LOGIC I 1. The Completeness Theorem 1.1. On consequences and

... model of T . Does the converse hold? The question was posed in the 1920’s by David Hilbert (of the 23 problems fame). The answer is that remarkably, yes, it does! This result, known as the Completeness Theorem for first-order logic, was proved by Kurt Gödel in 1929. According to the Completeness Th ...

... model of T . Does the converse hold? The question was posed in the 1920’s by David Hilbert (of the 23 problems fame). The answer is that remarkably, yes, it does! This result, known as the Completeness Theorem for first-order logic, was proved by Kurt Gödel in 1929. According to the Completeness Th ...

Introduction to Modal Logic - CMU Math

... then w1 “knows about” w2 and must consider it in making decisions about whether something is possible or necessary. V is a function mapping the set of propositional variables P to P(W ). The interpretation is the if P is mapped into a set contain w then w thinks that the variable P is true. ...

... then w1 “knows about” w2 and must consider it in making decisions about whether something is possible or necessary. V is a function mapping the set of propositional variables P to P(W ). The interpretation is the if P is mapped into a set contain w then w thinks that the variable P is true. ...

Prolog 1 - Department of Computer Science

... • A monk named Gaunilo, complained that if Anselm’s argument proved the existence of a greatest conceivable being, it also proved the existence of an island than which no greater island can be thought. • Kant argued that even if Anselm’s argument works for properties, it does not work for “existence ...

... • A monk named Gaunilo, complained that if Anselm’s argument proved the existence of a greatest conceivable being, it also proved the existence of an island than which no greater island can be thought. • Kant argued that even if Anselm’s argument works for properties, it does not work for “existence ...

P,Q

... Ex 20: [constructive proof] Show that there are n consecutive composite integers for every integer n >0. (I.e. for all n x (x+1,x+2,...x+n) are all composite. Sol: Let x = (n+1)! +1. => x+i = (n+1)! + (i+1) = (i+1)( (n+1)!/(i+1) +1) is composite for i = 1,..,n. QED. Ex 21: [nonconstructive proof] F ...

... Ex 20: [constructive proof] Show that there are n consecutive composite integers for every integer n >0. (I.e. for all n x (x+1,x+2,...x+n) are all composite. Sol: Let x = (n+1)! +1. => x+i = (n+1)! + (i+1) = (i+1)( (n+1)!/(i+1) +1) is composite for i = 1,..,n. QED. Ex 21: [nonconstructive proof] F ...

Default reasoning using classical logic

... and 5 we discuss how the models presented in Section 3 can be treated as classical models of propositional logic. We present algorithms that associate for each nite default theory a classical propositional theory that characterizes its extensions. Then, in Section 6 we use constraint satisfaction t ...

... and 5 we discuss how the models presented in Section 3 can be treated as classical models of propositional logic. We present algorithms that associate for each nite default theory a classical propositional theory that characterizes its extensions. Then, in Section 6 we use constraint satisfaction t ...

Canonicity and representable relation algebras

... • But, unlike in modal logic, knowing RRA is canonical doesn’t seem to help to axiomatise it. • It turns out that RRA is only barely canonical. The connection between A being representable and Aσ being representable is rather loose. The rest of the talk is mainly about this. ...

... • But, unlike in modal logic, knowing RRA is canonical doesn’t seem to help to axiomatise it. • It turns out that RRA is only barely canonical. The connection between A being representable and Aσ being representable is rather loose. The rest of the talk is mainly about this. ...

page 135 ADAPTIVE LOGICS FOR QUESTION EVOCATION

... Where W denotes the set of declarative wffs of L, a partition of L is a couple P = hT, F i such that T ∩ F = ∅ and T ∪ F = W. A declarative wff A is true in a partition P = hT, F i iff A ∈ T ; otherwise A is false. A partition is called admissible iff it is determined by the underlying semantics of ...

... Where W denotes the set of declarative wffs of L, a partition of L is a couple P = hT, F i such that T ∩ F = ∅ and T ∪ F = W. A declarative wff A is true in a partition P = hT, F i iff A ∈ T ; otherwise A is false. A partition is called admissible iff it is determined by the underlying semantics of ...

