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340 -----> example: 1. Fa 2. (x)(Fx Gx) 3. (x)(Gx Hx) -----> Pr Pr Pr / (x)(Hx v Kx) example: 1. (x)(Fx Gx) Pr / Fb (x)Gx 341 - UI, EG apply to WHOLE lines 1. (x)Fx (x)Gx 2. Fa Ga UI 1 WRONG 1. (x)(Fx Gx) 2. Fa Ga UI 2 OK 1. Fb Gb 2. Fb (x)Gx EG 1 WRONG 1. Fb Gb 2. (x)(Fx Gx) EG 1 OK 342 - REMEMBER to put in parens. to get scope right: 1. Fa 2. (x)(Fx Gx) 3. (x)(Gx Hx) 4. Fa Ga 5. Ga Ha 6. Fa Ha 7. Ha 8. Ha v Ka 9. (x)Hx v Kx Pr Pr Pr / (x)(Hx v Kx) UI 2 UI 3 HS 4, 5 MP 1, 6 Add 7 EG 8 WRONG - REMEMBER to put line numbers in justifications: 1. Fa Pr 2. (x)(Fx Gx) Pr 3. (x)(Gx Hx) Pr / (x)(Hx v Kx) 4. Fa Ga UI WRONG 5. Ga Ha UI WRONG - REMEMBER to put in individual variables in propositional functions and individual constants in instances: 1. Fa Pr 2. (x)(Fx Gx) Pr 3. (x)(Gx Hx) Pr / (x)(Hx v Kx) 4. F G UI 2 WRONG 343 RECALL that each quantifier (that is, (x) and (x)) has two rules: - one for GETTING RID of the quantifier - one for INTRODUCING the quantifier and so far, we've talked about: - the rule for getting rid of the universal quantifier: Universal Instantiation (UI) (x)x a - the rule for introducing the existential quantifier: Existential Generalization (EG) a (x)x 344 - note that: - "instantiation" involves getting rid of an quantifier by inferring an instance - "generalization" involves introducing a quantifier by generalizing from an instance 345 NOW the two remaining rules: - the rule for getting rid of the existential quantifier: Existential Instantiation (EI) (preliminary version) (x)x a - the rule for introducing the universal quantifier: Universal Generalization (UG) (preliminary version) a (x)x - these rules should seem much less intuitive than the first two--in fact, they should seem illegitimate as stated. In fact they are illegitimate as stated, so use of them is restricted in important ways. 346 The rule for getting rid of the existential quantifier: Existential Instantiation (EI) (preliminary version) (x)x a Where x stands for any simple or compound propositional function, and a stands for any substitution instance of the propositional function. - the idea is: from an existential statement you can infer an instance; this is ok, SO LONG AS the individual constant in the instance hasn't appeared in prior lines of the proof 347 - consider what goes wrong here (Nx x is a number; Ox x is odd; Ex x is even): 1. (x)(Nx Ox) 2. (x)(Nx Ex) 3. Na Oa 4. Na Ea 5. Oa 6. Na Ea Oa 7. (x)(Nx Ex Ox) Pr Pr EI 1 EI 2 Simp 3 Conj 4, 5 EG 6 WHAT?? - in line 4, we've said of an individual constant--that we've already said is an odd number in line 3--that it's an even number - to avoid this, in line 4 we can use an individual constant that we haven't used on prior lines 1. (x)(Nx Ox) 2. (x)(Nx Ex) 3. Na Oa 4. Nb Eb 5. Oa 6. Nb Eb Oa Pr Pr EI 1 EI 2 Simp 3 Conj 4, 5 - and the derivation of (x)(Nx Ex Ox) doesn't work (as it SHOULDN'T) because you can't apply EG to line 6 to get this existential statement 348 Existential Instantiation (EI) (final version) (x)x provided we flag a a - the flagging is just to help us identify use of individual constants so that we can apply the following restriction: R1 A letter being flagged must be new to the proof, that is, it may not appear, either in a formula or as a letter being flagged, previous to the step in which it gets flagged. 349 -----> example proof using EI Existential Instantiation (EI) (x)x provided we flag a a R1 A letter being flagged must be new to the proof, that is, it may not appear, either in a formula or as a letter being flagged, previous to the step in which it gets flagged. 1. (x)(Cx Lx) 2. (x)Cx 3. Ca 4. Ca La 5. La 6. (x)Lx Pr Pr / (x)Lx EI 2 (flag a) UI 1 MP 3, 4 EG 5 - use EI BEFORE UI since UI is unrestricted--you can use UI anywhere with no problem; if we had done: 1. (x)(Cx Lx) 2. (x)Cx 3. Ca La 4. Cb 5. ??? Pr Pr / (x)Lx UI 1 EI 2 (flag b) - we wouldn't have been able to use MP on line 5 350 The rule for introducing the universal quantifier: Universal Generalization (UG) (preliminary version) a (x)x Where x stands for any simple or compound propositional function, and a stands for any substitution instance of the propositional function. - the idea is: from an instance of a propositional function you can infer a universal statement with that propositional function; this is ok, SO LONG AS this instance was derived from universal statements 351 Universal Generalization (UG) (final version) flag a a (x)x - to ensure that the instance from which we infer a universal statement WAS derived from universal statements, we require that this instance is derived within a "flagged subproof" and apply the following restrictions: R1 A letter being flagged must be new to the proof, that is, it may not appear, either in a formula or as a letter being flagged, previous to the step in which it gets flagged. R2 A flagged letter may not appear either in the premises or in the conclusion of a proof. 