Where formal reasoning meets the street
If it rains → the street is wetIt rainsThe street is wet"If P then Q. P. Therefore Q." — The most fundamental rule of inference, tagged on every wall of rational thought.
| P | Q | P ∧ Q | P ∨ Q | P → Q |
|---|---|---|---|---|
| T | T | T | T | T |
| T | F | F | T | F |
| F | T | F | T | T |
| F | F | F | F | T |
"Every combination. Every possibility. The truth maps itself out right in front of you."
¬(P ∧ Q) ≡ (¬P ∨ ¬Q)¬(P ∨ Q) ≡ (¬P ∧ ¬Q)"Negate the whole, flip the operator, negate the parts. The duality written on every overpass."
∀x(P(x) → Q(x))
∃x(P(x) ∧ Q(x))
"How much of the world does your claim cover? All of it, or just a piece?"
Every premise demands a conclusion. The left side poses the structure — here, we trace the inference.
P ∧ Q
P → Q, Q ⊬ P
"The street is wet, therefore it rained" — ignoring sprinklers, floods, broken pipes.
P → Q, ¬P ⊬ ¬Q
"It didn't rain, so the street can't be wet" — a conclusion that doesn't follow.
All A are B, All C are B ⊬ All A are C
"All cats are animals. All dogs are animals. Therefore all cats are dogs." — The middle term connects nothing.
A proposition is either true or false. There is no middle ground.
P ∨ ¬P
Nothing can be both true and false at the same time.
¬(P ∧ ¬P)
If P implies Q, and Q implies R, then P implies R.
(P → Q) ∧ (Q → R) → (P → R)