Chapter 4. Predicate logic allows us to represent the internal properties of the statement. Example:

Transcription

1 4.1 Singular and General Propositions Chapter 4 Predicate logic allows us to represent the internal properties of the statement. Apples are red A Firetrucks are red F The previous symbols give us no indication of the attribute red. In predicate logic, Apples is the subject and red is the predicate term. We would denote the previous statements as follows: Apples are red Ra Firetrucks are red Rt Now we not only know we are dealing with apples and firetrucks but that they are also both red. Capital letters ( A Z) are predicates. Categories or properties that things can have Lower case letters ( a,b,c, w) are constants. They are the names of one specific and particular object. {x,,y, z} are individual variables. Socrates is Mortal Tokyo is populous Names to the right, predicate to the left. Mary is tall would be Tm Ms Pt not Mt If Paris is beautiful, then Andre told the truth Irene is either a doctor or a lawyer Bp Ta Di v Li Next we need to make a distinction between some and all. To do this we use to special quantifiers: (x) All is the universal quantifier ( ) Some is the existential quantifier All apples are red Some apples are red (x) Ra ( x) Ra Symbolic Logic Student Guide 63

2 Traditional logic emphasized four types of subject-predicate propositions. All skyscrapers are tall Given any individual thing, if it is a skyscraper then it is tall. Given any x, if x is a skyscraper then x is tall (x) (Sx Tx) No skyscraper is tall. Given any individual thing, it is a skyscraper then it is not tall. Given any x, if x is a skyscraper then x is not tall. (x) (Sx ~Tx) Some skyscrapers are tall There exists at least one thing that is a skyscraper and is tall There exists at least one x such that x is a skyscraper and x is tall. ( x)(sx Tx) Some skyscrapers are not tall There exists at least one thing that is a skyscraper and is not tall. There exists at least one x such that x is a skyscraper and x is not tall. ( x)(sx ~Tx) Note that existence is not necessary for universal quantifiers. Examples: Snakes are reptiles. (S R) Only executives have secretaries. (E S) All students passing logic worked hard. (S P H) Some logic tests are easy. ( L E) Symbolic Logic Student Guide 64

3 Beginning Translations If the English begins with All (every) Some Not all No Then the WFF begins with (x) ( x) ~(x) ~( x) All are Italian. = For all x, x is Italian = (x) Ix Someone is Italian. = For some x, x is Italian = ( x) Ix Not all are Italian = It is not the case that, for all x, x is Italian = ~(x) Ix No one is Italian = It is not the case that, for some x, x is Italian. = ~( x) Ix When the English begins with all, some, not all, or no, the quantifier must go outside all parentheses. No one is rich and non-italian. = ~( x)(rx ~Ix) not ~(( x)rx ~Ix) Use whatever propositional connective (, v,, or ) the English sentence specifies. All are German or Italian = (x) (Gx v Ix) It the sentence doesn t specify the connective use the following two rules: a) With the all... is.. form, use b) otherwise, if the English doesn t specify the connective use. All Italians are lovers = (x)(ix Lx) Some Italians are lovers = ( x)(ix Lx) Symbolic Logic Student Guide 65

4 In these examples, the expressions Sx Tx, Fx ~Bx, and so on are called statement functions. A statement function is a mere pattern for a statement. It makes no definite assertion about anything in the universe, has no truth value, and cannot be translated as a statement. The variables that occur in statement functions are called free variables because they are not bound by any quantifier. In contrast, the variables that occur in statements are called bound variables. A universal statement is a statement that makes an assertion about every member of its subject class. All S are P. (x) (Sx Px) For any x, if x is an S, then x is a P. A particular statement is a statement that makes an assertion about one or more unmanned members of the subject class. Some S are P. ( x) (Sx Px) There exists an x such that x is an S and x is a P. The convention for quantifiers is similar to the one adopted for the tilde operator. That is, the quantifier governs only the expression immediately following it. (x) (Ax Bx) (x) Ax Bx the universal quantifier governs the entire statement function the universal quantifier governs only the statement function Ax The operators of prepositional logic can be used to form compound arrangements of universal and particular statements. If Elizabeth is an historian, Then some women are historians If some cellists are music directors, Then some orchestras are properly led. He ( x)(wx Hx) ( x)(cx Mx) ( x)(ox Px) Symbolic Logic Student Guide 66

5 In Class Activity: 1) Elaine is a chemist. ( C) 2) All maples are trees. (M, T) 3) No novels are biographies (N, B) 4) If Gertrude is correct, then the Taj Mahal is made of marble. (C, M) 5) There are pornographic art works. (A, P) 6) Not every smile is genuine. (S, G) 7) There is trouble in River City. (T, R) 8) Everything is imaginable. (I) 9) A thoroughbred won the race. (T, W) 10) A few guests arrived late. (G, A) 11) Only talented musicians perform in the symphony. (T, M, P) 12) A good violin is rare and expensive. (G, V, R, E) 13) A room with a view is expensive. (R, V, E) 14) All who aren t logicians are evil. (E, L) 15) All are evil logicians. (E, L) Symbolic Logic Student Guide 67

