In logic, is 'P = Q,' where P and Q are propositions, ill-conceived?

Question

In formal/mathematical logic, what are the rules governing "identity" (equals sign) use? Why is it that a conditional (or in certain instances a biconditional) sign is used instead to show a relation between propositions.

I'm a high school student with a developing interest in logic, so any insight into the philosophy guiding the semantics of logical languages would be appreciated.

Thank you for your time.

Edit: Thank you for your responses. I have two fallacious arguments written propositionally: If P, then Q If P, then R Therefore: If Q, then R And If P, then Q If R, then Q Therefore: If P, then R However, if these particular propositions were interpreted as being connected not by a conditional sign but by an "=" (identity) sign, wouldn't we have examples of the transitive property (i.e. P=Q, P=R, so Q=R using the rule of identity elimination in Fitch calculus)? However, neither conditional/biconditional introduction/elimination would be able to prove this in Fitch. Is this why it's a bad idea to use "=" when translating to the more primitive language of propositional logic?

Answer 1

The equality sign could be used between propositions, but they would need to be equal, that is, they would need to be the same proposition.

One reason for using biconditional is that it can link propositions that are not equal, but which have the same truth value. For example, "English has 26 letters" and "Hydrogen has one electron" are both true, but they are not the same. So we would not write an equal sign between them, but we could write a biconditional between them.

In propositional logic, we are often much more concerned with whether propositions have the truth value than whether they are the same - to the point that the equality sign is often left out altogether.

Answer 2

If you mean you find it more sensible to use equality instead of $\leftrightarrow $ with the same purpose, it is mostly a matter of convenience; we already use the symbol of equality for other purposes.

If you mean you want the proposition "P = Q" to mean something like "P and Q are identical": in typical logical languages, formulas or propositions range over whatever the domain of the model is. For example, if we are considering the first order logic of graph theory (i.e. with one binary relation), a formula should have a truth value based on what graph it is interpreted in, and what vertices are assigned to its free variables.

If we allow a construction like "if P, Q are formulas, then P = Q is also a formula", then the interpretation of this sentence would have to depend not on the structure of a graph, but on the syntactic information of the formulas P and Q. This is simply not what we want the propositions to be "allowed" to express.

Answer 3

There is nothing ill-conceived about thinking of what is usually written as $P \Leftrightarrow Q$ or $P \leftrightarrow Q$ as an equation between the propositions denoted by $P$ and $Q$. However, it is conventional in mathematical logic not to use the equals sign for this relation.

This convention avoids various kinds of ambiguity and fits in well with the concepts of first-order logic (where we have a specific universe of discourse in mind, like natural number arithmetic, in which equality denotes equality of natural numbers, while bi-implication denotes equivalence of propositions about the natural numbers).

There are higher-order logics in which the world of propositions forms part of the universe of discourse and then bi-implication just becomes a special case of equality.

There is a separate question of notation for syntactic identity between linguistic constructs. I.e., the notation to use when we are thinking of $1 + 2$ or $A\land B$ as sequences of symbols not expressions that denote numbers or truth values. The symbol $=$ is often used for syntactic identity, but this usage can give rise to confusion: using $\equiv$ or $:=$ is a common way of avoiding this.

Answer 4

It looks like the first person to use the '=' sign was Robert Recorde in 1557. In translation Recorde says "And to avoid the tedious repetition of these words : is equal to : I will set a pair of parallel lines thus: =, because no two things can be more equal."

So one might ask, if you wrote an equals sign between logically different, yet truth-functionally equivalent, propositions would they qualify as two things which can't be more equal? Having studied the consequences of the forms of certain propositions, I've found that the consequences of those forms can end up as very different even when those forms qualify as having the same truth-value. So, I do not think it correct to write '=' between propositions.

A distinct answer: P and Q are not propositions, unless specified and oftentimes they are not specified. They serve as variables for propositions. Writing p = q or equivalently a = b or equivalently c = f would literally mean that the variables were equal.

There is no case where P and Q qualify as propositions. It just doesn't qualify as coherent or makes for wording that could use revision for clarity. P and Q may represent propositions, but that is all. Considering P and Q as propositions makes for a counterfactual situation.

Answer 5

First-order logic already uses = for identity on the domain under discussion (sets, integers or whatever). So even though we could conceivably use = for identity on truth values, it's clearer to have a different symbol $\leftrightarrow$.

Plus the notation $P \leftrightarrow Q$ has the advantage of suggesting $P \rightarrow Q$ and $Q \rightarrow P$.