DISCRETE MATHEMATICAL STRUCTURES

... elements is irrelevant, so {a, b} = {b, a}. If the order of the elements is relevant, then we use a different object called ordered pair, represented (a, b). Now (a, b) = (b, a) (unless a = b). In general (a, b) = (a!, b! ) iff a = a! and b = b! . Given two sets A, B, their Cartesian product A × B i ...

... elements is irrelevant, so {a, b} = {b, a}. If the order of the elements is relevant, then we use a different object called ordered pair, represented (a, b). Now (a, b) = (b, a) (unless a = b). In general (a, b) = (a!, b! ) iff a = a! and b = b! . Given two sets A, B, their Cartesian product A × B i ...

Chapter 9: Initial Theorems about Axiom System AS1

... from Γ looks like, but only that such a derivation exists. To see that the Deduction Theorem is not trivial, consider the simplest case imaginable. It is very easy to derive P from P (indeed, in any system!). The singleton sequence 〈P〉 is such a derivation; every line is either a premise or follows ...

... from Γ looks like, but only that such a derivation exists. To see that the Deduction Theorem is not trivial, consider the simplest case imaginable. It is very easy to derive P from P (indeed, in any system!). The singleton sequence 〈P〉 is such a derivation; every line is either a premise or follows ...

Proof theory for modal logic

... An axiom system for modal logic can be an extension of intuitionistic or classical propositional logic. In the latter, the notions of necessity and possibility are interdefinable by the equivalence 2A ⊃⊂ ¬3¬A. It is seen that necessity and possibility behave analogously to the quantifiers: In one in ...

... An axiom system for modal logic can be an extension of intuitionistic or classical propositional logic. In the latter, the notions of necessity and possibility are interdefinable by the equivalence 2A ⊃⊂ ¬3¬A. It is seen that necessity and possibility behave analogously to the quantifiers: In one in ...

A brief introduction to Logic and its applications

... Another reason why one could not prove P ∨ ¬P ? When you prove a statement such as A ∨ B you can extract a proof that answers whether A or B holds. If we were able to prove the excluded middle, we could extract an algorithm that, given some proposition tells us whether it is valid or not (Curry-Howa ...

... Another reason why one could not prove P ∨ ¬P ? When you prove a statement such as A ∨ B you can extract a proof that answers whether A or B holds. If we were able to prove the excluded middle, we could extract an algorithm that, given some proposition tells us whether it is valid or not (Curry-Howa ...

The Logic of Atomic Sentences

... What it takes for an argument to be good (correct): How to demonstrate that an inference is valid: a proof A proof breaks a non-obvious inference down into a series of trivial, obvious steps which lead you from the premises to the conclusion These steps are based on facts about the meaning of the te ...

... What it takes for an argument to be good (correct): How to demonstrate that an inference is valid: a proof A proof breaks a non-obvious inference down into a series of trivial, obvious steps which lead you from the premises to the conclusion These steps are based on facts about the meaning of the te ...

The Logic of Provability

... The syntax of GL is precisely the same as those systems outlined above, and so it is omitted here. The theorems of GL differ greatly from that of the other modal logics listed because of the addition of a new axiom: Definition: GL The logic generated by the following axioms, • All tautologies of pro ...

... The syntax of GL is precisely the same as those systems outlined above, and so it is omitted here. The theorems of GL differ greatly from that of the other modal logics listed because of the addition of a new axiom: Definition: GL The logic generated by the following axioms, • All tautologies of pro ...

The Foundations

... is true ? => The proposition:” It_is_raining” is true iff the condition (or fact) that the sentence is intended to state really occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now(the current situation), the statement It_is_r ...

... is true ? => The proposition:” It_is_raining” is true iff the condition (or fact) that the sentence is intended to state really occurs(happens, exists) in the situation which the proposition is intended to describe. =>Example: Since it is not raining now(the current situation), the statement It_is_r ...

