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Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Proving the Validity of an Argument Torben Amtoft Kansas State University Torben Amtoft Kansas State

  1. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Proving the Validity of an Argument Torben Amtoft Kansas State University Torben Amtoft Kansas State University Proving the Validity of an Argument

  2. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Torben Amtoft Kansas State University Proving the Validity of an Argument

  3. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Reminder In Fitch format, an argument takes the form P 1 premises · · · P n Q conclusion ◮ Such an argument is valid if conclusion Q is true whenever premises P 1 . . . P n are. ◮ A valid argument if sound if the premises are true. Torben Amtoft Kansas State University Proving the Validity of an Argument

  4. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Proving and Disproving Validity ◮ To show that an argument is not valid, use counterexamples: a world where the premises are true, but conclusion false. ◮ How to show that an argument is valid? (topic of Section 2.2-2.4) ◮ naive approach: consider all worlds where premise is true, and show that also conclusion is true. ◮ feasible approach: construct a proof Torben Amtoft Kansas State University Proving the Validity of an Argument

  5. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity A Valid Argument RightOf(b,c) LeftOf(d,e) b = d why? LeftOf(c,e) Torben Amtoft Kansas State University Proving the Validity of an Argument

  6. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity A Valid Argument RightOf(b,c) LeftOf(d,e) b = d why? LeftOf(c,e) Informal reasoning: (p.52) We are told that b is to the right of c . So c must be to the left of b , since right of and left of are inverses of one another. And since b = d , c is left of d , by the indiscernibility of identicals. But we are also told that d is left of e , and consequently c is to the left of e , by the transitivity of left of . This is our desired conclusion. Torben Amtoft Kansas State University Proving the Validity of an Argument

  7. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity A Formal Proof We establish a series of intermediate results: 1. RightOf(b,c) 2. LeftOf(d,e) 3. b = d 4. LeftOf(c,b) from 1, since LeftOf and RightOf are inverses 5. LeftOf(c,d) from 4 and 3, using identity elimination 6. LeftOf(c,e) from 5 and 2, since LeftOf is transitive Torben Amtoft Kansas State University Proving the Validity of an Argument

  8. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Identity Elimination (= Elim) ◮ If b is a cube and b equals c , then also c is a cube ◮ If John is happy and John is the father of Max, then the father of Max is happy In general (cf. p. 56) P ( t ) · · · t = u · · · P ( u ) ⊲ Torben Amtoft Kansas State University Proving the Validity of an Argument

  9. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity A Valid Arithmetic Argument x > 2 x · x > x + x why? Torben Amtoft Kansas State University Proving the Validity of an Argument

  10. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity A Valid Arithmetic Argument 1. x > 2 2. 2 > 0 fact 3. x > 0 from 1 and 2, since > is transitive 4. x · x > 2 · x from 1 and 3, using law of arithmetic: multiplying by positive number preserves inequalities 5. 2 · x = x + x fact 6. x · x > x + x from 4 and 5, using = Elim Torben Amtoft Kansas State University Proving the Validity of an Argument

  11. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity System F, vs. Fitch system F (p.54) Fitch (p.58) what is? formal system software package application domain general purpose tuned for Tarski’s world Proof rules: general = Elim , = Intro = Elim , = Intro ∧ Elim , ∧ Intro ∧ Elim , ∧ Intro etc. etc. shortcuts can be added Taut Con , FO Con specific can be added Ana Con (encodes block world laws) Torben Amtoft Kansas State University Proving the Validity of an Argument

  12. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Proof in Fitch Converting previous proof into Fitch: 1. RightOf(b,c) 2. LeftOf(d,e) 3. b = d 4. LeftOf(c,b) from 1, since LeftOf and RightOf are inverses 5. LeftOf(c,d) from 4 and 3, using identity elimination 6. LeftOf(c,e) from 5 and 2, since LeftOf is transitive Torben Amtoft Kansas State University Proving the Validity of an Argument

  13. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Proof in Fitch Converting previous proof into Fitch: 1. RightOf(b,c) 2. LeftOf(d,e) 3. b = d 4. LeftOf(c,b) Ana Con : 1 5. LeftOf(c,d) from 4 and 3, using identity elimination 6. LeftOf(c,e) from 5 and 2, since LeftOf is transitive Torben Amtoft Kansas State University Proving the Validity of an Argument

  14. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Proof in Fitch Converting previous proof into Fitch: 1. RightOf(b,c) 2. LeftOf(d,e) 3. b = d 4. LeftOf(c,b) Ana Con : 1 5. LeftOf(c,d) = Elim : 4, 3 6. LeftOf(c,e) from 5 and 2, since LeftOf is transitive Torben Amtoft Kansas State University Proving the Validity of an Argument

  15. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Proof in Fitch Converting previous proof into Fitch: 1. RightOf(b,c) 2. LeftOf(d,e) 3. b = d 4. LeftOf(c,b) Ana Con : 1 5. LeftOf(c,d) = Elim : 4, 3 6. LeftOf(c,e) Ana Con : 5, 2 Torben Amtoft Kansas State University Proving the Validity of an Argument

  16. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Limitations on Fitch ◮ Fitch is tuned for Tarski’s World, with Ana Con modeling the laws of the block world ◮ Fitch is not tuned for arithmetic. Thus Fitch cannot handle the proof that x · x > x + x Torben Amtoft Kansas State University Proving the Validity of an Argument

  17. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Limitations on Fitch ◮ Fitch is tuned for Tarski’s World, with Ana Con modeling the laws of the block world ◮ Fitch is not tuned for arithmetic. Thus Fitch cannot handle the proof that x · x > x + x ◮ Ana Con is very strong, so do not use it for exercises, unless explicitly allowed (neither use Taut Con or FO Con ) The Grade Grinder will report unauthorized use of Ana Con ! Torben Amtoft Kansas State University Proving the Validity of an Argument

  18. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Limitations on Fitch ◮ Fitch is tuned for Tarski’s World, with Ana Con modeling the laws of the block world ◮ Fitch is not tuned for arithmetic. Thus Fitch cannot handle the proof that x · x > x + x ◮ Ana Con is very strong, so do not use it for exercises, unless explicitly allowed (neither use Taut Con or FO Con ) The Grade Grinder will report unauthorized use of Ana Con ! ◮ System F is not tuned for Tarski’s world, and therefore has no special proof rules for the predicates there, like LeftOf ◮ But since the identity relation “=” is used in all domains of discourse, system F has special rules for that predicate Torben Amtoft Kansas State University Proving the Validity of an Argument

  19. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Identity Introduction (= Intro) Rule described p.55 is amazingly simple: t = t ⊲ This says that the identity relation is reflexive. Torben Amtoft Kansas State University Proving the Validity of an Argument

  20. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Properties of Identity Symmetry a = b b = a Torben Amtoft Kansas State University Proving the Validity of an Argument

  21. Outline Introduction Proving Validity: Examples The Notion of Proof Rules Proving Validity: More Examples Fitch Reasoning about Identity Properties of Identity Symmetry a = b b = a Transitivity a = b b = c a = c Torben Amtoft Kansas State University Proving the Validity of an Argument

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