6 4.2 Preliminary Quantification Rules Universal Instantiation (x) Fx Fa use any constant for a Also (x) Fx Fa If everyone is funny, then Al is funny, Bob is funny, and so forth. (x) Fx Fy can use any variable If everyone is funny, then an arbitrary person is funny. All economists are social scientists. Milton Friedman is an economist. Therefore, Milton Friedman is a social scientist Seems to follow by modus ponens. Universal instantiation makes this possible. Symbolized is 1. (x) (Ex Sx) 2. Em / Sm 3. Em Sm 1, UI 4. Sm 2,3, MP Universal Generalization Fy not Fa (x) Fx (x) Fx If you know everyone is friendly then an arbitrary variable is friendly. It is not that case: If Adam is friendly then everyone is friendly. Symbolic Logic Student Guide 68

7 Example All psychiatrists are doctors. All doctors are college graduates. Therefore, all psychiatrists are college graduates. Symbolized 1. (x) (Px Dx) 2. (x) (Dx Cx) / (x) Px Cx) 3. Py Dy 1, UI 4. Dy Cy 2, UI 5. Py Cy 3, 4 HS 6. (x) (Px Cx) 5, UG Existential generalization Fa ( x) Fx Fy ( x) Fx Fred is Friendly. Then you know there exists at least one friendly person. All tenors are singers. Placido Domingo is a tenor Therefore, there is at least one singer. 1. (x) (Tx Sx) 2. Tp / ( x) Sx 3. Tp Sp 1, UI 4. Sp 2,3 MP 5. ( x) Sx 4, EG Existential Instantiation ( x) Fx Fa can pick any constant not previously used. There is at least one thing that is friendly. Lets call that thing a. Not allowed: ( x) Fx Fy This implies that there is one thing that is friendly, therefore everything is friendly. Symbolic Logic Student Guide 69

8 All professors are college graduates Some professors are swimmers. Therefore, some swimmers are college graduates. 1. (x) (Px Cx) 2. ( x) (Px Sx) / ( x) (Sx Cx) 3. Pa Sa 4. Pa Ca 5. Pa 6. Ca 7. Sa 8. Sa Ca 9. ( x) (Sx Cx) The following is a defective proof: 1. ( x) (Fx Ax) 2. ( x) (Fx Ox) / ( x) (Ax Ox) 3. Fb Ab 4. Fb Ob 5. Ab 6. Ob 7. Ab Ob 8. ( x)(ax Ox) From above the translation would be: Suppose F stands for fruits, A for apples and O for Oranges. The argument is: Some fruits are apples. Some fruits are orange. Therefore, some apples are oranges. 1. (x) (Ax Bx) 2. (x) (Bx Cx) / (x) (Ax Cx) Symbolic Logic Student Guide 70

9 In Class Activity: Use the eighteen rule to derive the conclusions. 1) 1. (x)[ax (Bx v Cx)] 2. Ag ~Bg / Cg 2) 1. (x) [Ax (Bx v Cx)] 2. ( x) (Ax ~Cx) / ( x) Bx 3) 1. (x) (Bx v Ax) 2. (x) (Bx Ax) / (x) Ax 4) 1. ( x) Ax (x) (Bx) 2. ( x) Cx ( x) Dx 3. An Cn / ( x) (Bx Dx) Symbolic Logic Student Guide 71

10 5. 1. (x) (Ax Bx) 2. Am An / Bm Bn (x)(mx ~Lx) 2. ( x)(nx Lx) / ( x)(nx ~Mx) Symbolic Logic Student Guide 72

11 Translate the following arguments into symbolic form. Then use the rule of inference to derive the conclusion of each. 1. All logic students are smart. John is a logic student. Therefore John is smart. 2. Oranges are sweet. Also, oranges are fragrant. Therefore, oranges are sweet and fragrant. 3. Carrots are vegetables and peaches are fruit. Furthermore, there are carrots and peaches in the garden. Therefore, there are vegetables and fruit in the garden. Symbolic Logic Student Guide 73

12 The following examples illustrate invalid applications of the rules: 1. Fy Gy 2. (x) (Fx Gy) 1, UG 1. (x)fx Ga 2. Fx Ga 1, UI 1. Fc 2. ( x) Gx 3. Gc 2, EI 1. Fm Gm 2. (x) (Fx Gx) 1, UG 1. ( x) Fx 2. ( x) Gx 3. Fe 1, EI 4. Ge 2, EI 1. Fs Gs 2. ( x) Fx Gs 1, EG 1. ~(x) Fx 2. ~Fy 1, UI Symbolic Logic Student Guide 74

