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Proposition Calculus, Proposition variables, strings, formulas, unique readability theorem
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... is a very coarse language with limited ex-
pressive power
... allows you to break a complicated sentence
down into its subclauses, but not any further
... will be refined in PART II Predicate Calcu-
lus, the true language of 1st order logic
... is nevertheless well suited for entering for-
mal logic
e.g. x, y for real numbers or z, w for complex numbers or α, β for angles etc.
for sentences (propositions)
only their logical properties count, i.e. whether they are true or false (when we use variables for real numbers, we also don’t care about particular num- bers)
(i) → p 17 () (ii) ((p 0 ∧ p 1 ) → ¬p 2 ) (iii) ))¬)p 32
The notion of a formula of L is defined (re- cursively) by the following rules:
I. every propositional variable is a formula
II. if the string A is a formula then so is ¬A
III. if the strings A and B are both formulas then so are the strings
(A → B) read A implies B (A ∧ B) read A and B (A ∨ B) read A or B (A ↔ B) read A if and only if B
IV. Nothing else is a formula, i.e. a string φ is a formula if and only if φ can be obtained from propositional variables by finitely many applications of the formation rules II. and III.
Lemma If φ is a formula then
Proof: Induction on n := the length of φ:
any formula obtained via formation rules (II. and III.) has length > 1.
Suppose the lemma holds for all formulas of length ≤ n. Let φ have length n + 1
⇒ φ is not a propositional variable (n + 1 ≥ 2)
⇒ either φ is ¬ψ for some formula ψ - so φ begins with ¬
or φ is (ψ 1 ⋆ ψ 2 ) for some ⋆ ∈ {→, ∧, ∨, ↔} and some formulas ψ 1 , ψ 2 - so φ begins with (. 2
The unique readability theorem
A formula can be constructed in only one way:
For each formula φ exactly one of the follow-
ing holds
(b) φ is ¬ψ for some unique formula ψ;
(c) φ is (ψ⋆χ) for some unique pair of formulas
ψ, χ and a unique binary connective
⋆ ∈ {→, ∧, ∨, ↔}.
Proof: Problem sheet ♯1.
