cl-ch9

... denotations, but an interpretation must still specify a domain, and that specification makes a difference as to truth for closed formulas involving =. For instance, ∃x∃y ∼ x = y will be true if the domain has at least two distinct elements, but false if it has only one.) Closed formulas, which are a ...

... denotations, but an interpretation must still specify a domain, and that specification makes a difference as to truth for closed formulas involving =. For instance, ∃x∃y ∼ x = y will be true if the domain has at least two distinct elements, but false if it has only one.) Closed formulas, which are a ...

24.241 Logic I Problem set 04 solutions

... Inductive Step: If Claim is true for all P containing n or fewer connectives, it is true for all P containing n + 1 connectives. . . . as every sentence of SL contains n connectives, for some natural number n. Proof of Basis Clause: 1. [P](Q1 //P) is just Q1 . 2. So if Q1 is truth-functionally equi ...

... Inductive Step: If Claim is true for all P containing n or fewer connectives, it is true for all P containing n + 1 connectives. . . . as every sentence of SL contains n connectives, for some natural number n. Proof of Basis Clause: 1. [P](Q1 //P) is just Q1 . 2. So if Q1 is truth-functionally equi ...

The First Incompleteness Theorem

... The terminological situation is a bit messy. Some labels we’ll be using are entirely standard; some are used in importantly different ways by different authors; while some natural ideas seem to have no commonly used labels at all. It might be helpful, then, if I star my own non-standard terminology ...

... The terminological situation is a bit messy. Some labels we’ll be using are entirely standard; some are used in importantly different ways by different authors; while some natural ideas seem to have no commonly used labels at all. It might be helpful, then, if I star my own non-standard terminology ...

CHAPTER 1. SENTENTIAL LOGIC 1. Introduction In sentential logic

... • If 7 is not odd then 2 is odd • If 7 is odd and 2 is odd then 2 is not prime • (7 is odd and 2 is odd) or 2 is prime We have improved on English in the last example by using parentheses to resolve an ambiguity. And, or, not, if . . . then (or implies) are called (sentential) connectives. Using the ...

... • If 7 is not odd then 2 is odd • If 7 is odd and 2 is odd then 2 is not prime • (7 is odd and 2 is odd) or 2 is prime We have improved on English in the last example by using parentheses to resolve an ambiguity. And, or, not, if . . . then (or implies) are called (sentential) connectives. Using the ...

Reaching transparent truth

... call LPTT (for LP with transparent truth), is to take a countermodel to be a model that assigns some value greater than 0 to every member of Γ and 0 to every member of ∆. A third way, resulting in the logic we’ll call S3TT (for S3 with transparent truth), is to take a countermodel to be a model on w ...

... call LPTT (for LP with transparent truth), is to take a countermodel to be a model that assigns some value greater than 0 to every member of Γ and 0 to every member of ∆. A third way, resulting in the logic we’ll call S3TT (for S3 with transparent truth), is to take a countermodel to be a model on w ...

Between Semiotics and Pragmatics: Speaker

... language and Morris claims to have derived them from Peirce, who proposed a three-part division of semeiotic inquiry into speculative grammar, logic proper and speculative rhetoric. Speculative grammar studies the general characters of the signs themselves, and in this respect does not go beyond the ...

... language and Morris claims to have derived them from Peirce, who proposed a three-part division of semeiotic inquiry into speculative grammar, logic proper and speculative rhetoric. Speculative grammar studies the general characters of the signs themselves, and in this respect does not go beyond the ...

Between Truth and Falsity

... Hence, if a sentence is too vague to be given a determinate truth value, this is because it does not express a clear proposition (propositions, on this view), being the primary bearers of truth values. Similarly, if we do not know, say, whether the decimal expansion of π ever repeats, or exactly how ...

... Hence, if a sentence is too vague to be given a determinate truth value, this is because it does not express a clear proposition (propositions, on this view), being the primary bearers of truth values. Similarly, if we do not know, say, whether the decimal expansion of π ever repeats, or exactly how ...

What Classical Connectives Mean

... have no agency, because they are already settled. Tautologies about the future are also settled. SçS S = there will a sea battle at noon tomorrow. ...

... have no agency, because they are already settled. Tautologies about the future are also settled. SçS S = there will a sea battle at noon tomorrow. ...

