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
Syntax
A well-defined syntax allows us to specify, relative to a formal system L, which formulae are wffs (or well-formed formulae)
Formulae that are not admitted by the rules of syntax will be ill- rather than well-formed
Ill-formed formulae are excluded from the range of our considerations
RULE 1: If φ is a wff, then ∼φ is also a wff
EXAMPLE 1: ∼p is also a wff
EXAMPLE 2: p∼ is ill-formed
EXAMPLE 3: ∼∼∼∼∼∼∼p is also a wff, since RULE 1 can be applied iteratively
RULE 2: If φ and ψ are wffs, then φ ∧ ψ, φ ∨ ψ, φ → ψ, and φ ⟷ ψ are also wffs
EXAMPLE 1: (p → q) → (r → s) is also a wff, since RULE 2 can be applied iteratively
EXAMPLE 2: p ∨ qr is ill-formed
EXAMPLE 3: (p ∧ q) → ∼r is also a wff, since RULE 1 and RULE 2 can be applied in conjunction and iteratively
RULE 3: If R is an n-ary relation or n-place predicate and t_{1}, t_{2}, …, t_{n} are the individual terms, then R(t_{1}, t_{2}, …, t_{n}) is a wff
EXAMPLE 1: R(x, y) is a wff if R is a binary (2-ary or 2-place) relation and x and y are individual terms
EXAMPLE 2: R(x, y) is ill-formed if R is a unary (1-ary or 1-place) relation and x and y are individual terms
EXAMPLE 3: R(x_{1}, x_{2}, x_{3}, x_{4}) is a wff if R is a quaternary (4-ary or 4-place) relation and x_{1}, x_{2}, x_{3}, and x_{4} are individual terms
RULE 4: If t_{1} and t_{2} are both wffs, then t_{1} = t_{2} is also a wff
EXAMPLE 1: If neither t_{1} nor t_{2} is a wff, then t_{1} = t_{2} is ill-formed
EXAMPLE 2: If t_{1}, t_{2}, and t_{3} are wffs, then t_{1} = t_{2} = t_{3} is also a wff
RULE 5: If x is an individual variable and φ is a wff, then (∀x)φ and (∃x)φ are also wffs
EXAMPLE 1: x∀ is ill-formed
EXAMPLE 2: If P and Q are unary (1-ary or 1-place) relations, then ∀x(Px → Qx) is a wff
EXAMPLE 3: If L is a binary (2-ary or 2-place) relation, then ∀x∃y(L(x, y)) is a wff
RULE 6: If φ is a wff, then □φ and ◇φ are wffs
EXAMPLE 1: □p◇ is ill-formed
EXAMPLE 2: ∼□(p → q) is also a wff, since RULES 1, 2, and 6 can be applied in conjunction