Ch1 - COW :: Ceng

... circuit design There are efficient algorithms for reasoning in propositional logic Propositional logic is a foundation for most of the more expressive logics ...

... circuit design There are efficient algorithms for reasoning in propositional logic Propositional logic is a foundation for most of the more expressive logics ...

Handout 14

... provable is a tautology. Thus, the formal system of propositional logic is not only sound (i.e. generates only valid formulas) but also generates all of them. Theorem 5.2 (completeness of propositional logic). Let T be a set of formulas and A a formula. Then T (A ...

... provable is a tautology. Thus, the formal system of propositional logic is not only sound (i.e. generates only valid formulas) but also generates all of them. Theorem 5.2 (completeness of propositional logic). Let T be a set of formulas and A a formula. Then T (A ...

323-670 ปัญญาประดิษฐ์ (Artificial Intelligence)

... • Proof is a sequence of sentences First ones are premises (KB) Then, you can write down on line j the result of applying an inference rule to previous lines When f is on a line, you know KB f If inference rules are sound, then KB f ...

... • Proof is a sequence of sentences First ones are premises (KB) Then, you can write down on line j the result of applying an inference rule to previous lines When f is on a line, you know KB f If inference rules are sound, then KB f ...

.pdf

... Definition 1 ϕ is valid (or logically true, or true by virtue of logic, or a tautology) if v0 |= ϕ for all interpretations v0 . We typically write this as |= ϕ. Thus,|= (p ∧ q) ⇒ p is valid. In fact, more generally, for any ϕ and ψ, |= (ϕ ∧ ψ() ⇒ ϕ. How do you determine if a formula ϕ is valid? We n ...

... Definition 1 ϕ is valid (or logically true, or true by virtue of logic, or a tautology) if v0 |= ϕ for all interpretations v0 . We typically write this as |= ϕ. Thus,|= (p ∧ q) ⇒ p is valid. In fact, more generally, for any ϕ and ψ, |= (ϕ ∧ ψ() ⇒ ϕ. How do you determine if a formula ϕ is valid? We n ...

PDF

... theorem, and the only if part is the completeness theorem. We will prove the two parts separately here. We begin with the easier one: Theorem 1. Propositional logic is sound with respect to truth-value semantics. Proof. Basically, we need to show that every axiom is a tautology, and that the inferen ...

... theorem, and the only if part is the completeness theorem. We will prove the two parts separately here. We begin with the easier one: Theorem 1. Propositional logic is sound with respect to truth-value semantics. Proof. Basically, we need to show that every axiom is a tautology, and that the inferen ...

Programming and Problem Solving with Java: Chapter 14

... Not logically valid, BUT can still be useful. In fact, it models the way humans reason all the time: Every non-flying bird I’ve seen before has been a penguin; hence that non-flying bird must be a ...

... Not logically valid, BUT can still be useful. In fact, it models the way humans reason all the time: Every non-flying bird I’ve seen before has been a penguin; hence that non-flying bird must be a ...

Propositional Logic Predicate Logic

... We write N for the set of natural numbers. (In logic and computer science, 0 is a natural number.) Axiom (Induction Principle on Natural Numbers). For all unary predicates P on N, ...

... We write N for the set of natural numbers. (In logic and computer science, 0 is a natural number.) Axiom (Induction Principle on Natural Numbers). For all unary predicates P on N, ...

IntroToLogic - Department of Computer Science

... In any logical language expressive enough to describe the properties of the natural numbers, there are true statements that are undecidable -- their truth cannot be established by any algorithm. ...

... In any logical language expressive enough to describe the properties of the natural numbers, there are true statements that are undecidable -- their truth cannot be established by any algorithm. ...

.pdf

... Substitution A|pB is the replacement of all occurrences of the variable p in A by the formula B. There are a few issues, however, that one needs to be aware of. Variables that are bound by a quantifier, must not be replaced, as this would change the meaning. ((∃p)(p⊃∼q))|qp should not result in ((∃p ...

... Substitution A|pB is the replacement of all occurrences of the variable p in A by the formula B. There are a few issues, however, that one needs to be aware of. Variables that are bound by a quantifier, must not be replaced, as this would change the meaning. ((∃p)(p⊃∼q))|qp should not result in ((∃p ...

1

... (c) Prove the following by induction (on construction of formulas): if variable assignments s and s0 agree on all variables occurring in a formula φ, then |=U φ [s] iff |=U φ [s0 ] for all formulas φ and interpretations U. (You may assume that the language has only the connectives ¬ and → and that ∀ ...