PROPERTIES PRESERVED UNDER ALGEBRAIC

... Further related results are given by K. Bing, 1955. It seems probable t h a t every universal-existential sentence preserved under direct products is equivalent to a Horn sentence ; but an example by Chang and Anne Morel, of which we shall speak later, shows that the analogous statement for existent ...

... Further related results are given by K. Bing, 1955. It seems probable t h a t every universal-existential sentence preserved under direct products is equivalent to a Horn sentence ; but an example by Chang and Anne Morel, of which we shall speak later, shows that the analogous statement for existent ...

The logic and mathematics of occasion sentences

... occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical (Boolean) foundation for such a logic, thus ...

... occasion sentences their rightful central place in semantics and secondly, taking these as the basic propositional elements in the logical analysis of language, to contribute to the development of an adequate logic of occasion sentences and a mathematical (Boolean) foundation for such a logic, thus ...

How to Express Self-Referential Probability and Avoid the

... framework. We therefore do not face the same challenge. The rest of the paper is structured as follows. In the next section, Section 2, we shall introduce the language and the defined semantics. We shall focus on a single agent but our semantics can easily generalise to multiple agents. We in fact b ...

... framework. We therefore do not face the same challenge. The rest of the paper is structured as follows. In the next section, Section 2, we shall introduce the language and the defined semantics. We shall focus on a single agent but our semantics can easily generalise to multiple agents. We in fact b ...

Formal Theories of Truth INTRODUCTION

... I will take truth to apply to sentences of a fixed formal language only, but not to ‘foreign’ sentences. I’ll emphasize the axiomatic approach – at least in the beginning. This is not to say that truth cannot be defined in terms of correspondence etc. ...

... I will take truth to apply to sentences of a fixed formal language only, but not to ‘foreign’ sentences. I’ll emphasize the axiomatic approach – at least in the beginning. This is not to say that truth cannot be defined in terms of correspondence etc. ...

Logics of Truth - Project Euclid

... the class of propositions the truth predicate obeys the Tar ski criteria. We shall briefly review the theory of Frege structures and use it as a way into the theory of Kripke-Gilmore-Feferman. 1.2 The Kripke-Feferman-Gilmore theory Various approaches to the semantic paradoxes result in some logical ...

... the class of propositions the truth predicate obeys the Tar ski criteria. We shall briefly review the theory of Frege structures and use it as a way into the theory of Kripke-Gilmore-Feferman. 1.2 The Kripke-Feferman-Gilmore theory Various approaches to the semantic paradoxes result in some logical ...

Chapter 5: Methods of Proof for Boolean Logic

... of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from it. You repeat this process for each disjunct. So it doesn’t matter which disjunct is true—you get the ...

... of the disjuncts is true—you just don’t know which one. So you consider the individual “cases” (i.e., disjuncts), one at a time. You assume the first disjunct, and then derive your conclusion from it. You repeat this process for each disjunct. So it doesn’t matter which disjunct is true—you get the ...

Semantics of PL

... it is impossible for all of the premises to be true and for the conclusion to be false. Our first attempt at filling out ‘impossible’ was to translate the sentences into Sentential Logic and look at all possible truth-value assignments to the atomic sentences. Then since our connectives are truth-fu ...

... it is impossible for all of the premises to be true and for the conclusion to be false. Our first attempt at filling out ‘impossible’ was to translate the sentences into Sentential Logic and look at all possible truth-value assignments to the atomic sentences. Then since our connectives are truth-fu ...

Formal Logic, Models, Reality

... formal language. This is unavoidable because, by Tarski's theorem on truth definitions, the truth predicate cannot be represented in a consistent formal theory. Therefore the meaning of 'A B' must refer to something in the object language. But this contradicts the conclusion above that 'A B' ref ...

... formal language. This is unavoidable because, by Tarski's theorem on truth definitions, the truth predicate cannot be represented in a consistent formal theory. Therefore the meaning of 'A B' must refer to something in the object language. But this contradicts the conclusion above that 'A B' ref ...

IM_FA16-03-PredicateLogic

... 3) Therefore something has two syllables and two million people. Now what exactly is that thing? A word? A city? Something that is both? Really, there is no such weird thing which is both a word and a city. Compare this argument with the one above. 1) “Boston” has two syllables 2) Boston has two mil ...

