Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Logical possibility:A sentence-claim is logically possible if there is some (at least one) logically conceivable circumstance (situation or world) in which the claim is true. Jimmy Fallon is the president of the United States.To prove logical possibility:Simply imagine or describe a logical world (situation, state of affairs) in which the sentence is true.A narrower interpretation of logical possibility: Tarski’s World possibilitySentences that can be made true by at least one conceivable orientation of blocks in Tarski’s World can be called TW-possible.Cube(a) ᴧ Larger(a,b) But not: Ride(tet,dancingponies)To prove TW-possibility:Construct a world in Tarski’s World that makes the sentence true.Logical necessity:A sentence-claim is logically necessary if it is true in every logically possible circumstance (every logically conceivable world). Such a sentence is called a logical truth. Jimmy Fallon is Jimmy Fallon.A narrower interpretation of logical necessity: Tarski’s World necessitySentences that are true in every conceivable orientation of blocks in Tarski’s World can be called TW-necessary.Tet(a) ᴠ Cube(a) ᴠ Dodec(a)Before going any further, we need to look at the infamous Figure 4.1 on p. 102 of the textbook . . .Tautologies:A sentence is a tautology if its logical necessity derives strictly from its sentence structure and the meanings of the connectives that hold the sentence together.Tet(a) ᴠ ¬Tet(a)P ᴠ ¬P A tautology is a special kind of logical truth.We can still see the necessity when we ‘blind’ ourselves to the meaning of the atomic sentences.4 2, 7, 9 5, 8¬Tet(a) ᴠ ¬Cube(b) ᴠ a≠b¬(Small(a) ᴧ Small(b)) ᴠ Small(a)SameSize(a,b) ᴠ ¬(Small(a) ᴧ Small(b))Logically-possible(and in TW)TW-necessaryLogically-necessary(and in TW)Tautological(and in TW)1, 3, 6, 10Exercise 4.8 Problems #7-#9:Truth tables are truth-functional.Therefore, they ‘blind’ us to themeanings of the atomic sentences.P QP ᴧ QTRUE TRUE TRUETRUE FALSE FALSEFALSE TRUE FALSEFALSE FALSE FALSE¬(A ᴧ (¬A ᴠ (B ᴧ C))) ᴠ Bor“You couldn’t recognize a tautology if it knocked you up side the head”A B C¬ (A ᴧ (¬ A ᴠ (B ᴧ C))) ᴠ BT T TT T FT F TT F FF T TF T FF F TF F FA B C¬ (A ᴧ (¬ A ᴠ (B ᴧ C))) ᴠ BT T T TT T F FT F T FT F F FF T T TF T F FF F T FF F F FA B C¬ (A ᴧ (¬ A ᴠ (B ᴧ C))) ᴠ BT T T F TT T F F FT F T F FT F F F FF T T T TF T F T FF F T T FF F F T FA B C¬ (A ᴧ (¬ A ᴠ (B ᴧ C))) ᴠ BT T T F T TT T F F F FT F T F F FT F F F F FF T T T T TF T F T T FF F T T T FF F F T T FA B C¬ (A ᴧ (¬ A ᴠ (B ᴧ C))) ᴠ BT T T T F T TT T F F F F FT F T F F F FT F F F F F FF T T F T T TF T F F T T FF F T F T T FF F F F T T FA B C¬ (A ᴧ (¬ A ᴠ (B ᴧ C))) ᴠ BT T T F T F T TT T F T F F F FT F T T F F F FT F F T F F F FF T T T F T T TF T F T F T T FF F T T F T T FF F F T F T T FA B C¬ (A ᴧ (¬ A ᴠ (B ᴧ C))) ᴠ BT T T F T F T T TT T F T F F F F TT F T T F F F F TT F F T F F F F TF T T T F T T T TF T F T F T T F TF F T T F T T F TF F F T F T T F T TAUTOLOGY↑Truth-table necessity and possibilityTT-necessary – shown by main connective being True in all rows (= tautology)TT-possible – shown by main connective being True in at least one row.Another way to prove a tautology:If you can formally prove a conclusion without using any premises, the conclusion is a tautology.The truth-table method is unable to spot non-tautological logical necessities (standard logical truths that are not tautologies) whose necessity depends on the meanings of the atomic sentences:c = c ᴧ d = dIt looks like thissentence can beFALSE in 3 out of4 worlds, butactually those3 worlds couldnever logicallyexist becausethey violate thereflexivity ofIdentity. A (c=c)B (d=d)A ᴧ BTRUE TRUE TRUETRUE FALSE FALSEFALSE TRUE FALSEFALSE FALSE
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