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
Semantics
φ ⊨ I
General form of a semantic entailment relation between a logical formula and an interpretation of the world
A well-defined semantics allows us to associate formulae with their meanings
I denotes a model or interpretation of the world
φ denotes a logical formula
⊨ (double turnstile symbol) denotes a semantic entailment relation between φ and I
NOTE: ⊨ (for semantic consequence) is closely related to ⊦ (single turnstile symbol for syntactic or logical consequence)
φ ⊨ I denotes that φ is true on interpretation I
Truth table for ~p
p
~p
0
1
1
0
Assuming classical bivalence, any proposition will bear one of the following 2 truth values:
Truth (1) or falsity (0)
Where φ denotes ~p, it is true on 1 interpretation (i.e. 1 row of the truth table)
INTERPRETATION 1 (1^{st} row of the truth table): p = 0, ~p = 1
INTERPRETATION 2 (2^{nd} row of the truth table): p = 1, ~p = 0
According to INTERPRETATION 1, ~p is true when p is false
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)
Suppose that there are n possible worlds w_{1},w_{2},…, w_{n}
We may denote the actual world @ in terms of w_{i}, where i ∈ ℕ and i ∈ [1, n]
Where φ denotes □p (necessity), it is true in all possible worlds w_{1},w_{2},…, w_{n}
Where φ denotes ◇p (possibility), it is true in at least one possible world
Where φ denotes ◇p ∧ ∼□p (contingency), it is true in at least one possible world, though not all of them
Where φ denotes ∼◇p (impossibility), it is false in all possible worlds
Suppose that we live in a home world h
There are n morally acceptable alternative worlds w_{1},w_{2},…, w_{n} relative to h
We may denote the actual world @ in terms of w_{i}, where i ∈ ℕ and i ∈ [1, n]
Where φ denotes Op (obligatoriness), the action described by p is performed in all morally acceptable alternative worlds w_{1},w_{2},…, w_{n}
Where φ denotes Pp (permissibility), the action described by p is performed in at least one morally acceptable world
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
Where φ denotes ∼Pp (impermissibility), the action described by p is not performed in all morally acceptable alternative worlds