13 Change of Quantifier Rules (CQ) (x)fx ~( x) ~Fx Everyone is Funny There does not exist someone who is not funny ~(x)fx ( x) ~Fx Not everyone is funny There exists someone who is not funny ( x) Fx ~(x) ~Fx There exists someone who is funny Not everyone is not funny ~( x) Fx (x) ~Fx There does not exist anyone who is funny Everyone is not funny 1. ~( x) (Px ~Qx) 2. ~(x) (~Rx v Qx) / ( x) ~Px 3. (x) ~(Px ~Qx) 4. ( x) ~(~Rx v~qx) 5. ~(~Ra v Qa) 6. ~(Pa ~Qa) 7. Ra ~Qa 8. ~Pa v Qa 9. ~Qa 10. Qa v ~Pa 11. ~Pa 12. ( x) ~Px 1. ( x)(hx Gx) (x) Ix 2. ~Im / (x) (Hx ~Gx) 3. ( x) ~Ix 4. ~(x) Ix 5. ~( x) (Hx Gx) 6. (x) ~(Hx Gx) 7. (x) (~Hx v ~Gx) 8. (x) (Hx ~Gx) Symbolic Logic Student Guide 75

14 In Class Activity: 1) 1. (x) Ax ( x)bx 2. (x) ~Bx /( x)~ax 2) 1. ( x) Ax v ( x)(bx Cx) 2. ~( x) Bx / ( x) Ax 3) 1. (x)(ax Bx) 2. ~(x) Cx v (x) Ax 3. ~(x)bx /( x)~cx Symbolic Logic Student Guide 76

15 4. 1. (x) Fx 2. ~(x) Gx / ~(x)(fx Gx) (x)(fx v Gx) 2. ( x)~gx /( x) Fx (x)(fx Gx) 2. (x)(gx Hx) 3. ~( x)hx / ~(x)fx Symbolic Logic Student Guide 77

16 In Class Activity: 1) No snakes are men. So, no men are snakes. (S,M) 2) All mature crows are black. Therefore, all crows are black if they are mature. (M, C, B) 3) All People are logical. All unicorns are people. Therefore, all unicorns are logical. (P, L, U) 4) Only people over 21 are allowed. Bob is not over 21. So, Bob is not allowed. (T, A) 5) All cats are mammals. Some cats are not black. Therefore, some mammals are not black. (C, M, B) Symbolic Logic Student Guide 78

17 Conditional and Indirect Proofs Just as in propositional proofs, you can use the method of conditional and indirect proofs with predicate logic. 1. (x)(hx Ix) /( x) Hx ( x) Ix 2. ( x) Hx 3. Ha 4. Ha Ia 5. Ia 6. ( x) Ix 7. ( x) Hx ( x) Ix This next example differs from the previous one in that the antecedent of the conclusion is a statement function, not a complete statement. 1. (x) [(Ax v Bx) Cx] / (x) (Ax Cx) 2. Ax 3. Ax v Bx 4. (Ax v Bx) Cx 5. Cx 6. Ax Cx 7. (x) (Ax Cx) There is a restriction on these proofs: That is UG must not be used within the scope of an indented sequence if the instantial variable occurs free in the first line of that sequence. Line 2 is said to occur free while 1 7 are said to occur bound. 1 (x) Rx Sx / (x)(rx Sx) 2. Rx ACP 3. (x) Rx 2, UG (invalid) 4. (x) Sx 1,3, MP 5. Sx 4, UI 6. Rx Sx 2 5 CP 7. (x)(rx Sx) 6, UG Symbolic Logic Student Guide 79

18 Indirect proofs are subject to the same rule as Conditional proofs: 1, (x)[(px Px) (Qx Rx)] / (x)(qx Rx) 2. ~ (x)(qx Rx) AIP 3. ( x)~(qx Rx) 2, CQ 4. ~(Qm Rm) 3, EI 5. (Pm Pm) (Qm Rm) 4, UI 6. ~(Pm Pm) 4,5 MT 7. ~(~Pm v Pm) 6, Impl 8. Pm ~Pm 7, DM, DN 9. ~~(x)(qx Rx) 2 8, IP 10. (x)(qx Rx) 9, DN In Class Activity: Use indirect or conditional proof to derive the conclusions of the following: 1) 1. (x)(ax Bx) 2. (x)(ax Cx) / (x)[ Ax (Bx Cx)] 2) 1. (x) (Ax Cx) 2. ( x) Cx ( x) (Bx Dx) / ( x) Ax ( x)bx Symbolic Logic Student Guide 80

19 3) 1. (x)[(ax v Bx) Cx] 2. (x)[(cx v Dx) Ex] / (x)(ax Ex) (x)(ax Bx) 2. Am v An / ( x) Bx Symbolic Logic Student Guide 81