Logic Universe: Foundations & Reasoning

A single journey from the simplest idea of truth to full argument craft. Tap glowing chips for deeper rules, tables, and examples.

0) Symbol Key (friendly first)

We’ll say things in normal English first, then show the compact symbols once you’re comfortable.

• and ( ∧ ) — both must be true.
• or ( ∨ ) — at least one true.
• not ( ¬ ) — flips true ↔ false.
• if P then Q ( P → Q ) — only false when P is true and Q is false.
• if and only if ( P ↔ Q ) — true when both match.
• xor ( ⊕ ) — true when exactly one is true.
• for all x ( ∀x ) — every member of a domain.
• there exists x ( ∃x ) — at least one member of a domain.

1) Binary: 0 and 1, the skeleton of truth

Binary uses only two values: 0 = False and 1 = True. Truth tables are just binary counting through all possible worlds.

Here’s a taste of the “beat” inside binary:

Those rhythms become the skeleton of every truth table you will ever build.

2) Boolean Logic & Gates (with human intuition)

Gates are truth tables turned into machines. Memorise the *idea* first, then the 4-bit “signature” second.

NOT (negation)

not P means the opposite of P.

AND

P and Q is true only when both are true. Think: two people must press the launch button.

AND signature: 0001

OR

P or Q is true if at least one is true. Think: either keycard opens the door.

OR signature: 0111

XOR (exclusive or)

P xor Q is true when exactly one is true. Think: choose rice or noodles — but not both.

XOR signature: 0110

If and only if (biconditional)

P iff Q is true when P and Q match. Think: membership works both ways — you’re a member exactly when you have the pass.

Biconditional signature: 1001

Implication (“if P then Q”)

If P then Q is only false in one world: when P happens but Q doesn’t.

Implication signature: 1101

**Paris example:** If I’m in Paris, then I’m in France. The rule only breaks if I claim Paris but not France.

3) Propositional Logic (rules of inference)

Propositional logic treats whole statements as blocks. We test which moves preserve truth.

Modus Ponens (valid)

Modus Tollens (valid)

Affirming the consequent (invalid)

Form:

Why invalid? Because Q can happen for other reasons besides P.

Denying the antecedent (invalid)

Why invalid? Not-P doesn’t stop Q happening another way.

4) Predicate Logic (things in the world)

Predicate logic adds objects and domains. We’ll use English, with a one-time symbol view.

5) Categorical Logic: AEIO

Categorical logic describes how two circles (sets) overlap. AEIO are four “shapes of meaning”.

A: All S are P E: No S are P I: Some S are P O: Some S are not P

Minimum overlap rule (aliens example)

6) Square of Opposition

The square shows how AEIO statements “push” against each other.

7) Transformations: Obversion, Conversion, Contraposition

These are safe ways to rewrite categorical statements.

8) Syllogisms (chains of categories)

A syllogism links S and P through a middle term M. Validity is about distribution and overlap.

Classic valid form (Barbara)

Invalid form (foxes/cars)

9) Three domains, one family

Logic comes in domains:

• Boolean logic — operations on truth values (and/or/not).
• Propositional logic — whole statements as blocks (MP, MT).
• Categorical logic — sets/categories and overlaps (AEIO).

They overlap because they all track the same thing: which worlds allow truth to flow safely.

10) Building arguments (from natural English)

An argument is a bridge: premises → inference → conclusion. We build it in natural English first, then formalise if needed.

Domain example (your fridge story)

11) Fallacies (why they feel tempting)

Fallacies are arguments that *sound* right because the brain fills gaps.

12) Socratic method & debate maxims

13) Logic games (training ground)

Games build intuition faster than memorising lists. We’ll do them in dedicated tool pages, but the vibe is here: logic should feel like play.

Tools (after the course)

These are your revisit-anytime labs. We’ll build them next.