth value of its conjuncts are both true; otherwise, the truth value is false. • ...
Inquiry
An inquiry is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim.