Slide 1Truth table for the Material ConditionalSlide 3Slide 4Slide 5Slide 6Slide 7Truth table for the BiconditionalSlide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19The Material Conditional “If P, then Q”P Q antecedent consequentIn other words, whenever sentence P is true (in some world), sentence Q will also be true (in that world) . . . but not necessarily vice versa.Truth table for the Material ConditionalP Q P QT T TT F FF T T F F TP QVarious ways you may see it translated:Notice:“if P then Q” if P …“if P, Q” if P …“Q if P” … if P“Q provided P” … provided P“provided P, Q” provided P …How to think of the conditional arrow: P Q P Q if … then … NOT: … if …P Q(The above sentence of FOL may be translated in any of the ways below)Compare again:“if P then Q” if P …“if P, Q” if P …“Q if P” … if P“Q provided P” … provided P“provided P, Q” provided P …“P only if Q” . . . only if QNote “only if” is the one exception where you can read the arrow straight off as “only if”: “P only if Q” P Q“unless P, Q”“Q unless P” ¬ P Q“If not P, Q”“Q if not P”The Biconditional : ↔ P ↔ Q“P if and only if Q” or iff not a typo!Here P and Q have covarying truth values (always the same). This is like the material conditional except the conditional relation goes both directions.Truth table for the BiconditionalP QP ↔ QT T TT F FF T FF F Tbiconditional vs. logical equivalence •logical equivalence is a relation between multiple sentences (i.e., the truth values of sentence P and sentence Q covary)• the biconditional is a connective used within one sentence at a time (i.e., ‘P iff Q’)Two sentences P and Q are logically equivalent if and only if the biconditional formed from them, P ↔ Q is a logical truth (i.e., true in every world, or can never be false; note that this is not saying that P and Q are actually true in every world, only that their truth values necessarily covary because of the nature of the two sentences)1. Affirming the Consequent: From A B and B, infer A.2. Modus Tollens: From A B and ¬B, infer ¬A.3. Strengthening the Antecedent: From B C, infer (A ᴧ B) C.4. Weakening the Antecedent: From B C, infer (A ᴠ B) C.5. Strengthening the Consequent: From A B, infer A (B ᴧ C).6. Weakening the Consequent: From A B, infer A (B ᴠ C).7. Constructive Dilemma: From A ᴠ B, A C, and B D, infer C ᴠ D.8. Transitivity of the Biconditional: From A ↔ B and B ↔ C, infer A ↔ C.(9.) Modus Ponens: From A B and A, infer B.1. Affirming the Consequent: From A B and B, infer A.2. Modus Tollens: From A B and ¬B, infer ¬A.3. Strengthening the Antecedent: From B C, infer (A ᴧ B) C.4. Weakening the Antecedent: From B C, infer (A ᴠ B) C.5. Strengthening the Consequent: From A B, infer A (B ᴧ C).6. Weakening the Consequent: From A B, infer A (B ᴠ C).7. Constructive Dilemma: From A ᴠ B, A C, and B D, infer C ᴠ D.8. Transitivity of the Biconditional: From A ↔ B and B ↔ C, infer A ↔ C.(9.) Modus Ponens: From A B and A, infer B.= VALID INFERENCE PATTERN = LOGICAL FALLACY1. Affirming the Consequent: From A B and B, infer A.2. Modus Tollens: From A B and ¬B, infer ¬A.3. Strengthening the Antecedent: From B C, infer (A ᴧ B) C.4. Weakening the Antecedent: From B C, infer (A ᴠ B) C.5. Strengthening the Consequent: From A B, infer A (B ᴧ C).6. Weakening the Consequent: From A B, infer A (B ᴠ C).7. Constructive Dilemma: From A ᴠ B, A C, and B D, infer C ᴠ D.8. Transitivity of the Biconditional: From A ↔ B and B ↔ C, infer A ↔ C.(9.) Modus Ponens: From A B and A, infer B. = VALID INFERENCE PATTERN = LOGICAL FALLACY1. Affirming the Consequent: From A B and B, infer A.2. Modus Tollens: From A B and ¬B, infer ¬A.3. Strengthening the Antecedent: From B C, infer (A ᴧ B) C.4. Weakening the Antecedent: From B C, infer (A ᴠ B) C.5. Strengthening the Consequent: From A B, infer A (B ᴧ C).6. Weakening the Consequent: From A B, infer A (B ᴠ C).7. Constructive Dilemma: From A ᴠ B, A C, and B D, infer C ᴠ D.8. Transitivity of the Biconditional: From A ↔ B and B ↔ C, infer A ↔ C.(9.) Modus Ponens: From A B and A, infer B.1. Affirming the Consequent: From A B and B, infer A.2. Modus Tollens: From A B and ¬B, infer ¬A.3. Strengthening the Antecedent: From B C, infer (A ᴧ B) C.4. Weakening the Antecedent: From B C, infer (A ᴠ B) C.5. Strengthening the Consequent: From A B, infer A (B ᴧ C).6. Weakening the Consequent: From A B, infer A (B ᴠ C).7. Constructive Dilemma: From A ᴠ B, A C, and B D, infer C ᴠ D.8. Transitivity of the Biconditional: From A ↔ B and B ↔ C, infer A ↔ C.(9.) Modus Ponens: From A B and A, infer B.1. Affirming the Consequent: From A B and B, infer A.2. Modus Tollens: From A B and ¬B, infer ¬A.3. Strengthening the Antecedent: From B C, infer (A ᴧ B) C.4. Weakening the Antecedent: From B C, infer (A ᴠ B) C.5. Strengthening the Consequent: From A B, infer A (B ᴧ C).6. Weakening the Consequent: From A B, infer A (B ᴠ C).7. Constructive Dilemma: From A ᴠ B, A C, and B D, infer C ᴠ D.8. Transitivity of the Biconditional: From A ↔ B and B ↔ C, infer A ↔ C.(9.) Modus Ponens: From A B and A, infer B.1. Affirming the Consequent: From A B and B, infer A.2. Modus Tollens: From A B and ¬B, infer ¬A.3. Strengthening the Antecedent: From B C, infer (A ᴧ B) C.4. Weakening the Antecedent: From B C, infer (A ᴠ B) C.5. Strengthening the Consequent: From A B, infer A (B ᴧ C).6. Weakening the Consequent: From A B, infer A (B ᴠ C).7. Constructive Dilemma: From A ᴠ B, A C, and B D, infer C ᴠ D.8. Transitivity of the Biconditional: From A ↔ B and B ↔ C, infer A ↔ C.(9.) Modus Ponens: …
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