Intermed Symbolic Logic
Intermed Symbolic Logic PHI 112
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This 9 page Class Notes was uploaded by Marlee Kulas on Wednesday September 9, 2015. The Class Notes belongs to PHI 112 at University of California - Davis taught by Staff in Fall. Since its upload, it has received 31 views. For similar materials see /class/191928/phi-112-university-of-california-davis in PHIL-Philosophy at University of California - Davis.
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Date Created: 09/09/15
Review of Sentence Logic G J Mattey Winter 2009 Philosophy 112 Sentence Logic 0 Sentence logic deals with sentences of a natural language that are either true or false L 5 o Sentence logic ignores the internal structure of simple sentences L 5 o Sentence logic is concerned with sentences which are compounded in a certain way 0 A primary goal of sentence logic is to enable the evaluation of a certain class of arguments in natural language L 2 o In an argument a sentence that is the argument s conclusion is claimed to be supported by a set of sentences that are its premises Deductive Validity o The kind of support investigated by sentence logic is that of Deductive Validity 0 Valid Deductive Argument An argument in which without fail if the premises are true the conclusion will also be true L 3 In general the source of deductive validity in sentence logic lies in the way in which the sentences in the argument are compounded 0 So the main item of business in sentence logic is to investigate the properties of the devices which allow the formation of compound sentences from simpler sentences 1 Syntax of Sentence Logic I 11 Notions of Syntax Syntax and Sentence Logic o A fact of Syntax is a fact which concerns symbols or sentences insofar as the fact can be determined from the form of the symbols or sentences from the way they are written IL 161 0 There are two central syntactical facts investigated by sentence logic IL 161 Whether or not a string of symbols is a Sentence of sentence logic The application of Rules of Inference to sentences of sentence logic 0 Sentences of sentence logic are denoted by boldface Metavariables Q through Z L 0 Sets of sentences of sentence logic are denoted by italicized boldface metavari ables X through 2 IL 158 12 Formation Rules Vocabulary of Sentence Logic 0 The vocabulary of sentence logic consists of three kinds of items Sentence Letters L 56 6 A B C Z with or without integer subscripts l falsum Connectives L 6 50 54 6 Sign ofNegation 6 V Sign of Disjunction 6 amp Sign of Conjunction 6 3 Sign ofthe Conditional 6 E Sign ofthe Biconditional Punctuation marks L 1112 C l TC 11 C Sentences of Sentence Logic 0 The sentences wellformed formulas or wffs of sentence logic are determined by the following Formation Rules L 16 55 i All capital letters A B C Z with or without integer sub scripts and l are wffs Atomic Sentences ii If X is a wff then so is X Negated Sentence iii le and Y are wffs then so is X amp Y Conjunction iv le and Y are wffs then so is X V Y Disjunction v le and Y are wffs then so is X D Y Conditional vi le and Y are wffs then so is X E Y Biconditional vii Nothing else is a wff of sentence logic 0 Conventions L 1214 Square or curly brackets may replace parentheses Outermost punctuation marks may be dropped if there is no further com pounding Punctuation marks around negations may be dropped 2 Semantics of Sentence Logic 21 Truth Tables Semantics and Sentence Logic 0 A fact of Semantics concenis the referents interpretation or insofar as we understand this notion the meaning of symbols and sentences IL 161 0 There are two distinct ways in which we interpret the symbols of sentence logic Informally as standins for Transcriptions of natural language sentences L Chs 2 4 Formally as having one of two Truth Values true or false t or f respec tively L 8 o The formal interpretation of sentence logic will serve as a guide to how to tran scribe sentences of natural language into sentence logic Truth Values 0 We can study the semantical facts about sentence logic without knowing any thing about the naturallanguage sentences for which they might stand in 0 Any atomic sentence may be interpreted either as being true or as being false 0 The assignment of truth values to atomic sentences is called a Case L 9 o The truth value of a compound sentence is strictly determined by the truth values of its component parts Sentence logic is TruthFunctional o The way in which the truth value of a compound sentence is determined can be summarized in a table called a Truth Table L 8 Truth Table for Falsum The symbol 1 Which is intended to stand for any sentence that cannot be true is always assigned the truth value f all cases f Truth Table for Negation The negation of X takes the opposite of the truth value assigned to X in the given case f t X X case 1 t case 2 f Truth Table for Conjunction A conjunction X amp Y is true in a case if and only if both X and Y are true in that case X Y X amp Y case 1 t t t case 2 t f f case 3 f t f case 4 f f f Truth Table for Disjunction A disjunction X V Y is true in a case if and only either X or Y is true in that case X Y X V Y case 1 t t t case 2 t f t case 3 f t t case 4 f f f Truth Table for the Conditional A conditional X D Y is true in a case if and only if either X is false in that case or Y is true in that case case 1 case 2 case 3 case 4 Truth Table for the Biconditional A biconditional X E Y is true in a case if and only if both X and Y have the same truth value in that case case 1 case 2 case 3 case 4 Deductive Validity in Sentence Logic 0 An argument in sentence logic consists of a set X of Wffs premises and a sen tence Y conclusion 0 To say that an argument expressed With sentences of sentence logic is Valid is to say that any assignment of truth values to sentence letters Which makes all of the premises true also makes the conclusion true L 47 0 We symbolize this relation of validity as