LOGICAL.DAY

An Alchemist's Logic Workshop

Specimen #01

Propositional Logic

The foundation of all logical reasoning. A proposition is a declarative statement that is either true or false — a specimen of absolute certainty pinned to the workshop wall.

P: "It is raining" Q: "The ground is wet" P → Q "If it rains, the ground is wet"

In this workshop, propositions are the raw ingredients. Each one a dried herb, hung from the rafter, waiting to be combined into more potent formulae.

P P P
Specimen #02

AND Gate ( ∧ )

Conjunction. True only when both operands are true. Two reagents must combine for the reaction to occur.

P ∧ Q True AND True = True True AND False = False
∧ ∧ ∧
Specimen #03

OR Gate ( ∨ )

Disjunction. True when at least one operand is true. Like the branching mycelium, any path leads to truth.

P ∨ Q True OR False = True False OR False = False
∨ ∨ ∨
Specimen #04

NOT ( ¬ )

Negation. Inverts the truth value. The alchemical mirror that turns gold to lead.

¬P NOT True = False
¬ ¬ ¬
Specimen #05

Implication ( → )

Material conditional. False only when the antecedent is true and the consequent is false.

P → Q True → False = False False → anything = True

The most treacherous operator. "Ex falso quodlibet" — from falsehood, anything follows.

→ → →
Specimen #06

Truth Table

A complete enumeration of all possible truth values for a compound proposition. The alchemist's recipe book, listing every possible combination of ingredients and their results.

P Q P ∧ Q P ∨ Q P → Q P ⇔ Q
T T T T T T
T F F T F F
F T F T T F
F F F F T T
Specimen #07

Tautology & Contradiction

Tautology

Always true, regardless of inputs. The philosopher's stone of logic.

P ∨ ¬P ≡ T

Contradiction

Always false, regardless of inputs. A volatile compound that self-destructs.

P ∧ ¬P ≡ F
⊥ ⊥ ⊥
Specimen #08

Modus Ponens

The fundamental rule of inference: if P implies Q, and P is true, then Q must be true. The alchemist's most reliable transmutation.

Premise 1: P → Q Premise 2: P ──────────────────── ∴ Q

Latin: "the way that affirms by affirming." The most ancient and trusted tool in the logician's workshop.

⇒ ⇒ ⇒
Specimen #09

Modus Tollens

Denying the consequent: if P implies Q, and Q is false, then P must be false. A reverse distillation, extracting truth from absence.

Premise 1: P → Q Premise 2: ¬Q ──────────────────── ∴ ¬P

Latin: "the way that denies by denying." The contrapositive in action.

&nArr; &nArr; &nArr;
Specimen #10

Hypothetical Syllogism

Chaining implications: if P implies Q, and Q implies R, then P implies R.

P → Q Q → R ──────────── ∴ P → R
Specimen #11

Biconditional ( ⇔ )

If and only if. True when both sides share the same truth value. A perfectly balanced equation.

P ⇔ Q (P → Q) ∧ (Q → P)
⇔ ⇔ ⇔
Specimen #12

De Morgan's Laws

The dual nature of conjunction and disjunction under negation. An alchemical transmutation where AND becomes OR through the crucible of NOT.

¬(P ∧ Q) ≡ (¬P) ∨ (¬Q) ¬(P ∨ Q) ≡ (¬P) ∧ (¬Q)
¬(P ∧ Q) ¬(P ∨ Q) ¬P ∨ ¬Q ¬P ∧ ¬Q

Observe: negation distributes and flips the operator, like an alchemical transmutation — lead into gold, gold into lead.

≡ ≡ ≡
Specimen #13

Exclusive OR ( ⊕ )

True when exactly one operand is true, but not both. A jealous compound that refuses to share.

P ⊕ Q (P ∨ Q) ∧ ¬(P ∧ Q)
⊕ ⊕ ⊕
Specimen #14

Proof by Contradiction

Reductio ad absurdum: assume the negation of what you wish to prove, and derive a contradiction. The alchemist's trial by fire.

1. Assume ¬P 2. Derive Q from ¬P 3. Derive ¬Q from ¬P 4. Q ∧ ¬Q — Contradiction! ⚠ 5. Therefore P — QED ✓

The most dramatic proof technique: destroy your own hypothesis to reveal the truth beneath.

Specimen #15

The Contrapositive

The logical mirror: a statement and its contrapositive are always equivalent. If the original is true, so is its reflection.

Original:      P → Q Contrapositive: ¬Q → ¬P
"If it rains, ground is wet" "If ground not wet, it didn't rain"

The alchemist's looking glass. Every truth casts a contrapositive shadow.

↔ ↔ ↔
Specimen #16

Quantifiers

Universal ( ∀ )

"For all" — a claim about every element in the domain.

∀x P(x) "All mushrooms glow"

Existential ( ∃ )

"There exists" — at least one element satisfies the condition.

∃x P(x) "Some mushroom glows"
¬∀x P(x) ≡ ∃x ¬P(x) "Not all glow" = "Some don't glow"
∀ ∀ ∀