Slides / Bullets

• ‘And’, ‘Or’ and ‘Not’
  • January 22nd, 2004
• ∧, ∨, ¬
  • ‘∧’ is the conjunction symbol — read ‘and’.
  • ‘∨’ is the disjunction symbol — read ‘or’ or ‘either...or...’
  • ‘¬’ is the negation symbol — read ‘not’ or ‘it is not the case that’.
    • Collectively, these are known as the Boolean connectives, after the 19th century logician George Boole
• Syntax
  • ‘∧’ and ‘∨’ are binary connectives: they take two sentences and make a new sentence.
    • Whenever P and Q are sentences, (P) ∧ (Q) and (P) ∨ (Q) are also sentences.
    • Contrast the English words ‘and’ and ‘or’, which can also conjoin verb phrases (‘Lassie barked and howled’) and noun phrases (‘Dean or Kerry will win’).
  • ‘¬’ 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 ‘not’, which can occur in all sorts of places.
• • Thus, we can build up arbitrarily complicated sentences:
    • ¬(White(snow))
    • (¬(White(snow))) ∨ (Green(grass))
    • (¬(White(snow))) ∨ (Green(grass)) ∧ (¬(¬(¬(Red(blood)))))
    • etc.
• Semantics
  • (P) ∧ (Q) is true when both P and Q are true; it is false when either P or Q is false.
    • Unlike the English ‘and’, there’s never any suggestion about temporal order.
  • (P) ∨ (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 ‘or’, there’s never any suggestion that it’s not the case that both P and Q are true. (∨ is ‘inclusive’ rather than ‘exclusive’)
  • ¬(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’ve just described is quite hard to read because of all the parentheses.
  • But we can’t just leave out the parentheses altogether. This would leave us with apparently ambiguous sentences like
    • State(guam) ∧ State(puertorico) ∨ State(pennsylvania).
  • We’ll adopt a more liberal notation that requires parentheses only when they’re required to avoid ambiguity.
• • We allow ‘¬’ to be followed by a sentence not in parentheses. In these circumstances ‘¬’ takes narrowest scope:
    • ¬State(guam) ∧ State(idaho) is equivalent to ¬(State(guam)) ∧ State(idaho).
    • Like ‘-’ in algebra.
  • We allow ‘∧’ and ‘∨’ to link sentences not in parentheses, when those sentences start with the negation symbol.
  • We allow multiple sentences to be joined by ‘∧’ and ‘∨’, thus:
    • State(guam) ∨ State(puertorico) ∨ State(alaska)
    • Just like ‘+’ and ‘×’ in alegebra.
• The Boolean connectives in Tarski’s 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’s false, you can play the game to figure out where you’re 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)