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Truth, deduction, computation; lecture 3
This document contains lecture notes on logic and proofs. It introduces logical arguments with premises and conclusions. It explains valid and sound arguments. It also introduces Fitch notation for representing logical proofs and shows examples of basic rules like introduction and elimination rules for identity. It discusses what constitutes a proof and provides examples of proofs in Fitch notation. It also discusses proving that conclusions do not follow from premises by providing counterexamples.
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Truth, deduction, computation; lecture 3
- 1. Truth, Deduction, Computation Lecture 3 TheLogic of Atomic Sentences Vlad Patryshev SCU 2013
- 2. Introducing Arguments... Premise1, premise2…conclusion! Or: conclusion - because premise1,... E.g. ● All men are mortal; Superman is a man, hence Superman is mortal ● Pavlova is a man: after all, Pavlova is mortal, and all men are mortal
- 3. Introducing Arguments... Premise1, premise2…conclusion! Or: conclusion - because: premise1,... E.g. ● All men are mortal; Superman is a man, hence Superman is mortal ● Pavlova is a man: after all, Pavlova is mortal, and all men are mortal
- 4. Arguments ● Valid arguments(true, assuming premises are true) ● Sound arguments (valid, and premises are true)
- 5. Fitch Notation (LPLversion) All cactuses have needles Prickly pear is a cactus Prickly pear has needles Fitch Bar Conclusion Premises
- 6. What is aProof? Definition. Proof is a step-by-step demonstration that a conclusion follows from premises. Counterexample: I ride my bicycle every day The probability of an accident is very low I will never have an accident
- 7. Good Example ofa Proof 1. Cube(c) 2. c=b 3. Cube(b) = Elim: 1,2
- 8. Elimination Rule Aka theIndiscernibility of Identicals Aka Substitution Principle (weaker than Liskov’s) Aka Identity Elimination If P(a) and a = b, then P(b). E.g. x2 - 1 = (x+1)*(x-1) x2 > x 2 - 1 x2 > (x+1)*(x-1)
- 9. Introduction Rule Aka Reflexivityof Identity P1 P2 … Pn x = x
- 10. Symmetry of Identity Ifa = b then b = a a = b a = a b = a
- 11. Transitivity of Identity Ifa = b and b = c then a = c a = b b = c a = c
- 12. Other relationships maybe transitive If a < b and b < c then a < c a b c a < < < < b c d d
- 13. F-notation (specific toLPL book) (Has nothing to do with System F) We include in intermediate conclusions For example: P1 P2 … Pn S1 S2 … Sm S 1. a = b 2. a = a 3. b = a = Intro = Elim: 2, 1
- 14. Introduction Rule inFitch P1 P2 … Pn x = x
- 15. Introduction Rule (=Intro) in F = Intro x = x
- 16. Elimination Rule inF = Elim P(n) … n = m … P(m)
- 17. Reiteration Rule inF = Reit P … … … P
- 18. “Bidirectionality of Between”in F Between(a,b,c) … … … Between(a,c,b)
- 19. Now, How DoesIt Work? From premises SameSize(x, x) and x = y, prove SameSize(y, x). 1. SameSize(x, x) 2. x = y … ?. SameSize(y, x)
- 20. Now, How DoesIt Work? (take 2) From premises SameSize(x, x) and x = y, prove SameSize(y, x). 1. 2. … ?. ?. SameSize(x, x) x = y y = x SameSize(y, x) = Elim: 1, ?
- 21. Now, How DoesIt Work? (take 3) From premises SameSize(x, x) and x = y, prove SameSize(y, x). 1. 2. … 3. 4. 5. SameSize(x, x) x = y y = y y = x SameSize(y, x) = Intro = Elim: 3, 2 = Elim: 1, 4
- 22. Analytical Consequence inFitch This is something like a rule, but is based on “common sense” and external knowledge. E.g. Cube(a) SameShape(a, b) Cube(b) =Ana Con (“because we know what Cube means”) Can be used to prove anything as long as we believe in our rules. It’s okay.
- 23. Proving Nonconsequence E.g. Are allbinary operations associative? Addition is, multiplication is, even with matrices or complex number 1. op(a, b) = x 2. op(b, c) = y ?. op(a, y) = op(x, c)
- 24. Proving Nonconsequence E.g. Are allbinary operations associative? Addition is, multiplication is, even with matrices or complex number 1. op(a, b) = x 2. op(b, c) = y ?. op(a, y) = op(x, c) No!!! Take binary trees. Take terms (from Chapter 1)
- 25. Proving Nonconsequence Given premisesP1,...,Pn, and conclusion Q. Q does not follow from P1,...,Pn if we can provide a counterexample.
- 26. References What Fitch actuallyis: Fitch Online: http://en.wikipedia.org/wiki/Fitch-style_calculus http://logik.phl.univie.ac.at/~chris/gateway/formular-uk-fitch.html LPL software online (in Java Applets) http://softoption.us/content/node/339