Die Logik ist keine Lehre, sondern ein Spiegelbild der Welt.
Logik ist transzendental.

- Ludwig Wittgenstein's (1921, 6.13) Tractatus Logico-Philosophicus in the original German

Logic is not a body of doctrine, but a mirror-reflection of the world.
Logic is transcendental.
- Pears/McGuinness translation

φ ⊨ I

General form of a semantic entailment relation between a logical formula and an interpretation of the world

1. A well-defined semantics allows us to associate formulae with their meanings

2. I denotes a model or interpretation of the world
3. φ denotes a logical formula
4. ⊨ (double turnstile symbol) denotes a semantic entailment relation between φ and I
5. NOTE: ⊨ (for semantic consequence) is closely related to ⊦ (single turnstile symbol for syntactic or logical consequence)

6. φI denotes that φ is true on interpretation I

Truth table for ~p

 p ~p 0 1 1 0

1. Assuming classical bivalence, any proposition will bear one of the following 2 truth values:
2. Truth (1) or falsity (0)

3. Where φ denotes ~p, it is true on 1 interpretation (i.e. 1 row of the truth table)
4. INTERPRETATION 1 (1st row of the truth table): p = 0, ~p = 1
5. INTERPRETATION 2 (2nd row of the truth table): p = 1, ~p = 0

6. According to INTERPRETATION 1, ~p is true when p is false
7. There are no other alternative interpretations of the world

Truth table for p ∧ q

 p q p ∧ q 0 0 0 1 0 0 0 1 0 1 1 1

Where φ denotes p ∧ q, it is true when p is true and q is true (INTERPRETATION 4)

Truth table for p ∨ q

 p q p ∨ q 0 0 0 1 0 1 0 1 1 1 1 1

Where φ denotes p ∨ q, it is true when either p or q is true (INTERPRETATION 2 and INTERPRETATION 3) or both p and q are true (INTERPRETATION 4)

Truth table for p → q

 p q p → q 0 0 1 1 0 0 0 1 1 1 1 1

Where φ denotes p → q, it is only false when p is true and q is false (INTERPRETATION 2)

Truth table for p ⟷ q

 p q p ⟷ q 0 0 1 1 0 0 0 1 0 1 1 1

Where φ denotes p ⟷ q, it is true when p and q have the same truth values (INTERPRETATION 1 and INTERPRETATION 4)
1. Suppose that there are n possible worlds w1,w2,…, wn
2. We may denote the actual world @ in terms of wi, where i ∈ ℕ and i ∈ [1, n]

3. Where φ denotes □p (necessity), it is true in all possible worlds w1,w2,…, wn
4. Where φ denotes ◇p (possibility), it is true in at least one possible world
5. Where φ denotes ◇p ∧ ∼□p (contingency), it is true in at least one possible world, though not all of them
6. Where φ denotes ∼◇p (impossibility), it is false in all possible worlds
1. Suppose that we live in a home world h
2. There are n morally acceptable alternative worlds w1,w2,…, wn relative to h
3. We may denote the actual world @ in terms of wi, where i ∈ ℕ and i ∈ [1, n]

4. Where φ denotes Op (obligatoriness), the action described by p is performed in all morally acceptable alternative worlds w1,w2,…, wn
5. Where φ denotes Pp (permissibility), the action described by p is performed in at least one morally acceptable world
6. Where φ denotes Pp ∧ ∼Op (optionality), the action described by p is performed in at least one morally acceptable world, though not all of them
7. Where φ denotes ∼Pp (impermissibility), the action described by p is not performed in all morally acceptable alternative worlds