Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Use of truth tables: Proving or demonstrating tautological relations (i.e., relations such as tautological consequence that derive strictly from the effects of the truth-functional connectives)Limitations of truth tables:1) Their potentially inordinate size2) Because truth tables are strictly truth-functional, they can demonstrate only tautological relationsNew proof rules based on Boolean connectives (allow us to carry out proofs sensitive to non-tautological relations):Conjunction Introduction: ᴧ Intro Conjunction Elimination: ᴧ Elim Disjunction Introduction: ᴠ Intro Disjunction Elimination: ᴠ ElimConjunction Introduction: ᴧ Intro We can conjoin any two or more sentences that we already know (or can assume) are true, thus building a new conjunction out of known parts.1 A2 BA ᴧ Bᴧ Intro: 1,2 If we these are each true separately, then we know they are true together.Conjunction Elimination: ᴧ ElimIf we already know (or can assume) that a particular conjunction is true, we can ‘break’ the conjunction apart and assert any of the individual conjuncts as a new step. 1 A ᴧ B A ᴧ Elim: 1 If we these are true together, then we know each one is true by itself.Disjunction Introduction: ᴠ IntroWe can take any sentence already known (or assumed) to be true and make it one disjunct of a larger disjunction (whether or not we know the other disjunct(s) to be true).01 AA ᴠ Bᴠ Intro: 1We can do this because only one disjunct of a disjunction need be true to make the entire disjunction true (recall the truth table for ᴠ). If we this is true by itself, then we know that any disjunction containing it is true.Disjunction Elimination: ᴠ Elim (‘proof by cases’)If we have a disjunction we already know to be true, and we seek a conclusion that we can prove to be a logical consequence of each of the disjuncts separately, then even if we don’t know which individual disjuncts are actually true, we can assert the conclusion (because it follows separately from each and every one of the disjuncts—we’ve ‘covered all our bases’, so to speak).TonyKing GeorgeQueen ElizabethThe DukeTony: “Fish dead.”Therefore . . . . . . Tony sadTony and the dead goldfish:King George or Queen Elizabeth or The Duke has died.King George dead Tony sadQueen Elizabeth dead Tony sadThe Duke dead Tony sadRegardless which fish is (are) dead, the outcome is the same: Tony is sad (in fact, he’s devastated; hence, he is absent from class). Proof by cases.1 A ᴠ B must have a disjunction to begin with2 A one disjunct becomes premise of a subproof3 C must prove this from the premise (A) 4 B other disjunct becomes premise of another subproof5 C must prove this from the premise (B)6 C ᴠ Elim: 1, 2-3, 4-5conclusion here must be same as conclusion of each subproof above Notice how to cite stepsMandatory: Know how to use each of these four rules in the Fitch program (be sure to do the ‘You Try It’ sections and know how to input the rules)Optional: You don’t need to know the “default and generous uses” of each
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