LESSON #21: MATERIAL CONDITIONAL (7.1); BICONDITIONAL (7.2); INFORMAL METHODS OF PROOF (8.1)Assigned reading pp. 178-185I’m going to skip some of the discussion that’s in the text about the nature of truth-functional connectives, though it is still important and you should read and understand it. But I think our time here is best spent if I focus on the most important ideas in the text.POWERPOINT SLIDE #1The Material Conditional Translates as “If P, then Q”, where P and Q represent two sentences of FOL, and where P is the ‘antecedent’ or the condition that—when met or satisfied—ensures that Q (the ‘consequent’) follows as as a logical consequence. You can also read this less formally as “Whenever sentence P is true (in some world), sentence Q will also be true (in that world).”P Qantecedent consequentPOWERPOINT SLIDE #2Discuss the truth table for the material condition relation. Notice that the sentence is false in only one case (i.e., when the condition is met but the consequence doesn’t follow), and that . . . and this may seem a little weird . . . a material conditional with a false antecedent is always true as a whole sentence.POWERPOINT SLIDE #3There are other English expressions that can be translated with the material conditional:P Q“if P then Q” “if P, then Q”“if P, Q” “if P, Q”“Q if P” “Q if P”“Q provided P” “Q provided P”“provided P, Q” “provided P, Q”Notice that in all these cases, P occurs in the English translation immediately after the word “if” (or its equivalent “provided”)—whereas in the FOL sentence P comes before the conditional arrow. Obviously, then, you can’t simply translate the material conditional arrow symbol solely with the word “if” without getting youself in trouble. It’s safer to think of it as indicating the whole “if … then …” relation. Like this:POWERPOINT SLIDE #4 P Q P Q if … then … NOT: … if …POWERPOINT SLIDE #5“only if”: There is one exception to what I’ve said above, namely, when you see “only if”. Notice that here the order of P and Q relative to the word “if” is reversed, which means that “only if” is the only translation of the bunch where you can read the conditional arrow straight off before the consequent:“P only if Q” P QThis is because, as the book explains on p. 182, adding the word “only” highlights the fact that there is a necessary conditional relationship holding between P and Q versus merely a sufficient conditional relationship. I’m not going to go further into this in class because it can be confusing and the point is not important enough to spend time on. Just remember that “only if” is exceptional.POWERPOINT SLIDE #6The English word “unless” translates the into FOL as an “if not” relation:“unless P, Q”“Q unless P”“If not P, Q”“Q if not P”All these are translations of: ¬P QPOWERPOINT SLIDE #7The Biconditional: P ↔ QThis is when P and Q always have the same truth values (i.e., when one is T, the other is T, and when one is F, the other is F).This is just like the regular material conditional above, except the relation works both ways. Usually translated as “if and only if” and abbreviated as “if” (notice that there are two f’s in the abbreviated form. This is not a typo!). Also can be phrased “just in case” (mathematician’s usage).POWERPOINT SLIDE #8The truth table for the biconditional reflects the fact that the truth values of the two halves of a biconditional statement always covary. So, a biconditional is true only when P and Q are both true (row 1), or when they are both false (the last row).POWERPOINT SLIDE #9Note that the biconditional bears a close resemblance to the logical equivalence relation we’ve already learned, except that whereas equivalence is a relation between multiple sentences, the biconditional material is a connective used within one sentence at a time.There is, however, a way to express the relationship between the biconditional and the equivalence relation: 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).Assigned reading pp. 199-204This section of the reading (8.1) prepares the way for the formal proof rules involving conditionals that we’ll study next class. The part of the reading that I want to focus on here in today’s class concerns the discussion of various inference patterns that have played an important role in the classical study of logic. Some of these patterns of inference are invalid and some are valid, and exercise 8.1 asks you to decide which are which. Let’s talk through them together:POWERPOINT SLIDE #101. 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.POWERPOINT SLIDES #11-191. 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
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