How are logical statements defined?

Question

I know that if we have $A \implies B$ and $B \implies A$ we can write $A \iff B$, where $B$ and $A$ are logical statements of some sort. An example of this is $x + y =3 \iff x = 3 - y$. However, I would like to ask, what exactly defines $A$ and $B$? Can we make the argument that $x + y = 3$ and $x = 3 - y$ convey the same information and therefore $A = B$? Is it not then usless to write $A \iff A$? Why do we make the distinction with $A$ and $B$? Obviously $x + y =3$ and $x = 3 -y$ are equivalent (hence the $\iff$), but what makes them different enough to call one $A$ and one $B$. Thanks.

Answer 1

To understand what A and B are, we have to look at how they are defined in the field of logic. Specifically, we look at the syntax (formal language) of propositional logic, which is the simplest form of logic.

A propositional formula is defined as follows:

1. Any propositional atom (p, q, r, etc.) is a propositional formula. Atoms are like variables, that can only get a truth value (true or false).
2. $\top$ and $\bot$ are formulas. They represent truth or falsity
3. If A is a formula then so is $(\lnot A)$ where $\lnot$ represents "not" i.e. the unary operation of negation
4. If A, B are formulas then so are $(A \land B), (A \lor B), (A \rightarrow B), (A \leftrightarrow B)$ where these symbols between A and B are boolean connectives (boolean operations) that represent and, or, implies and if and only if respectively.
5. Nothing is a propositional formula unless it's built using these rules

So A and B are actually quite strictly defined. They are propositional formulas which can be constructed only through the above definition.

The elements that make up a formula can of course be extended, to create a language with a different syntax. The immediate extension of propositional logic, called predicate (or first-order) logic adds to the above: structured atomic formulas, quantifiers, variables, equality, and function symbols. Defining first-order logic formulas is a bit more involved so I will refrain from doing it here.

Note the following crucial element of any logical system: Above, I gave the meaning of the symbols $\land, \lor, \lnot, \rightarrow, \leftrightarrow$ in English as and, or etc. That was in fact not necessary. That is because above, so far we have only been dealing with the syntax of propositional logic, that is, how you can mechanically construct formulas using the formal language of the logical system.

The meaning assigned to the symbol comes from the semantics of the logical system. To assign meaning, we look at the evaluation of the propositional formulas in a given situation. You might have seen that done through truth tables. We say that A and B (in the language we just defined that is $A \land B$) is true if A and B are both true, otherwise, it is false.

As an aside, any logical system is usually made up of syntax, semantics as well as a proof theory which is a purely systematic way of identifying and obtaining valid statements in the language. That is, it's a way of arguing in the formal language.

To answer the question in full, we now know what A and B are on either side of a logical connective: formulas. For your specific example, $x + y = 3$ and $x = 3 - y$ contain variables as well as equality, so they are first-order formulas. These are indeed different formulas. However, they are what we call equivalent.

Defining equivalence in first-order logic is once again a bit more involved (uses things called L-structures). But I can give an example in propositional logic whose syntax we defined above. In propositional logic, we say two formulas are logically equivalent if they have the same truth value in any situation. Roughly speaking they mean the same thing. For example, $A \land B$ is equivalent to $B \land A$. Famous equivalences you may have encountered are De Morgan's laws.