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I am a newbie in logic and I am reading the book Logic and Structure by Dirk van Dalen. In the first chapter the author gives the following definition:

Definition 1.1.2: The set PROP of propositions is the smallest set X with the properties

(i) pi ∈ X(i ∈ N), ⊥∈X,

(ii) ϕ,ψ ∈ X ⇒ (ϕ∧ψ), (ϕ∨ψ), (ϕ →ψ), (ϕ ↔ψ)∈X,

(iii) ϕ ∈ X ⇒ (¬ϕ) ∈ X.

I can make sense of (ii) and (iii). (i), however, is not so clear to me. I understand it as 'propositions pi belong to X and ⊥ (falsity) also belongs to X', meaning that it is possible for propositions in X to be false. I know that it doesn't necessarily excludes the possibility of a proposition being ¬⊥, or true, but it also doesn't make this possibility explicit. So my question is... shouldn't it? If it is not necessary to make explicit that veracity is possible in X why is it necessary to do so for falsity?

I apologise if this is too basic a question.

Anyway, thank you very much for your time.

NeonGabu
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  • True and False as truth value are not relevant here: we are defining the syntax of the language, i.e. the rule to produce correct (i.e. meaningful) expressions. We have to start from a basic set of symbols: the propositional variables: $p_1, p_2, \ldots$, the falsum sign: $\bot$ and the logical connectives: $\lnot, \land, \ldots$. – Mauro ALLEGRANZA May 25 '22 at 07:29
  • Then we define the set $\text{PROP}$ of propositional formulas: rule (i) says that the propositional variables $p_i$ as well as falsum $\bot$ are propositional formulas. Rule (ii) states how to use binary connectives: thus $p_1 \land p_2$ and $p_1 \to \bot$ are examples of formulas, according to rule (ii). Rule (iii) states how to use the unary connective (negation): thus, according to this rule, $\lnot p_1$ and $\lnot \bot$ and $\lnot (p_1 \to \bot)$ are examples of formulas, – Mauro ALLEGRANZA May 25 '22 at 07:32
  • See also this post as well as this one. – Mauro ALLEGRANZA May 25 '22 at 09:28

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The third rule says that $\phi \in X\to (\neg \phi \in X)$.Since $\bot\in X$ we have that $(\neg \bot)\in X$.

Vivaan Daga
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