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
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
▯ It is conventional to let
3 Something kinda like the con- ?Q
– P ▯ Q
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