ON LOVELY PAIRS OF GEOMETRIC STRUCTURES 1. Introduction

... Note that such a theory T extends DLO so any lovely T -pair (M, P (M )) is a dense pair. It is proved in [9, Theorem 2.5] that the common theory of dense pairs is complete, and thus it coincides with TP . Thus, the study of TP can be seen as a generalization of van den Dries’ work on dense pairs of ...

... Note that such a theory T extends DLO so any lovely T -pair (M, P (M )) is a dense pair. It is proved in [9, Theorem 2.5] that the common theory of dense pairs is complete, and thus it coincides with TP . Thus, the study of TP can be seen as a generalization of van den Dries’ work on dense pairs of ...

The Project Gutenberg EBook of The Algebra of Logic, by Louis

... Venn ) as Symbolic Logic; and Symbolic Logic is, in essentials, the Logic of Aristotle, given new life and power by being dressed up in the wonderful almost magicalarmour and accoutrements of Algebra. In less than seventy years, logic, to use an expression of De Morgan's, has so thriven upon sym ...

... Venn ) as Symbolic Logic; and Symbolic Logic is, in essentials, the Logic of Aristotle, given new life and power by being dressed up in the wonderful almost magicalarmour and accoutrements of Algebra. In less than seventy years, logic, to use an expression of De Morgan's, has so thriven upon sym ...

Gentzen`s original consistency proof and the Bar Theorem

... wanted a way to understand the truth of a sentence of number theory that is in some sense ‘finitary’ but at the same time supported classical reasoning in number theory. In this respect, the original paper as well as the 1936 paper go beyond the original Hilbert program of finding finitary consisten ...

... wanted a way to understand the truth of a sentence of number theory that is in some sense ‘finitary’ but at the same time supported classical reasoning in number theory. In this respect, the original paper as well as the 1936 paper go beyond the original Hilbert program of finding finitary consisten ...

The Model-Theoretic Ordinal Analysis of Theories of Predicative

... though they differ from the "standard"assignments only slightly. Note that different notations can denote the same ordinal, as is the case with so and cowo.Further note that equivalent notations need not have equivalent limit sequences; for example, eo[n] = On+2, but co60[n] = e)'n+2+l. We assume th ...

... though they differ from the "standard"assignments only slightly. Note that different notations can denote the same ordinal, as is the case with so and cowo.Further note that equivalent notations need not have equivalent limit sequences; for example, eo[n] = On+2, but co60[n] = e)'n+2+l. We assume th ...

Lecture Notes on Stability Theory

... consists of an underlying set M , together with some distinguished relations Ri (subsets of M ni , ni ∈ N), functions fi : M ni → M , and constants ci (distinguished elements of M ). We refer to the collection of all these relations, function symbols and constants as the signature of M. For exampl ...

... consists of an underlying set M , together with some distinguished relations Ri (subsets of M ni , ni ∈ N), functions fi : M ni → M , and constants ci (distinguished elements of M ). We refer to the collection of all these relations, function symbols and constants as the signature of M. For exampl ...

Completeness or Incompleteness of Basic Mathematical Concepts

... natural to regard this way of being determined as being implied by the concept. Whether I am right or wrong about Gödel’s use, I will always use “implied be the concept” in a non-epistemic sense.56 (4) The concept of the natural numbers is first-order complete: it determines truth values for all s ...

... natural to regard this way of being determined as being implied by the concept. Whether I am right or wrong about Gödel’s use, I will always use “implied be the concept” in a non-epistemic sense.56 (4) The concept of the natural numbers is first-order complete: it determines truth values for all s ...

Belief Revision in non

... In general, different translation mechanisms can be defined from a given object logic to classical logic, depending on the underlying semantic structures used by the object logic. In the case of non-classical logics that are extensions of classical logic, the translation mechanism essentially maps t ...

... In general, different translation mechanisms can be defined from a given object logic to classical logic, depending on the underlying semantic structures used by the object logic. In the case of non-classical logics that are extensions of classical logic, the translation mechanism essentially maps t ...

Conjunctive normal form - Computer Science and Engineering

... An important set of problems in computational complexity involves finding assignments to the variables of a 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 i ...

... An important set of problems in computational complexity involves finding assignments to the variables of a 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 i ...