... 3) Therefore something has two syllables and two million people. Now what exactly is that thing? A word? A city? Something that is both? Really, there is no such weird thing which is both a word and a city. Compare this argument with the one above. 1) “Boston” has two syllables 2) Boston has two mil ...

The Foundations: Logic and Proofs

... raining.” then p →q denotes “If I am at home then it is raining.” In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). ...

... raining.” then p →q denotes “If I am at home then it is raining.” In p →q , p is the hypothesis (antecedent or premise) and q is the conclusion (or consequence). ...

A Partially Truth Functional Approach to

... truth assignments --- themselves, maps from S to {T,F} --- to [0,1] with unit sum. We define the assertibility of a statement to be the probability of its truth given that it has a truth value. In other words, relative to a given distribution Pr of probability, the assertibility of statement A is: Σ ...

... truth assignments --- themselves, maps from S to {T,F} --- to [0,1] with unit sum. We define the assertibility of a statement to be the probability of its truth given that it has a truth value. In other words, relative to a given distribution Pr of probability, the assertibility of statement A is: Σ ...

Propositional Logic

... contained. Also, logical reasoning methods are designed to work no matter what meanings or values are assigned to the logical “variables” used in sentences. Although the values assigned to variables are not crucial in the sense just described, in talking about logic itself, it is sometimes useful t ...

... contained. Also, logical reasoning methods are designed to work no matter what meanings or values are assigned to the logical “variables” used in sentences. Although the values assigned to variables are not crucial in the sense just described, in talking about logic itself, it is sometimes useful t ...

Scharp on Replacing Truth

... Leibniz’s law we can infer that L is true if and only if it isn’t. If we furthermore assume the classical laws of logic we can derive from this any conclusion we like. Something has clearly gone wrong somewhere. But saying where it went wrong is only half the problem. It is surely also important tha ...

... Leibniz’s law we can infer that L is true if and only if it isn’t. If we furthermore assume the classical laws of logic we can derive from this any conclusion we like. Something has clearly gone wrong somewhere. But saying where it went wrong is only half the problem. It is surely also important tha ...

10a

... • Amazingly, this is the only interference rule you need to build a sound and complete theorem prover – Based on proof by contradiction and usually called resolution refutation ...

... • Amazingly, this is the only interference rule you need to build a sound and complete theorem prover – Based on proof by contradiction and usually called resolution refutation ...

Advanced Topics in Propositional Logic

... 1.Start out as though you were going to build a truth table, by listing all the atomic sentences in in a row, followed by S. But do not write T or F beneath any of them yet. 2.If there is a conjunct of the form Ai, assign T to Ai, i.e., write T in the reference column under Ai. Repeat this as long a ...

... 1.Start out as though you were going to build a truth table, by listing all the atomic sentences in in a row, followed by S. But do not write T or F beneath any of them yet. 2.If there is a conjunct of the form Ai, assign T to Ai, i.e., write T in the reference column under Ai. Repeat this as long a ...

1 The Easy Way to Gödel`s Proof and Related Matters Haim Gaifman

... The choice of an arithmetical language is not essential to the theorem. We can consider a theory T in any language, provided that a sufficient portion of the syntax (including notions relating to proofs) can be expressed, so that various properties are provable. We started by enumerating the wffs an ...

... The choice of an arithmetical language is not essential to the theorem. We can consider a theory T in any language, provided that a sufficient portion of the syntax (including notions relating to proofs) can be expressed, so that various properties are provable. We started by enumerating the wffs an ...

FIRST DEGREE ENTAILMENT, SYMMETRY AND PARADOX

... induction on the complexity of formulas that this then extends to all of the formulas in the language: for any formula A, if Aρ0 then Aρ′ 0 too, and if Aρ1 then Aρ′ 1 too. The evaluations ρ and ρ′ may still differ, because ρ might leave a gap where ρ′ fills in a value, 0 or 1, or where ρ assigned on ...

... induction on the complexity of formulas that this then extends to all of the formulas in the language: for any formula A, if Aρ0 then Aρ′ 0 too, and if Aρ1 then Aρ′ 1 too. The evaluations ρ and ρ′ may still differ, because ρ might leave a gap where ρ′ fills in a value, 0 or 1, or where ρ assigned on ...