lec-notes-conditionals-2.pdf

Conditionals Benj Hellie January 22, 2014 2. (a) But why not? Suppose Fred thinks (b) But not total universality: the anyone at ▯ Terminology: goats eat cans. Does he have to think issue is ‘anyone under consideration’. – A conditional is an if-then state- if goats don’t eat cans, you have (you (This is pertinent to the McGee stu▯.) don’t have) a million bucks, and if it is A conditional is kind of like a lite en- ment (isn’t) snowing today, goats eat cans? tailment. – We call the ‘if’-part the an- Evidently not. tecedent and the ‘then’-part the (b) One reason this would be bad is he 1.4 We get the right answers consequent would then find himself ready to argue – Modus ponens (MP) is this rule: The theory is purpose-built to avoid the entailments like this to persuade us that goats eat ▯ If P, Q; P ‘ Q cans: in (1). Moreover, i. if goats don’t eat cans, you have a – Modus tollens (MT) is this rule: 4. (a) Modus ponens: if I think everyone who ▯ If P, Q; :Q ‘ :P million bucks; and if goats don’t believes P should believe Q and I be- eat cans, you don’t have a million lieve P, then I believe Q bucks: so if goats don’t eat cans, you both do and don’t have a mil- (b) Modus tollens: if I think everyone who believes P should believe Q and I be- 1 The conditional lion bucks—and no contradiction lieve :Q, then I’d better not believe P— is true, so goats eat cans! ii. if it’s snowing outside, goats eat still, could I perhaps suspend judgement 1.1 Some entailments cans; and if it isn’t snowing out- on P? i. It looks like this theory doesn’t get Modus ponens is a real genuine entailment. Modus side, goats eat cans: so if it is tollens is a more problematic case. or isn’t snowing outside, goats eat full-blown modus tollens cans, and it’s gotta be one or the ii. But that is a good thing, as we shall other—so no matter what, goats see 1.2 Some nonentailments eat cans! Surely none of these is a real entailment, right? But those arguments are totally unper- 2 Something that isn’t the con- 1. (a) goats eat cans 0 if goats don’t eat cans, suasive, because we don’t accept the relevant conditionals ditional you have a million bucks (b) goats eat cans 0 if goats don’t eat cans, 2.1 Some entailments 1.3 What the conditional means you don’t have a million bucks Going by the meaning of ‘_’ (‘or’), each of these is (c) goats eat cans 0 if it’s snowing today, 3. (a) In general, it seems as if a conditional goats eat cans has universality built into it: an entailment, right? (d) goats eat cans 0 if it isn’t snowing today, ▯ When we believe ‘if P, Q’, we 5. (a) goats eat cans ‘ goats eat cans _ you goats eat cans think anyone who believes P have a million bucks should believe Q (b) goats eat cans ‘ goats eat cans _ you Good! Glad we’re on the same page. don’t have a million bucks Unless I am unreasonable—but in logic, we ignore that case. 1 (c) goats eat cans ‘ it isn’t snowing today 2.4 ▯ , if 3.2 ▯ in pictures _ goats eat cans (d) goats eat cans ‘ it’s snowing today _ ▯ and ‘if’ can’t be the same thing. Let P be ‘goats 3.2.1 Wondering whether P and whether Q eat cans’ and Q be ‘you have a million bucks’ and goats eat cans compare the (a) and (b) of (1) and (7); then let Q be We can represent the situation of someone wonder- ‘goats eat cans’ and P be ‘it’s snowing today’ and ing about the question whether P and the question After all, P _ Q is just less specific than P and less whether Q (e.g., ‘do goats eat cans?’ and ‘do horses specific than Q; and whenever I believe something, compare the (c) and (d) of (1) and (7). ▯ is involved eat hay?’) with the following sort of diagram: in entailments that ‘if’ just plain ain’t. They have I believe anything less specific than it; and entail- di▯erent meanings. ment is just if you believe this, you gotta believe that—or, in the abstract, these are just instances of ?P _-introduction X ▯ ▯ For this reason, we call the entailments in (7) the bad material entailments. 2.2 An abbreviation X ▯ It is conventional to let 3 Something kinda like the con- ?Q – P ▯ Q ditional abbreviate ▯ Standardly ▯ is called the material conditional. In logic courses, it is ordinarily o▯ered as the connec- – :P _ Q tive that means ‘if’. As we have seen, ▯ has a dif- ferent meaning from ‘if’. The top left box is a▯rmative answers to both (If we are being super-careful, we would write whether P and whether Q (namely, P^ Q); the bot- ‘((:P) _ Q)’—the claim is that at least one of :P, tom right is negative to both (namely, :P ^ :Q); 3.1 The good news the bottom left is a▯rmative to P, negative to Q Q is true) If the bad news about the material conditional is its (namely, P ^ :Q); the top right is a▯rmative to Q, overgenerating entailments, the good news is that it negative to P (namely, :P ^ Q). At this stage, they 2.3 Some entailments at least gets modus ponens and modus tollens accu- leave open all four possibilities. rately: If they give an a▯rmative answer to P, that 6. (a) P ‘ P _ Q (b) P ‘ P _ :Q