Slides / Bullets
 The material conditional →
 → is a binary truthfunctional connective.
 It has the following truthtable:
 The sentence on the left of → is called the antecedent, the one on the right is called the consequent.

 P→Q is tautologically equivalent to ¬P ∨ Q, as can be easily verified using a truth table.
 So having → in the language doesn’t let us express anything we couldn’t have expressed without it—but it’s convenient nevertheless.
 Translations
 Suppose I say: ‘If I left my scarf in the coffee shop, I left my cellphone there too’
 If I left the scarf there and didn’t leave the cellphone there, it’s clear that I’ve said something false. If I left both of them there, it seems pretty clear that I haven’t.
 What if it turns out I didn’t leave the scarf there? In this case it sounds a bit odd to suggest that I’ve said something false: I might have had no good reason to say what I said, but that’s not the same thing.

 So there’s a case to be made that ‘P → Q’ is a correct translation into FOL of an English sentence ‘If P, then Q’.
 Think of it in terms of what one rules out: in saying ‘If P then Q’, one is ruling out the case where P is true and Q isn’t, and it’s not clear that one is ruling out anything else.
 When we’re doing translations in this course, we will translate ‘If..then...’ using the material conditional.

 But is this really correct? If it were, the following sentences would all be true:
 ‘If pigs can fly, the moon is made of green cheese’
 ‘If pigs can fly, the moon isn’t made of green cheese’
 ‘If pigs can fly, pigs can’t fly’
 This seems pretty strange!

 On the other hand, there’s some evidence that ‘if...then...’ really does express the material conditional.
 The argument ‘P or Q; therefore if notP, then Q’ seems valid.
 But if this is valid, so is ‘notP or Q; therefore if P then Q’. So the English conditional is true whenever the material conditional is.
 A vexed question in ‘philosophical logic’.

 Other English expressions we’ll translated using ‘P→Q’:
 Q if P (this is obviously equivalent to ‘If P then Q’
 Q provided that P
 P only if Q
 ‘You will pass the course only if you pass the final exam’
 ‘Unless P, Q’ and ‘Q unless P’ are translated as ‘¬P→Q’

 It’s important to distinguish the conditional symbol—which is part of FOL—from the notion of logical consequence which is a relation between sentences of FOL.
 A conditional can be true even if the consequent is not a logical consequence of the antecedent.
 However, for a conditional is logically true, the consequent does have to be a logical consequence of the antecedent.
 The material biconditional ↔
 → is a binary truthfunctional connective.
 It has the following truthtable:
 The biconditional is true when the left hand side and right hand side have the same truthvalue; otherwise it’s false.

 P↔Q is tautologically equivalent to (P→Q)∧(Q→P).
 It’s also tautologically equivalent to (P∧Q)∨(¬P∧¬Q).

 We use ‘↔’ to translate the English expression ‘if and only if’, often abbreviated by mathematicians and philosophers as ‘iff’.
 ‘Iff’ is sometimes read as ‘just in case’—a special bit of jargon.