352 -----> example proof using UG Universal Generalization (UG) flag a a (x)x - FINALLY we can prove our first example argument: P1 All whales are mammals. P2 All mammals are animals. C All whales are animals. 353 1. (x)(Wx Mx) 2. (x)(Mx Ax) 3. flag a 4. Wa Ma 5. Ma Aa 6. Wa Aa 7. (x)(Wx Ax) Pr Pr / (x)(Wx Ax) FS (UG) UI 1 UI 2 HS 4, 5 UG 6 - the first line of the flagged subproof is "flag a" (or "flag b" or "flag c". . .); call this the flagging step - the justification for the first line of the subproof is FS (UG) (for Flagging Step, Universal Generalization) - the last line of the subproof is the instance from which we infer a universal statement - the application of UG in the line after the subproof discharges the flagging step - all the terminology (scope, scope marker, etc.) that applies to subproofs using CP and IP applies here as well 354 NOTICE how use of the flagged subproof and application of the restrictions: R1 A letter being flagged must be new to the proof, that is, it may not appear, either in a formula or as a letter being flagged, previous to the step in which it gets flagged. R2 A flagged letter may not appear either in the premises or in the conclusion of a proof. PREVENTS us from using an instance derived from an existential statement to infer a universal statement: 1. (x)Cx Pr / (x)Cx 2. flag a FS (UG) 3. Ca EI (flag a) WRONG: violates R1 1. (x)Cx Pr / (x)Cx 2. flag a FS (UG) 3. Cb EI (flag b) - line 3. is ok, but it won't help us get (x)Cx--the instance from which we infer a universal statement has got to be the letter flagged in the flagging step, line 2. 355 Proof strategies in predicate logic: I. For proofs where premises/conclusion include quantifier statements II. For proofs where premises/conclusion include truth functional compounds of quantifier statements A. For proofs where premises/conclusion include NEGATIONS of quantifier statements B. For proofs where premises/conclusion include conjunctions, disjunctions, conditionals, and biconditionals of quantifier statements 356 Proof strategies in predicate logic: I. For proofs where premises/conclusion include quantifier statements Follow this general pattern: - FS (UG) (if needed to get universal statement) - EI and/or UI (to get rid of quantifiers in premises) - use EI before UI - 20 sentential rules (to get instance you want to generalize over) - that's the 8 inference rules, 10 replacement rules, and CP and IP - EG or UG (to get a quantifier statement) 357 -----> example: 1. (x)(Fx Gx) 2. (x)Fx -----> Pr Pr / (x)Gx example: 1. (x)(Kx Lx) 2. (x)((Kx Lx) Mx) Pr Pr / (x)(Kx Mx) 358 Strategies for proofs in predicate logic (cont.) II. For proofs where premises/conclusion include truth functional compounds of quantifier statements A. For proofs where premises/conclusion include NEGATIONS of quantifier statements Follow this general pattern: - QN or CQN rules (if needed to turn negated quantifier statements in premises into quantifier statements) - FS (UG) (if needed to get universal statement) - EI and/or UI (to get rid of quantifiers in premises or results of QN or CQN rule applications) - 20 sentential rules (to get instance you want to generalize over) - EG or UG (to get a quantifier statement) - QN or CQN rules (if needed to turn quantifier statement into a negated quantifier statement) 359 -----> example: 1. (x)(Fx (Hx v Ix)) Pr 2. ~(x)(Fx Hx) Pr 3. (x)(Ix (~Zx Hx)) Pr / ~(x)(Fx ~Zx) 360 -----> example: 1. ~(x)(Mx Nx) 2. (x)(Px Nx) Pr Pr / ~(x)(Mx Px) 361 Strategies for proofs in predicate logic (cont.) B. For proofs where premises/conclusion include conjunctions, disjunctions, conditionals, and biconditionals of quantifier statements COMBINE: - strategies you've learned to get conjunctions, disjunctions, conditionals, and biconditionals - in particular, REMEMBER to use CP to get conditionals also, remember strategies such as: - if conclusion is a conjunction, you'll probably get the conclusion by getting each conjunct on a line by itself and using Conj - if conclusion is a disjunction, try to get one disjunct on a line by itself and use Add WITH: - patterns that you've learned to get quantifier statements and negations of quantifier statements 362 -----> example: 1. (x)(Fx Gx) Pr / ~(x)Fx v (x)Gx 363 -----> example: 1. (x)Ax (x)Bx 2. (x)Cx (x)Dx 3. An Cn Pr Pr Pr / (x)(Bx Dx) 364