If there is no atom A how can be the both A and ¬A satisfy the L?

Question

I was reading a definition in the logic lecture notes of Oxford University. In Definition 1.2

Definition 1.2.

A string of L is a formula of L if and only if

1) it is a propositional variable (i.e. it consists of a single symbol, which is a propositional variable),

2)it is of the form ¬A, where A is a formula,

3) it is of the form (A → B), (A ∧ B), (A ∨ B) or (A ↔ B), where A and B are formulas. The set of all formulas of L is denoted Form(L).

I did not understand the 2nd rule,

it is of the form ¬A, where A is a formula,

if there is no atom A how can be the both A and ¬A satisfy the L? Can someone explain to me.

Answer 1

This is a definition that tells us how we are allowed to build formulae. The set of formulae that we can build is called $L$.

We can build a formula using the following formation rules:

• By taking a propositional variable; this in itself is a formula
• By taking a formula $A$ that we have already built and prefixing it with a negation sign $\neg$; this gives us the formula $\neg A$
• By taking two formulae $A$ and $B$, that we have already built, and putting $\rightarrow$ (or $\vee$ or $\wedge$ or $\leftrightarrow$) between them; this gives us the formula $A \rightarrow B$ (or $A \vee B$ or $A \wedge B$ or $\leftrightarrow$)

There are no other ways that we can build formulae.

An example of the use of the formation rules is that $p \rightarrow \neg q$ is a formula, if $p$ and $q$ are propositional variables. We show this by using the formation rules as follows:

• $q$ is a propositional variable, so it is a formula.
• Since $q$ is a formula, then so is $\neg q$
• $p$ is a propositional variable, so it is a formula.
• Since $p$ and $\neg q$ are formulae, then so is $p \rightarrow \neg q$.

Answer 2

By 1), any atomic statement (like, $P$) is a formula.

Now that we know that $P$ is a formula, we can use 2), meaning that $\neg P$ is also a formula.

So, both $P$ and $\neg P$ end up being formulas.

Remember, we are just trying to define the acceptable (grammatica) expressions of the language. This is completely different from saying that both expressions are true.

Indeed, the same goes for English: "It rains" is a grammatical expression, and so is "it does not rain". They are not both true, but that is not the point. The definition you are looking at merely defines what expressions in logic are grammatical.