Phil 2440 Chapter 5: Predicate Logic Symbolizations
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1 Phil 2440 Chapter 5: Predicate Logic Symbolizations To discuss today: Atomic sentences in predicate logic Quantifiers Some important kinds of sentences Well-formed formulas About the predicate calculus What is the predicate calculus? Why is the predicate calculus better than the propositional calculus? Atomic propositions and their structure Predicates and subjects All cats are furry Logical vs. grammatical predicates & subjects It is raining Relations Symbolizing things Individuals: a, b,..., t Predicates: A, B,... Atomic sentences: Fa, Cb, Rca,... Variables & open sentences Individual variables: x, y, z, u, v, w Open sentences: Fx, Rxy,... Quantified sentences Quantifiers in English. Examples: All cats are furry. Some cats are furry. Most cats are furry. No cats are furry. Quantifiers in predicate logic: Universal quantifier: ( x), (x), ( y), (y),... Existential quantifier: ( x), ( y),... Examples: ( x) Fx (x) Cx Domain of a quantifier: Drinks for everyone! Interpretations: (x) Dix (x) ((Px & Rx) Dix) Multiple quantifiers: Someone loves everyone
2 = ( x)(x loves everyone) = ( x)(y)(x loves y) = ( x)(y) Lxy Quantifier scope: A quantifier goes with the first complete sentence following it. Bound vs. free variables Examples: ( x) (Cx & Fx) ( x) Cx & Fx ( x) Cx & (x) Fx (x) Lxy Important kinds of sentences and how to symbolize them All A s are B = (x)(ax Bx) Some A s are B = ( x)(ax & Bx) Some A s are non-b = ( x)(ax & ~Bx) No A s are B = (x)(ax ~Bx) = ~( x)(ax & Bx) Existential import Only and unless : Only A s are B = (x)(bx Ax) = (x)(~ax ~Bx) A thing is A unless it is B = (x)(~bx Ax) Times and places: Someday I ll be famous = ( x)(dx & Fix) God is everywhere = (x)(px Lgx) Well-formed formulas Include open sentences & complete sentences. Rules for wff s: 1. Atomic formulas are wff s. 2. If is a wff, then ~ is a wff. 3. If and are wff s then ( ), ( & ), ( ), and ( ) are wff s. 4. If is a wff, then (x), ( x), (y), ( y), etc., are wff s. Examples: (Hx P) ~Acac (x) y Fx (Ha Fy) Ax & Fy Ba (Ha (x)fy) Ca ( x) ( y)(z) Ax
3 Phil 2440 Chapter 6: Predicate Logic Proofs To discuss today: 5 new inference rules Strategy for predicate logic proofs Applying the old rules to predicate logic sentences All the old rules still apply. Implicational rules: only apply to whole lines. Examples: which of the following are good? 1. ( x) Fx Ga 2. ( x) Fx 3. Ga 1,2 MP Example 2: 1. ( x) (Fx Ga) 2. ( x) Fx 3. Ga 1,2 MP Example 3: 1. ( x) (Fx Gx) 2. ( x) ~Fx 3. ( x) Gx 1,2 DS Example 4: 1. (x) (~Fx & ~Gx) 2. (x) ~(Fx Gx) 1 DeM Quantifier Negation (QN) Rule: One kind of quantifier can be switched to the other kind, while adding/subtracting a ~ on both sides of it. Thus: (x) ~( x) ~ ( x) ~(x) ~ ( x) ~ ~(x) (x) ~ ~( x) Example: ~( x) Fx ~( x) ~Gx (x) ~Fx (x) Gx 1. ~( x) Fx ~( x) ~Gx p 2. (x) ~Fx p 3. (x) ~Fx ~( x) ~Gx 1 QN 4. ~( x) ~Gx 2,3 MP 5. (x) Gx 4 QN Existential Instantiation (EI)
4 Rule: Remove existential quantifier and substitute for every variable under its scope an unknown symbol: ( v) (v) (u) Using unknown symbols. Examples: 1. ( x) (Fx & Gx) 2. Fa & Ga 1 EI Example 2: 1. ( x) (Fx & Gx) 2. Fb & Gb 1 EI Example 3: 1. ( x) [Fx & (y) (Fy Gx)] 2. Fa & (y) (Fy Ga) 1 EI Restriction on EI: u cannot appear previously in the proof Example 4: 1. ( x) Hx p 2. ( x) Px p 3. Ha 1 EI 4. Pa 2 EI 5. Ha & Pa 3,4 conj. Example 5: 1. ( x) Hx premise 2. ( x) Px premise 3. Ha 1 EI 4. Pb 2 EI Note: cannot apply EI to part of a line. Example 6: 1. ( x) (Fx & Gx) ( y) Ay 2. (Fa & Ga) ( y) Ay 1 EI Example 7: 1. (x)( y) Myx 2. (x) Mbx 1 EI Existential Generalization (EG) Rule: Replace one or more occurrences of a constant/unknown with a variable, and add an existential quantifier to the sentence. (u) (c) ( v) (v) ( v) (v) 1. Fc 2. ( x) Fx 1 EG