follows X I Y 0 An argument from X to Y is invalid X12 Y if and only if it is not valid ie if in some case all the sentences in X are true and Y is false 0 A Counterexample is a case Which shows an argument to be invalid by making all the premises true and the conclusion false L 478 Other Semantical Properties of Sentence Logic 0 Sentences X and Y of sentence logic are Logically Equivalent if and only if in all possible cases they have the same truth value L 2930 0 Sentence X of sentence logic is a Logical Truth or Tautology if and only if it is true in all possible cases L 38 o Sentence X of sentence logic is a Contradiction if and only if it is false in all possible cases L 38 22 Application of Sentence Logic Transcription and Connectives o The semantical facts about compound sentences of sentence logic suggest how to use them to transcribe compound sentences of the natural language not it is not the case that amp and but V or inclusive D ifthen material conditional E if and only if material biconditional o The transcriptions for negation conjunction and disjunction are less controver sial than those for the conditional and biconditional IL Ch 4 o Ordinarily we do not transcribe natural language sentences as l as this symbol is useful only within sentence logic itself Using the Semantics of Sentence Logic 0 The semantics of sentence logic can be used to show the validity or invalidity of some naturallanguage arguments o Validity or invalidity of naturallanguage arguments can be shown using the se mantics only if the sentences making up the argument are adequately transcribed L 25 o If the sentences of the argument are adequately transcribed and the argument in sentence logic is valid then the naturallanguage argument is valid 0 If the argument is adequately transcribed the argument in sentence logic is in valid and the transcription reveals all of its logical structure then the natural language argument is invalid o Predicate logic is needed because sentence logic does not reveal all the logical structure of many naturallanguage arguments IL 12 3 Syntax of Sentence Logic II 31 Rules of Inference Natural Deduction o It is possible to determine the validity or invalidity of naturallanguage argu ments using the sentences of sentence logic purely syntactically ie without interpreting them at all 0 This is done using Rules of Inference which relate sets of sentences to a given sentence L 60 0 We here use the technique known as Natural Deduction after Gerhard Gentzen which was originally formulated in 1929 by Stanislaw Jankowski o The formulation of natural deduction rules used here is due to Frederick Fitch 1952 o The distinctive feature of Fitch s rules is their use of Subderivations L 65 Rules of Inference o Roughly a Derivation is the result of the application of inference rules 0 A rule of inference allows one to write down a sentence Y given that one has already written down some set of sentences X For example given A and A D B one may write down B 0 We want our rules of inference to be TruthPreserving I 62 A rule is truthpreserving or sound if and only if there is no possible case in which all the sentences of X are true and Y is fa se Classifying the Rules 0 In any system of sentence logic some rules of inference are primitive while others are derived 1 98 A primitive rule is taken as basic A derived rule is a shortcut which gives the same result of a more compli cated combination of uses of primitive rules 0 For each connective there is one primitive rule which results in it being intro duced and another which results in its being eliminated 0 There is a further rule which allows any sentence to be repeated subject to re strictions 32 Scope Lines Scope Lines 0 A Scope Line is a device used to keep track of the premises of an argument or of any assumptions made in the course of the argument I 62 o In the following schema the scope line indicates two premises of an argument Assumptions 0 Some rules of inference require making an Assumption Which must eventually be Discharged I 67 0 When an assumption has been made and discharged in the course of a derivation the segment of the derivation is called a Subderivation I 65 o The derivation Within Which a subderivation occurs is called an Outer Derivation of the subderivation 33 Preservation of Truth Rules Without Assumptions 0 The truthpreserving character of rules of inference that do not require an as sumption can be seen by considering cases 0 For example by Conjunction Elimination if a conjunction X amp Y occurs at any point of a derivation either of its two conjuncts may be written down This rule is truthpreserving because if X amp Y is true then X is true and Y is true I 7010pt case 1 case 2 case 3 case 4 Negation Introduction 0 The rule of Negation Introduction requires an assumption of a sentence X and a derivation of a contradiction Y and Y from it o The assumption can then be discharged and the negation of X written 0 One of Y and Y is false so given they both follow by truthpreserving rules from X and other premises or assumptions X itself must be false and X true I 71 Conditional Introduction 0 The rule of Conditional Introduction requires an assumption of a sentence X and a derivation of a sentence Y from it o The assumption can then be discharged and the conditional X D Y written X Y X D Y 0 Since Y follows from truthpreserving rules from X and other premises or as sumptions there is no way for X to be true and Y false I 67 Falsum Introduction 0 The rules for the falsum sentence letter are not given in Teller s text since he does not use the symbol T in the syntax of sentence logic 0 The introduction rule allows that T may be written down any time that a sen tence and its negation occur to the immediate right of a given scope line X X T 0 Since there is no possible case in which both X and X are true there is no possible case in which X and B are true and T is false
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