Logic
\[\vee, \lor
\wedge
\colon
\oplus
\otimes
\lnot
\forall
\exists\]
Formal Notation
- Inference: agent will use knowledge and logic to provide inference
- Soundness: Only valid conclusions can be proven
-
Completeness: All valid conclusions can be proven
- Predicate: a function that maps object args to T/F
- Feathers(bluebird) »> True
Implication x implies y, $\Rightarrow$
Conjunctions, Disjunctions, Negations, Implications
Conjunction
if an animal lays eggs and an animal flies than the animal is a bird
\[\text{If Lays-eggs(animal)} \wedge \text{Flies(animal)} \Rightarrow \text{Then Bird(animal)}\]Disjunction
if an animal lays eggs or an animal flies than the animal is a bird \(\text{Lays-eggs(animal)} \vee \text{Flies(animal)} \Rightarrow \text{Then Bird(animal)}\)
If an animal flies and is not a bird, it is a bat \(\text{Flies(animal)} \wedge \lnot\text{Bird(animal)} \Rightarrow \text{Bat(animal)}\)
Truth Tables
- Demorgan’s law $\lnot(A\wedge B) == \lnot A \vee \lnot B$
- The outer not flips the inner operation
Kinda weird logic for “Implies” $\Rightarrow$
| A | B | $A\Rightarrow B$ |
|---|---|---|
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
$A \Rightarrow B == \lnot A \vee B$
Rules of Inference
Modus Ponens
| x | Logic |
|---|---|
| Sentence 1 | $P \Rightarrow Q$ |
| Sentence 2 | $P$ |
| Sentence 3 | $Q$ |
Modus Tollens
| x | Logic |
|---|---|
| Sentence 1 | $P \Rightarrow Q$ |
| Sentence 2 | $\lnot Q$ |
| Sentence 3 | $\lnot P$ |
Resolution Theorem Proving
- Prove the opposite of what we’re trying to prove
- Start by eliminating the thing you’re trying to prove with the contrapositive in the full sentence