5 Example 2: 1. Fa & Ga 2. ( x) (Fx & Gx) 1 EG Example 3: 1. Fa & Ga 2. ( x) (Fx & Ga) 1 EG Universal Instantiation (UI) Rule: Remove a universal quantifier and replace all variables under its scope with a constant/unknown symbol. (v) (v) (v) (v) (u) (c) Example: 1. (x) Fx 2. Fa 1 UI Universal Generalization (UG) Rule: Replace every occurrence of an unknown symbol with a variable, and add a universal quantifier to the sentence. (u) (v) (v) Restrictions: Does not work on constants. u does not occur previously in a line obtained by EI u does not occur in an undischarged assumption Examples: which of these are correct uses of UG? 1. (x) (Sx Dx) p 2. (x) Sx p 3. ~Da a 4. Sa Da 1 UI 5. Sa 2 UI 6. Da 4,5 MP 7. Da & ~Da 3,6 conj. 8. Da 3-7 RAA 9. (x) Dx 8 UG Example 2: 1. (x) (Sx Dx) p 2. (x) Sx p 3. Sa Da 1 UI 4. Sa 2 UI 5. Da 3,4 MP 6. (x) Dx 5 UG
6 Example 3: 1. ( x) Fx p 2. Fb 1 EI 3. (x) Fx 2 UG Example 4: 1. ~(x) Fx p 2. Fb a 3. (x) Fx 2 UG 4. (x) Fx & ~(x) Fx 1,3 conj. 5. ~Fb 2-4 RAA 6. (x) ~Fx 5 UG Miscellaneous Stuff Remembering the names of the rules. General predicate-logic proof strategy. Use EI first (before UI). (x) (Ax Bx) (x) (Bx Ax) (x) Ax 1. (x) (Ax Bx) p 2. (x) (Bx Ax) p 3. Aa Ba 1 UI 4. Ba Aa 2 UI 5. ~Aa a 6. ~Ba 4,5 MT 7. Aa 3,6 DS 8. Aa & ~Aa 7,5 conj 9. Aa 5-8 RAA 10. (x) Ax 9 UG Example 2: ( x)ax (x)(bx Cx) Am & Bm Cm 1. ( x)ax (x)(bx Cx) p 2. Am & Bm p 3. Am 2 simp 4. Bm 2 simp 5. ( x) Ax 3 EG 6. (x) (Bx Cx) 1,5 MP 7. Bm Cm 6 UI 8. Cm 4,7 MP
7 Phil 2440 Chapter 7: Relations and Identity To Discuss Today: Logical properties of relations Symbolizations involving identity Logical laws of identity Properties of relations Symmetry: Symmetric: Rxy Ryx. x is next to y Asymmetric: Rxy ~Ryx x is bigger than y Non-symmetric: (neither symmetric nor asymmetric) x hits y Transitivity: Transitive: Rxy & Ryz Rxz x is bigger than y Intransitive: Rxy & Ryz ~Rxz x is the daughter of y Non-transitive: (neither transitive nor intransitive) x is a friend of y Reflexivity: Reflexive: Rxy Rxx x lives in the same house as y Irreflexive: ~Rxx x is older than y Non-reflexive: (neither reflexive nor irreflexive) x loves y Equivalence relations: Have the properties of equivalence : Reflexive, symmetric, transitive Identity General points about identity: Numerical vs. type identity Properties of identity: Equivalence relation Symbol for identity: x = y : x is numerically identical with y Note: = is a 2-place predicate. Numerical statements: There is exactly 1 cat: ( x)[cx & (y)(cy y=x)]
8 There are exactly 2 cats: ( x)( y) [(Cx & Cy) & x y & (z) (Cz [z=x z=y])] There are exactly 3 cats: ( x)( y)( z) [(Cx & Cy & Cz) & (x y & y z & x z) & (w)(cw (w=x w=y w=z))] There is at most 1 cat: ~( x)( y) [(Cx & Cy) & x y] Definite descriptions: The King of France is bald: ( x)[kxf & (y)(kyf y=x)] & (x)(kxf Bx) ( x)[(kxf & Bx) & (y)(kyf y=x)] Aside: Strawson s criticism of Russell s analysis Logical laws of identity The Law of Identity (Id): Intuitive statement: everything is identical to itself. Rule: Write down = at any stage, where is any constant or unknown symbol. Leibniz Law (LL): If x=y, then any property of x is a property of y and vice versa. Rule: From ( ) and =, deduce ( ). Example: To prove: (x)(y)(x=y y=x). (Identity is symmetric.) 1. a=b a. (for CP) 2. a=a Id 3. b=a 1,2 LL 4. a=b b=a 1-3 CP 5. (y) (a=y y=a) 4 UG 6. (x)(y) (x=y y=x) 5 UG
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