Slides / Bullets

• ’ÄòAnd’Äô, ’ÄòOr’Äô and ’ÄòNot’Äô
  • January 22nd, 2004
• ’àß, ’à®, ¬¨
  • ’Äò’àß’Äô is the conjunction symbol ’Äî read ’Äòand’Äô.
  • ’Äò’à®’Äô is the disjunction symbol ’Äî read ’Äòor’Äô or ’Äòeither...or...’Äô
  • ’Äò¬¨’Äô is the negation symbol ’Äî read ’Äònot’Äô or ’Äòit is not the case that’Äô.
    • Collectively, these are known as the Boolean connectives, after the 19th century logician George Boole
• Syntax
  • ’Äò’àß’Äô and ’Äò’à®’Äô are binary connectives: they take two sentences and make a new sentence.
    • Whenever P and Q are sentences, (P) ’àß (Q) and (P) ’à® (Q) are also sentences.
    • Contrast the English words ’Äòand’Äô and ’Äòor’Äô, which can also conjoin verb phrases (’ÄòLassie barked and howled’Äô) and noun phrases (’ÄòDean or Kerry will win’Äô).
  • ’Äò¬¨’Äô is a unary connective: it takes one sentence and makes a new sentence.
    • Whenever P is a sentence, ¬¨(P) is a sentence.
    • Contrast the English word ’Äònot’Äô, which can occur in all sorts of places.
• • Thus, we can build up arbitrarily complicated sentences:
    • ¬¨(White(snow))
    • (¬¨(White(snow))) ’à® (Green(grass))
    • (¬¨(White(snow))) ’à® (Green(grass)) ’àß (¬¨(¬¨(¬¨(Red(blood)))))
    • etc.
• Semantics
  • (P) ’àß (Q) is true when both P and Q are true; it is false when either P or Q is false.
    • Unlike the English ’Äòand’Äô, there’Äôs never any suggestion about temporal order.
  • (P) ’à® (Q) is true when one or both of P and Q is true; it is false when both of them are false.
    • Unlike the English ’Äòor’Äô, there’Äôs never any suggestion that it’Äôs not the case that both P and Q are true. (’à® is ’Äòinclusive’Äô rather than ’Äòexclusive’Äô)
  • ¬¨(P) is true when P is false; it is false when P is true.
• • We can sum this up in the form of truth-tables:
• Parentheses
  • The notation I’Äôve just described is quite hard to read because of all the parentheses.
  • But we can’Äôt just leave out the parentheses altogether. This would leave us with apparently ambiguous sentences like
    • State(guam) ’àß State(puertorico) ’à® State(pennsylvania).
  • We’Äôll adopt a more liberal notation that requires parentheses only when they’Äôre required to avoid ambiguity.
• • We allow ’Äò¬¨’Äô to be followed by a sentence not in parentheses. In these circumstances ’Äò¬¨’Äô takes narrowest scope:
    • ¬¨State(guam) ’àß State(idaho) is equivalent to ¬¨(State(guam)) ’àß State(idaho).
    • Like ’Äò-’Äô in algebra.
  • We allow ’Äò’àß’Äô and ’Äò’à®’Äô to link sentences not in parentheses, when those sentences start with the negation symbol.
  • We allow multiple sentences to be joined by ’Äò’àß’Äô and ’Äò’à®’Äô, thus:
    • State(guam) ’à® State(puertorico) ’à® State(alaska)
    • Just like ’Äò+’Äô and ’Äòˆó’Äô in alegebra.
• The Boolean connectives in Tarski’Äôs World
• The Henkin-Hintikka game
  • The point of this is to give you another way to understand why your sentences get the truth-values they do in TW.
  • If you think a sentence should be true in a world, but TW says it’Äôs false, you can play the game to figure out where you’Äôre going wrong.
• For next week
  • Read: sections 3.1-3.8, and if you want to read ahead, 4.1-4.4; do the You try it exercises.
  • Do: exercises 2.22, 2.24 - 2.27, 3.6, 3.9, 3.13 - 3.15 (10% each)