Slides / Bullets

• The material conditional ’Üí
  • ’Üí is a binary truth-functional connective.
  • It has the following truth-table:
  • The sentence on the left of ’Üí is called the antecedent, the one on the right is called the consequent.
• • P’ÜíQ is tautologically equivalent to ¬¨P ’à® Q, as can be easily verified using a truth table.
    • So having ’Üí in the language doesn’Äôt let us express anything we couldn’Äôt have expressed without it’Äîbut it’Äôs convenient nevertheless.
• Translations
  • Suppose I say: ’ÄòIf I left my scarf in the coffee shop, I left my cellphone there too’Äô
    • If I left the scarf there and didn’Äôt leave the cellphone there, it’Äôs clear that I’Äôve said something false. If I left both of them there, it seems pretty clear that I haven’Äôt.
    • What if it turns out I didn’Äôt leave the scarf there? In this case it sounds a bit odd to suggest that I’Äôve said something false: I might have had no good reason to say what I said, but that’Äôs not the same thing.
• • So there’Äôs a case to be made that ’ÄòP ’Üí Q’Äô is a correct translation into FOL of an English sentence ’ÄòIf P, then Q’Äô.
    • Think of it in terms of what one rules out: in saying ’ÄòIf P then Q’Äô, one is ruling out the case where P is true and Q isn’Äôt, and it’Äôs not clear that one is ruling out anything else.
  • When we’Äôre doing translations in this course, we will translate ’ÄòIf..then...’Äô using the material conditional.
• • But is this really correct? If it were, the following sentences would all be true:
    • ’ÄòIf pigs can fly, the moon is made of green cheese’Äô
    • ’ÄòIf pigs can fly, the moon isn’Äôt made of green cheese’Äô
    • ’ÄòIf pigs can fly, pigs can’Äôt fly’Äô
  • This seems pretty strange!
• • On the other hand, there’Äôs some evidence that ’Äòif...then...’Äô really does express the material conditional.
    • The argument ’ÄòP or Q; therefore if not-P, then Q’Äô seems valid.
    • But if this is valid, so is ’Äònot-P or Q; therefore if P then Q’Äô. So the English conditional is true whenever the material conditional is.
  • A vexed question in ’Äòphilosophical logic’Äô.
• • Other English expressions we’Äôll translated using ’ÄòP’ÜíQ’Äô:
    • Q if P (this is obviously equivalent to ’ÄòIf P then Q’Äô
    • Q provided that P
    • P only if Q
      • ’ÄòYou will pass the course only if you pass the final exam’Äô
  • ’ÄòUnless P, Q’Äô and ’ÄòQ unless P’Äô are translated as ’Äò¬¨P’ÜíQ’Äô
• • It’Äôs important to distinguish the conditional symbol’Äîwhich is part of FOL’Äîfrom 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 ’Üî
  • ’Üí is a binary truth-functional connective.
  • It has the following truth-table:
  • The biconditional is true when the left hand side and right hand side have the same truth-value; otherwise it’Äôs false.
• • P’ÜîQ is tautologically equivalent to (P’ÜíQ)’àß(Q’ÜíP).
    • It’Äôs also tautologically equivalent to (P’àßQ)’à®(¬¨P’à߬¨Q).
• • We use ’Äò’Üî’Äô to translate the English expression ’Äòif and only if’Äô, often abbreviated by mathematicians and philosophers as ’Äòiff’Äô.
    • ’ÄòIff’Äô is sometimes read as ’Äòjust in case’Äô’Äîa special bit of jargon.