Intermed Symbolic Logic
Intermed Symbolic Logic PHI 112
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Date Created: 09/09/15
Atomic Sentences of Predicate Logic G J Mattey Winter 2009 Philosophy 112 Predicate Logic and Sentence Logic 0 Predicate logic is an extension of Sentence Logic All items of the vocabulary of Sentence Logic are items of the vocabulary of Predicate Logic All sentences of Sentence Logic are sentences of Predicate Logic All conventions for sentencelogic syntax apply to predicatelogic syntax o Predicate logic contains vocabulary items and sentences that are not part of Sen tence Logic 0 This feature makes Predicate Logic a proper extension of Sentence Logic Syntax Unique to Predicate Logic 0 Several new vocabulary items are found in Predicate Logic Predicates Names Function symbols Variables Quanti er symbols 0 In this segement of the course we will restrict our attention to the rst three new vocabulary items Subject and Predicate o Predicate logic deals with sentences which say something about someone or something ll 0 The part of the sentence signifying someone or something about which some thing is said is the subject of the sentence o The part of the sentence that says something about someone or something is the predicate of the sentence 0 Example Adam is blond Blond is the predicate of the sentence Adam is the subject of the sentence Object and Property 0 The subject of a subjectpredicate sentence is intended to refer to an object Adam refers to a person Adam 0 The predicate of a subjectpredicate sentence is intended to signify a property or a relation Blond is intended to signify the property being blond o The property intended to be signi ed by the predicate is attributed to the object intended to be referred to by the name Predicate signi es Subject signi es Property Attributed to Object Being blond Property of Adam 0 In the semantics of Predicate Logic all names signify objects and predicates signify sets rather than properties Relations 0 In some sentences a predicate has multiple subjects Eve loves Adam 0 Each subject is intended to refer to an object Adam is intended to refer to Adam Eve is intended to refer to Eve 0 The twoplace predicate of such a sentence indicates a relation holding between What the subjects are intended to refer to Predicate sign es Subjectl signi es Subjeth signi es Relation Between Obj ect1 and Objeth Loving Relation of Eve to Adam Symbolization of Subjects and Predicates o Naturallanguage predicates are symbolized in Predicate Logic using uppercase Roman letters with or without integer subscripts which are called Predicates B might stand in for the oneplace English predicate blond L might stand in for the twoplace English predicate loves o Naturallanguage subjects are symbolized in Predicate logic using lowercase Roman letters with or without integer subscripts which are called Names a might stand in for the English name Adam e might stand in for the English name Eve 0 The choice of predicates and names of Predicate logic to stand in for natural language predicates and names is arbitrary except that it must be consistent in a given context Atomic Sentences of Predicate Logic 0 The most basic sentences of Predicate Logic are Atomic Sentences 0 There are two types of atomic sentences An atomic sentence of Sentence Logic which is a predicate followed by no names A predicate followed by any number of names which are called Argu ments 0 Examples S Ba Lea Interpreting Atomic Sentences 0 As merely syntactical entities atomic sentences of Predicate Logic have no meaning 0 The assignment of meanings to sentences is called semantics 0 Atomic sentences can be given meaning as with Sentence Logic Informally as standins for Transcriptions of natural language sentences Formally as having truthvalues t or f Transcriptions o Transcriptions have two components A Transcription Guide which associates vocabulary of Predicate Logic with naturallanguage expressions and either 96 A sentence or sentences of Predicate Logic which is supposed to cap ture the structure of the naturallanguage sentence or sentences or x A sentence or sentences in natural language which is supposed to cap ture the structure of the Predicate Logic sentence or sentences 0 The transcription guide itself contains one or more of these components A sentence for every sentence letter of Predicate Logic used in the tran scription An object for every name of Predicate Logic used in the transcription A speci cation of a property or relation for each predicate of Predicate Logic used in the transcription An Example of a Transcription 0 Suppose we want to transcribe the sentence Adam is blond but Eve is not 0 We choose to let a stand for Adam and e for Eve a Adam e Eve 0 We specify a oneplace predicate B by writing Bx and a twoplace predicate L by writing ny etc followed by a quasiEnglish expression specifying a property or relation Bx x is blond o The transcription is Ba amp Be Interpretations of Sentences o A transcription guide functions as an interpretation of the Predicate Logic ex pressions it contains and of the sentences that can be composed of them 0 Just as a truthtable only speci es truthvalues for the sentences that appear in it an interpretation in Predicate Logic may be limited to only a part of its syntax o The range of application of an interpretation is determined by which members of the vocablulary it contains Sentence Letters Names Predicates o If all the names predicates and