Review

first-order logic is a logical system for reasoning about properties of objects.

• $\exists$ - there exists (existential operator)
• $\forall$ - for any (universal quantifier)
• “Some A is a B” translates as $\forall x.~(A(x) \land B(x))$
  • useful intuition: existentially quantified statements are false unless there’s a positive example
• “All A’s are B’s” translates as $\forall x.~(A(x) \to B(x))$
  • useful intuition: universally quantified statements are true unless there’s a counterexample

The Aristotelian Forms

• “All As are Bs” → $\forall x.~(A(x) \to B(x))$
• “Some As are Bs” → $\exists x.~((A(x) \land B(x))$
• “No As are Bs” → $\forall x.~(A(x) \to \neg B(x))$
• “Some As aren’t Bs” → $\exists x.~(A(x) \land \neg B(x))$
• it is worth committing these patterns to memory.

Exercises

• Using the predicates

  • Person(p), which states that p is a person
  • Loves(x, y), which states that x loves y

• write a sentence in first-order logic that means “every person loves someone else.”

  • my attempt: $\forall x.~(\text{Person}(x) \to (\exists y.~\text{Person}(y) \land \neg (x = y) \land~\text{Loves}(x,y)))$
  • answer:

  $$ \begin{align*} &\forall p.~(\text{Person}(p) \to \\ &~\exists q.~(\text{Person}(q) \land p \neq q~\land\\ &~~\text{Loves}(p,q)\\ &~)\\ &) \end{align*} $$

• Using the same predicates, write a sentence that means “there is a person that everyone else loves

  • my attempt: $\exists p.~(\text{Person}(p) \to ( \forall q.~(\text{Person}(q) \land p \neq q \land \text{Loves}(q,p)))$
    • incorrect!
  • answer: $\exists p.~(\text{Person}(p) \land (\forall q.~(\text{Person}(q) \land p \neq q \to \text{Loves}(q,p)))$
    • I switched some $\land$s and $\to$s.

Combining Quantifiers

• combining quantifiers makes this more interesting; like the exercises above
• order of the quantifiers matters

Exercise