Slides / Bullets

• Translation and the quantifiers
  • The four Aristotelian forms, and their standard translations.
• • Do not make the common mistake of translating ‘Some Fs are Gs’ by ‘∃x(F(x)→G(x))’.
    • The latter sentence is true if there is any object that is either not an F, or a G!
• • Some points to note
    • ‘Some Fs are Gs’ is treated as equivalent to ‘Some F is a G’, despite the fact that some people have the intuition that ‘Some Fs are Gs’ would be false if only one F was a G.
      • Are they right?
    • ‘∃x(F(x)∧G(x))’ and ‘∀x(F(x)→G(x)’ are not inconsistent: both could easily be true. But some people have the intuition that ‘Some Fs are Gs’ and ‘All Fs are Gs’ are inconsistent.
      • Are they right?
• • • If ‘∀x(¬F(x))’ is true, ‘∀x(F(x)→G(x))’ and ‘∀x(F(x)→¬G(x))’ are both true — in this case they are called vacuously true. So ‘∀x(F(x)→G(x))’ does not entail ‘∃x(F(x)∧G(x))’. But some people have the intuition that ‘All Fs are Gs’ and ‘No Fs are Gs’ are inconsistent, and the ‘All Fs are Gs’ entails ‘Some F is a G’.
      • Are they right?
  • Arguably, none of the intuitions I’ve just been talking about is right: they all arise from confusing conversational implicature with entailment.
• • Sometimes the roles of ‘F’ and ‘G’ in the translations we’ve just been looking at will be played by complex predicates like ‘happy dog’ or ‘black dog owned by Clinton’. So in our translations, the role of F(x) will be played by a complex open formula.