sentenceletters occuring in a sentence X occur in the interpretation then it is an Interpretation of the Sentence X TruthConditions for Atomic Sentences 0 Another kind of interpretation can be made without the use of a transcription guide 0 Let us call a proper atomic sentence of Predicate Logic one which contains names and predicates and therefore which is not a sentence of Sentence Logic 0 We could assign proper atomic sentences truthvalues in the same way as in Sentence Logic 0 This is done by expanding SentenceLogic cases to apply to atomic sentences of Predicate Logic 0 Each atomic sentence would then be given a single truth value by the interpreta tion Minimal Interpretations o In a minimal interpretation the truthvalues of proper atomic sentences are as signed just as in Sentence Logic 0 Consider atomic sentences of Predicate Logic which contain B e and a and nothing else 0 There are exactly two such sentences to which a minimal interpretation could apply Ba and Be 0 This allows us to list four possible interpretations in the guise of a truthtable interpretation 1 interpretation 2 interpretation 3 interpretation 4 Ba 39 mm Be t f t f Syntactic Representation of Minimal Interpretations 0 Each of the four minimal interpretations just listed can be represented in the form of a sentence of Predicate Logic The sentence is unnegated if the interpretation assigns it the value true The sentence is negated if the interpretation assigns it the value false a Be Syntactic Representation B interpretation 1 t t interpretation 2 t f Ba amp Be interpretation 3 f t Ba amp Be interpretation 4 f f Ba amp Be Domains 0 To get more informative interpretations of sentences of Predicate Logic we need a way to link the predicates and names occurring in the sentences with objects in the world 0 A minimal interpretation says nothing about objects 0 A more robust interpretation should include a speci cation of the objects of which each predicate is true and the objects of which each predicate is false IL 17 0 We can make an interpretation more robust by supplementing it with A Domain consisting of at least one object An association of each name in the interpretation with at least one object in the domain An Example 0 Suppose the domain of an interpretation consists of Adam and Eve 0 This can be written as D 2 Adam Eve 0 The interpretation might assign Adam to a and Eve to e 0 Suppose further that it is the case that Adam is blond and Eve is not blond 0 Given our earlier remarks we might want to say that B stands for the property of being blond 0 Then we would conclude that on this interpretation Ba is hue and Be is false Intensional and Extensional Interpretations 0 An interpretation that makes predicates stand for properties is called inten sional o Intensional interpretations were studied by Leibniz in the eighteenth century 0 Modern logic however mostly does not use intensional interpretations 0 Instead it takes a given predicate to stand for a set of objects that have a given property 0 Such an interpretation is extensional OnePlace Predicates 0 Suppose we have interpretations Which include the predicate B and the names a and e and nothing else 0 Consider a class of interpretations Whose domain is Adam Eve and Where a stands for Adam e for Eve 0 There are four possible extensions for B and therefore four possible interpre tations in this class a e B interpretation 1 Adam Eve Adam Eve interpretation 2 Adam Eve Adam interpretation 3 Adam Eve Eve interpretation 4 Adam Eve Nobody Determining TruthValues o It is easy to see how the truthvalues of Ba and Be are determined given these interpretations interpretation 1 Adam Eve interpretation 2 Adam Eve Adam interpretation 3 Adam Eve Eve interpretation 4 Adam Eve Nobody mama t t f f o The truthcondition for atomic sentences With oneplace predicates is if the ob ject designated by the name is in the extension of the predicate then the sentence is true otherwise it is false Extensions of Relation Symbols 0 Predicates of twoplaces or more are called relation symbols 0 The extensions of relation symbols are sets of pairs triples quadruples etc n tuples depending on the number of arguments the predicate has 0 We represent these ntuples With angled brackets The pair Ztuple consisting of Adam and Eve is represented as Adam Eve The triple 3tuple consisting of Adam Eve and Cid is represented as Adam Eve Cid o Strictly speaking the extension of a oneplace predicate is a set of ltuples The ltuple consisting of Adam is represented as Adam An Example 0 Let an interpretation listing the twoplace predicate L have a domain consisting of Adam and Eve 0 There are four possible items in the extension of L Adam Adam Adam Eve Eve Adam Eve Eve 0 An interpretation representing the case where Adam and Eve love only them selves would specify the extension of L as consisting of Adam Adam and Eve Eve General TruthCondition 0 With the given interpretation we can once again see what the truthvalues of various sentences are a e Laa Lae Lea Lee Adam Eve ltAdamAdamgt ltEveEve t f f t 0 An atomic sentence with an nplace predicate is true in an interpretation if the n tuple consisting of the referents of the arguments from left to right is a member of the extension of the predicate the sentence is false otherwise L a a i i Adam Adam Functions 0 Predicate Logic may be enhanced by the addition of a way of forming namelike expressions Function Symbols 0 Function symbols consist of italicized lowercase Roman