... (c) Prove the following by induction (on construction of formulas): if variable assignments s and s0 agree on all variables occurring in a formula φ, then |=U φ [s] iff |=U φ [s0 ] for all formulas φ and interpretations U. (You may assume that the language has only the connectives ¬ and → and that ∀ ...

Slides - UCSD CSE

... Assume, to the contrary that ______________________ (~p) Then, __________________________________ (formula that follows from p) Now, _________________________ (p " ~p) ...

... Assume, to the contrary that ______________________ (~p) Then, __________________________________ (formula that follows from p) Now, _________________________ (p " ~p) ...

Lecture_ai_3 - WordPress.com

... • Interpretation of implication is T if the previous statement has T value • Interpretation of Biconditionalis T only when symbols on the both sides are either T or F ,otherwise F ...

... • Interpretation of implication is T if the previous statement has T value • Interpretation of Biconditionalis T only when symbols on the both sides are either T or F ,otherwise F ...

Definition - Rogelio Davila

... (A1) (X (Y X)) (A2) ((X (Y Z)) ((X Y) (X Z))) (A3) ((X Y) ((X Y) X)) ...

... (A1) (X (Y X)) (A2) ((X (Y Z)) ((X Y) (X Z))) (A3) ((X Y) ((X Y) X)) ...

Bound and Free Variables Theorems and Proofs

... • syntax: what are the valid formulas • semantics: under what circumstances is a formula true • proof theory/ axiomatization: rules for proving a formula true Truth and provability are quite different. • What is provable depends on the axioms and inference rules you use • Provability is a mechanical ...

... • syntax: what are the valid formulas • semantics: under what circumstances is a formula true • proof theory/ axiomatization: rules for proving a formula true Truth and provability are quite different. • What is provable depends on the axioms and inference rules you use • Provability is a mechanical ...

Predicate Calculus pt. 2

... finite subset of T is satisfiable and enlarging T to a maximal set of propositional formulas T ∗ (in the same variables) so that every finite subset of T ∗ is satisfiable and let µ(p) = W ⇐⇒ p ∈ T ∗ . Show that µ makes all formulas in T true. Exercise 2 A (symmetric, irreflexive) graph G = (V, E) co ...

... finite subset of T is satisfiable and enlarging T to a maximal set of propositional formulas T ∗ (in the same variables) so that every finite subset of T ∗ is satisfiable and let µ(p) = W ⇐⇒ p ∈ T ∗ . Show that µ makes all formulas in T true. Exercise 2 A (symmetric, irreflexive) graph G = (V, E) co ...

Propositional Logic

... (A1) (X (Y X)) (A2) ((X (Y Z)) ((X Y) (X Z))) (A3) ((X Y) ((X Y) X)) ...

... (A1) (X (Y X)) (A2) ((X (Y Z)) ((X Y) (X Z))) (A3) ((X Y) ((X Y) X)) ...

Homework 5

... (1) Reduce these P 2 formulas to purely propositional formulas. (a) (∀p) p ⊃ ⊥ (b) (∀p)(∀q) ((∼p ∨ q)⊃(p⊃q)) (c) (∀p)(∀q) ((p⊃p ∨ q) ∧ (p ∧ q⊃p)) (2) Give Refinement logic rules for P 2 . (3) Construct an example of a formula that is satisfiable in a denumerable universe but not in any finite one (e ...

... (1) Reduce these P 2 formulas to purely propositional formulas. (a) (∀p) p ⊃ ⊥ (b) (∀p)(∀q) ((∼p ∨ q)⊃(p⊃q)) (c) (∀p)(∀q) ((p⊃p ∨ q) ∧ (p ∧ q⊃p)) (2) Give Refinement logic rules for P 2 . (3) Construct an example of a formula that is satisfiable in a denumerable universe but not in any finite one (e ...

Book Question Set #1: Ertel, Chapter 2: Propositional Logic

... d. ( A IMPLIES B ), where A and B are a propositional variable An implication that, ‘if A then B’ (also known as material implication) e. ( A EQUIVALENT-TO B ), where A and B are a propositional value A statement of equivalence where, ‘A if and only if B’ 6.) What does it mean for two propositional ...

... d. ( A IMPLIES B ), where A and B are a propositional variable An implication that, ‘if A then B’ (also known as material implication) e. ( A EQUIVALENT-TO B ), where A and B are a propositional value A statement of equivalence where, ‘A if and only if B’ 6.) What does it mean for two propositional ...

4 slides/page

... • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the following: Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence thi ...

... • epistemic logic: for reasoning about knowledge The simplest logic (on which all the rest are based) is propositional logic. It is intended to capture features of arguments such as the following: Borogroves are mimsy whenever it is brillig. It is now brillig and this thing is a borogrove. Hence thi ...