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# Week1-Chapter1.pdf Phil 215

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This 4 page Class Notes was uploaded by Hunter Cooper on Monday September 7, 2015. The Class Notes belongs to Phil 215 at Western Kentucky University taught by Sam McMyler in Fall 2015. Since its upload, it has received 412 views. For similar materials see Symbolic Logic in PHIL-Philosophy at Western Kentucky University.

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Date Created: 09/07/15

Philosophy 215 Symbolic Logic Week 1 Chapter 1 0 First Order Logic FOL o A language that uses quantified variables I These variables as a general rule are letters af and n1 n2 n3 etc I These variables are known as constants and are not capitalized in FOL 0 Terms I Simple individual constants names I Complex function symbols 0 REMEMBER I Every individual constant must name an actual object I No constant can name more than one object I Objects can have more than one name or no name at all 0 Arity the number of logical arguments related to a predicate o Unary is one o Binary is two 0 Ternary is three 0 Predicates used to express properties of objects or relations between objects ie Larger Smaller Cube LeftOf would be a few in the Tarski s World block language 0 Predicates always start with a capital letter 0 Every predicate has a determinate arity ie LeftOf is binary while Cube is unary 0 Languages that use function symbols 0 Function Symbols use name like terms in order to express complex claims in atomic sentences that could not be expressed using names and predicates I ex father mother fatherfathermax is Max s grandfather I The term father is a function symbol I The whole sentence though fatherfathermax is a complex term 0 Complex terms only refer to one object I All function symbols start with lower case letters 0 First Order Arithmetic o This language allows statements to be expressed about the natural numbers 0 1 2 3 and so on as well as the usual operations of addition and multiplication I Uses infix notation ab the addition symbol is in between the functions 0 Predicates I equal to I lt less than 0 Functions addition I x multiplication 0 Terms of firstorder arithmetic are formed by the following ways I The names 0 and 1 are terms I If t1 and t2 are terms then t1 t2 and t1x t2 are also terms I Nothing can be a term unless it can be obtained by repeated application of 1 and 2 0 List Notation O EX t1 t2 t3 I The terms refer to the objects in the set t1 refers to a specific object in the world that is in this set I The terms in a set are call elements or members 0 The size of a set is called the cardinality I t1 t2 t3 has a cardinality of 3 I t1 t2 has a cardinality of 2 o It is possible to have a list with no members I Ex which would have a cardinality of 0 I The symbol for a list with no members is a circle with a line through it 0 s o REMEMBER s I s s o A list with the symbol a in it refers to a set that includes a set that does not have any members This is somewhat confusing but is important to note As a general rule if you d like to cite a set with no members use either the notation or the symbol a not both 0 Naming Sets I d a b c 0 Membership Predicate I E is the symbol used to show an object s membership in a set I Ex in the set d a b c if we wanted to say that object a is a member of the set titled d we would say a e d 0 With the membership predicate the object comes first and then the set title comes second Philosophy 215 Symbolic Logic Week 1 Chapter 1 0 First Order Logic FOL o A language that uses quantified variables I These variables as a general rule are letters af and n1 n2 n3 etc I These variables are known as constants and are not capitalized in FOL 0 Terms I Simple individual constants names I Complex function symbols 0 REMEMBER I Every individual constant must name an actual object I No constant can name more than one object I Objects can have more than one name or no name at all 0 Arity the number of logical arguments related to a predicate o Unary is one o Binary is two 0 Ternary is three 0 Predicates used to express properties of objects or relations between objects ie Larger Smaller Cube LeftOf would be a few in the Tarski s World block language 0 Predicates always start with a capital letter 0 Every predicate has a determinate arity ie LeftOf is binary while Cube is unary 0 Languages that use function symbols 0 Function Symbols use name like terms in order to express complex claims in atomic sentences that could not be expressed using names and predicates I ex father mother fatherfathermax is Max s grandfather I The term father is a function symbol I The whole sentence though fatherfathermax is a complex term 0 Complex terms only refer to one object I All function symbols start with lower case letters 0 First Order Arithmetic o This language allows statements to be expressed about the natural numbers 0 1 2 3 and so on as well as the usual operations of addition and multiplication I Uses infix notation ab the addition symbol is in between the functions 0 Predicates I equal to I lt less than 0 Functions addition I x multiplication 0 Terms of firstorder arithmetic are formed by the following ways I The names 0 and 1 are terms I If t1 and t2 are terms then t1 t2 and t1x t2 are also terms I Nothing can be a term unless it can be obtained by repeated application of 1 and 2 0 List Notation O EX t1 t2 t3 I The terms refer to the objects in the set t1 refers to a specific object in the world that is in this set I The terms in a set are call elements or members 0 The size of a set is called the cardinality I t1 t2 t3 has a cardinality of 3 I t1 t2 has a cardinality of 2 o It is possible to have a list with no members I Ex which would have a cardinality of 0 I The symbol for a list with no members is a circle with a line through it 0 s o REMEMBER s I s s o A list with the symbol a in it refers to a set that includes a set that does not have any members This is somewhat confusing but is important to note As a general rule if you d like to cite a set with no members use either the notation or the symbol a not both 0 Naming Sets I d a b c 0 Membership Predicate I E is the symbol used to show an object s membership in a set I Ex in the set d a b c if we wanted to say that object a is a member of the set titled d we would say a e d 0 With the membership predicate the object comes first and then the set title comes second

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