letters followed by a set of parentheses and containing from O to n commas where n is a nite number f g 7 h 7 7 o Filling the spaces in the function symbols are either names or other lledin function symbols a fa7fb hgfa7gb7c Interpretation of Function Symbols 0 Function symbols are intended to represent functions 0 We illustrate the notion of a function by beginning with oneplace function sym bols where what lls the blank is a name for example f a o In an interpretation the name designates exactly one member of the domain which is an Argument of the function that the function symbol signi es o The lledin function symbol itself designates exactly one member of the domain the Value of the function An Example 0 Consider a domain with two members Adam and Eve 0 Informally we might consider f to designate the spouse function which presumes monogamy 0 Adam is the spouse of Eve and Eve is the spouse of Adam 0 So when Adam is the argument Eve is the value and when Eve is the argument Adam is the value a e fa fe Adam Eve Eve Adam Applying the Example 0 The interpretation of atomic sentences can be applied to the interpretation of function symbols 0 Speci cally we can determine values of the following sentences B Bf a Bf e t e f a ife Eve Adam Adam Eve t a interp 1 Adam Eve interp 2 Adam Eve Eve Adam Adam f t interp 3 Adam Eve Eve Adam Eve t f interp 4 Adam Eve Eve Adam Nobody f f ManyPlace Functions 0 Functions of more than one place are treated similarly to functions of one place 0 There is always a single value no matter how many arguments the function might have 0 For example consider a domain which consists of the positive integers 1 2 3 0 An addition function takes any two positive integers as arguments and retunis a single value the sum of the two numbers 0 If a designates l b designates 2 and f designates the addition function then fab designates 3 Values of Functions as Arguments of Functions 0 Since the value of a function is an object in the domain it can serve as the argument of a function even the same one o A natural example is the application of the addition function to the result of addition 1 2 1 o If we use the interpretations just given we can represent this as f f aba o fab represents the number 3 so ffaba represents the result of the ad dition of 3 and l ie 4 Constant Terms 0 We shall call a name a Constant or Constant Term 0 A function symbol which is lled in by names is also a constant term 0 More generally a function symbol which is lled in by constant terms is also a constant term 0 Nothing else is a constant term The Identity Predicate o A special twoplace predicate is the Identity predicate 2 o By convention the identity predicate is placed between two constant terms rather than in front of them a b a gfbfc o The identity predicate is always interpreted in the same way 0 If both constant terms designate the same member of the domain the identity sentence containing them is true otherwise it is false An Example 0 In the following example we look at four interpretations that differ in the func tion designated by the function symbol f a e fa me a e afe fe fa i 1 Adam Eve Adam Adam f t t i 2 Adam Eve Adam Eve f f f i 3 Adam Eve Eve A am f t f i 4 Adam Eve Eve Eve f f t Metavariables for Atomic Sentences o The metavariables s and t designate any constant terms name or lledin function symbol 0 Note that in the text 5 and t are restricted to designating names 0 The metavariables P Q and R indicate sentences of Predicate Logic containing terms 0 Ps indicates a sentence of Predicate Logic containing the term designated by s and Pt indicates a sentence containing the term designated by t If 5 indicates the constant a then Lea can be represented as Ps o Pth indicates the result of substituting t in all the places where 5 occurs Introduction Rule for the Identity Predicate 0 Identity Introduction 1 t t may be written on any line of a derivation o The rule is truthpreserving because the designation of any constant term is xed by the interpretation so that the designation of t is always the same in which case the designation of t the designation of t is in the extension of 2 no matter what t designates Elimination Rule for the Identity Predicate 0 Identity Elimination 2E when s t occurs on an accessible earlier line then given Ps it follows that Pth a b Lae Lbe a b Lbe Lae o The rule is truthpreserving because the truth of s t requires that s and t des ignate the same individual and by the semantics anything holding of what 5 designates holds of what t designates Summary 0 The subject of a subjectpredicate English sentence is represented in a tran scribed Predicate Logic sentence by a constant term of Predicate Logic Whose designation is determined by the interpretation of the sentence A name A lledin function symbol 0 The predicate of a subjectpredicate English sentence is represented in a tran scribed Predicate Logic sentence by a predicate of Predicate Logic An nplace predicate Whose extension is determined by the interpretation A twoplace identity predicate 2 Whose extension is the same on any in terpretation 0 An atomic sentence of Predicate Logic consists of an nplace predicate followed by n constant terms Objectual Semantics o The semantics that has been introduced here is sometimes called objectual o The original semantics for Predicate Logic due to Alfred Tarski is objectual in character 0 The semantics is called objectual because the truthconditions for atomic sen tences are based on the