Recap

Operator Precedence

How do we parse $\neg x \to y \lor z \to x \lor y \land z$
Operator precedence for propositional logic:
- $\neg$
- $\land$
- $\lor$
- $\to$
- $\lrarr$
all operators are right-associative, e.g. $a\to (b \to c)$
- in the example above: $(\neg x) \to ((y \lor z) \to (x \lor (y \land z)))$
main points
- $\neg$ binds to whatever immediately following it
- $\land$ and $\lor$ bind more tightly than $\to$
- this course will commonly write expressions like $p \land q \to r$ without adding parentheses

what is first order logic? a logical system for reasoning about properties of objects. augments logical connectives from propositional logic with
- predicates that describe properties of objects
- functions that map objects to one another
- quantifiers that allow us to reason about multiple objects
Every first order statement returns either true or false.

Example: $\text{Likes(You, Eggs)} \land \text{Likes(You, Tomato)} \to \text{Likes(You, Shakshuka)}$
- You, Eggs, Tomato, and Shakshuka are constant symbols: refer to objects, not predicates (true/false).
- the things that look like function calls are called predicates. Predicates take objects as arguments and evaluate to true or false.
  - typically, you’ll have a list of predicates, what they stand for, and how many arguments they take. It’ll be given separately than the formulas you write.
- what remains are traditional propositional connectives.

Examples: Cute(a) $\to$ Dikdik(a) $\lor$ Kitty(a) $\lor$ Puppy(a)
Another example: $x < 8 \to x < 137$
- the less-than sign is just another predicate. Binary predicates are sometimes written in infix notation this way. Numbers are not “built-in”, they’re constant objects just like “You”