Chapter 8: Syntax and Semantics I : Logic & Proofs July 21, 2009 Karin Howe

Recall the Kangaroo Argument 1.All kangaroos can fly. 2.Jim is a kangaroo.____  Jim can fly. In "standard form:" 1.If it is a kangaroo, it can fly.K  F 2.Jim is a kangaroo._______K____  Jim can fly.  F

Isn't this a bit fishy? 1.If it is a kangaroo, it can fly.K  F 2.Jim is a kangaroo._______K____  Jim can fly.  F oNote the mismatch between the antecedent on line 1, and the statement on line 2! oAlso note that we've changed line 1 - used to read "All kangaroos can fly"!

The Kangaroo Argument done "right" 1.All kangaroos can fly. 2.Jim is a kangaroo.____  Jim can fly. 1.K 2.J___ FF oThere - that's better! Right? oWhoops! ….. now it's invalid! oYet, the Kangaroo Argument is clearly valid!! oNeed a way of representing the structure of the argument that can highlight the internal structure of the statements, and their relationship to each other. oPredicate logic to the rescue!

The Kangaroo Argument done RIGHT! 1.All kangaroos can fly. 2.Jim is a kangaroo.____  Jim can fly. Read this off as: 1.For all x, if x is a kangaroo, then x can fly. 2.Jim is a kangaroo.____________________  Jim can fly. Symbolized: 1.(  x)(K(x)  F(x))K(x) = x is a kangaroo 2.K(j)____________F(x) = x can fly  F(j)j = Jim

The language of predicate logic New symbols: –  : for all Use to symbolize words like "all," "every" –  : there exists Use to symbolize "some" (means at least one) Punctuation:, Keep old symbols: , &, , ,  Previous punctuation: ( )

Predicates n-ary predicates (n > 0) –Examples: x is a dog: D(x) x loves y: L(x,y) x gave y the z: G(x,y,z) In theory, you can have any number of places in a predicate: R(x 1, x 2, …, x 8, …. x 27, …) Can also have 0-place predicates: atomic formulae (e.g., P, Q, R, etc.)

Constants (Singular terms) Can also have constants in our language, that refer known individuals with a certain quality. Convention: these constants are restricted to letters a - t Use these constants to plug "holes" in predicates with the known individuals who have the indicated quality: _____ laughedL(x) m = MaryL(m) n = NancyL(n) o = OscarL(o) a = UrsulaL(a)

Truth and Falsity in Predicate Logic In an important sense, no different than in propositional logic Consider the atomic formula: A –A = Alligators live in Florida True or false? –A = Alligators live in Pittsburgh True or false? Consider the predicate formula: –K(x) = x is a kangaroo; j = Jim K(j) – true or false? –K(x) = x is a logic professor; j = Jim K(j) – true or false? All of the usual truth table rules (for the connectives) will still apply

Interpretations Definition: Truth and falsity with respect to an interpretation: 1.If  is a 0-place predicate letter, then  is true iff I(  ) = T. 2.If  is of the form  (x 1, …, x n ) where  is a n-place predicate letter (with n > 0), and x 1, …, x n are n terms, then  is true on I iff is in I(  ) Example: –S(x,y,z) = the sum of x and y is z –P(x,y,z) = the product of x and y is z –Domain: {0,1,2,3,4,5} –Interpretation: I(S) = {,,,,,,, …} I(P) = {,,,,,,, …} –True or False? S(1,3,4) S(0,1,1) S(3,1,4) S(4,3,5) P(0,1,0) P(1,1,1) P(2,3,5) P(5,3,4)

Change the Interpretation, Change the Truth Value Example: –S(x,y,z) = x is sitting between y and z –P(x,y,z) = x is next to y and two places to the left of z –Domain: {Amelie, Chris, Daniel, Nathan, Sungwoo, Tomasz} –Interpretation: I(S) = {,,,,, } I(P) = {,,,,, } –True or False? S(a,n,s) S(d,a,c) S(s,t,n) S(n,c,t) P(a,c,d) P(d,a,t) P(d,s,n) P(n,c,a)

Practice with Symbolization Hey diddle diddle, The cat and the fiddle, The cow jumped over the moon, Dictionary: J(x,y) = x jumped over y c = the cow m = the moon Symbolization: J(c,m) The little dog laughed to see such sport, Dictionary: L(x) = x laughed to see such sport d = the little dog Symbolization: L(d) And the dish ran away with the spoon Dictionary: R(x,y) = x ran away with y a = the dish s = the spoon Symbolization: R(a,s)

Jack and Jill went up the hill To fetch a pail of water. Dictionary: W(x,y,z) = x went up y to do z a = Jack i = Jill h = the hill f = fetch a pail of water Symbolization: W(a,h,f) & W(i,h,f) Jack fell down and broke his crown, Dictionary: F(x) = x fell down B(x) = x broke x's crown a = Jack Symbolization: F(a) & B(a) And Jill came tumbling after Dictionary: T(x) = x came tumbling after i = Jill Symbolization: T(i)

Hickory, dickory, dock, The mouse ran up the clock. Dictionary: R(x,y) = x ran up the y m = the mouse c = the clock Symbolization: R(m,c) The clock struck one, Dictionary: S(x,y) = x struck y c = the clock o = one Symbolization: S(c,o) The mouse ran down, Dictionary: R(x) = x ran down m = the mouse Symbolization: R(m) Hickory, dickory, dock

The itsy bitsy spider went up the water spout. Dictionary: W(x,y) = x went up the y s = the itsy bitsy spider a = the water spout Symbolization: W(s,a) Down came the rain, and washed the spider out. Dictionary: C(x) = x came down A(x,y) = x washed y out r = the rain s = the itsy bitsy spider Symbolization: C(r) & A(r,s) Out came the sun, and dried up all the rain Dictionary: O(x) = x came out D(x,y) = x dried up y b = the sun r = the rain Symbolization: O(b) & D(b,r) And the itsy bitsy spider went up the spout again Dictionary: B(x,y) = x went up the y again s = the itsy bitsy spider a = the water spout Symbolization: B(s,a)

Peter, Peter pumpkin eater, Had a wife but couldn't keep her; Dictionary: H(x,y) = x had y K(x,y) = x kept y p = Peter, Peter pumpkin eater a = Peter's wife Symbolization: H(p,a) &  K(p,a) He put her in a pumpkin shell Dictionary: P(x,y,z) = x put y in z p = Peter, Peter pumpkin eater a = Peter's wife s = pumpkin shell Symbolization: P(p,a,s) And there he kept her very well. Dictionary: W(x,y,z) = x kept y in z very well p = Peter, Peter pumpkin eater a = Peter's wife s = pumpkin shell Symbolization: W(p,a,s)

Pussy cat, pussy cat, where have you been? I've been to London to visit the Queen. Dictionary: V(x,y,z) = x went to y to visit the z p = pussy cat l = London q = the Queen Symbolization: V(p,l,q) Pussy cat, pussy cat, what did you do there? I frightened a little mouse, under her chair. Dictionary: F(x,y,z) = x frightened y under z p = pussy cat m = little mouse c = the Queen's chair Symbolization: V(p,m,c)

Proofs Involving Predicate Formulas Practice CPL Problems Lab #5