designation of objects in the domain by constant terms and on the objects or ordered ntuples of objects in the extensions of predicates Presentation of the Domain 0 The formulation of interpretations in Exercises 21 and 22 differs from that in 25 With the latter being closer the the formulation presented here 0 In the earlier exercises the domain is speci ed using the names given it by the interpretation Dbd o In practice we often specify the domain of an interpretation simply by giving the interpretations name for those objects IL 16 o In the 25 exercises the domain is speci ed directly All integers l 2 3 4 Determining TruthValues o The speci cation of which one place predicates apply to which individual ob jects which two place predicates apply to which pairs of object and so on is presented in Exercises 21 to 24 in a syntactical way 0 There is a conjunction of unnegated and negated atomic sentences which sum marizes the result of the atomic sentences getting a truthvalue For example Tb amp Td amp Kbb amp Kbd amp de amp Kdd o In Exercises 25 which requires the application of predicates to objects the actual application is left to the general knowledge of the reader For example it is presumed that the reader knows that 1 is odd and that not all integers are greater than 17 Tarski Semantics 0 We have developed semantics for atomic sentences in the style of Tarski o This style most closely resembles the later exercises o The Tarskistyle semantics makes an explicit assignment of members of the do main to names The domain is Betty Dylan with b designating Betty and d desig nating Dylan 0 The extensions of predicates of Predicate Logic and the meanings of function symbols are given The extension of T is ltDylangt 0 From this we can determine truthvalues Betty is not in the extension of T so Tb is false and Tb is true From Syntactic Representations to Explicit Intepretations o The syntactic representation of the semantics requires that an interpretation have a domain which is speci ed by names of Predicate Logic 0 If we specify the domain in this way we have not said what the items in the domain are unless we associate the Predicate Logic names with naturallanguage names as in a transcription guide D bd b Betty d Dylan 0 Conjunctions of negated and unnegated atomic sentences along with the tran scription guide can be used to determine which ntuples of objects are in the extensions of which predicates o Tb indicates that the onetuple of the object in the domain designated by b Betty is not in the extension of T Object Language and Metalanguage Formal Languages 0 Sentence Logic and Predicate Logic are formal languages 0 A formal language is a set of sentences generated by rules of formation from a vocabulary o The sentences of Sentence Logic and Predicate Logic are not part of natural language though some may resemble naturallanguage sentences 0 The formal languages Sentence Logic and Predicate Logic are the objects of our study and as such they are called object languages The Metalanguage o If we are going to state anything about an object language we must make use of a language 0 We call a language used to study an object language a metalanguage o In theory the metalanguage may be identical to or include the obj ect language We use English to study English in linguistics 0 We will strictly separate our metalanguage English with some extra technical vocabulary from our obj ect languages 0 Keeping the languages separate allows us to avoid sem antical paradox Tarski Use and Mention 0 When we talk about an item of language we are said to mention it 0 Whenever an item of any object language is mentioned it must be placed within single quotation marks 0 We may use English to mention an item of English Bush has four letters and starts with a B George W Bush was born in Texas is false This sentence is false is true Metavariables 0 We may also use English to mention items of Sentence Logic and Predicate Logic 3 is a connective of Sentence Logic P D Q is a conditional If P is true and P D Q is true then Q is true 0 To state general facts about Sentence Logic and Predicate Logic we must use expressions that designate classes of items of the object language 0 Such expressions are called metavariables Metavariables for Sentences and Sets of Sentences 0 To mention metavariables them selves single quotation marks must be used 0 Q through 2 will be used as metavariables for sentences of both Sentence Logic and Predicate Logic 0 X through 2 will be used as metavariables for sets of sentences of both Sen tence Logic and Predicate Logic Metavariables and Connectives o In English we refer to connectives of Sentence Logic using such expressions as sign of negation or sign of the conditional 0 We may also refer to them by mentioning them 3 0 We must have a way to combine our use of metavariables with reference to con nectives Names of Themselves 0 We could make general statements about connectives using English If X is a sentence of Sentence Logic then the result of pre xing the sen tence X refers to with a N or sign of negation and surrounding the result with parentheses is a sentence of Sentence Logic 0 This kind of statement is obviously very cumbersome 0 We want to say If X is a sentence then NX is a sentence 0 Strictly speaking this mixes the object language with the metalanguage 0 So we say without paradox that in NX N is used as a name of itself or autonymously Carnap
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