Does bivalence hold in principle of explosion?

Question

Bivalence states that statements without free variables are either true or false, not both.

On Wikipedia, there is a demonstration of the principle of explosion:

1. We know that "Not all lemons are yellow", as it has been assumed to be true.

2. We know that "All lemons are yellow", as it has been assumed to be true.

3. Therefore, the two-part statement "All lemons are yellow OR unicorns exist” must also be true, since the first part is true.

4. However, since we know that "Not all lemons are yellow" (as this has been assumed), the first part is false, and hence the second part must be true, i.e., unicorns exist.

In particular, step 3 treats the statement "All lemons are yellow" as being true, and step 4 treats it as being false.

So I wonder if a statement could be both true and false within an argument involving contradiction (like above)?

(Please note that I am confining the considerations within the argument that involves contradiction. I understand that if we are in a consistent system, then each statement should have only one truth value with respect to some specific interpretation; and I understand that, in a consistent system with the interpretation, if some statement implies contradiction then this statement is exactly false and cannot be true. But I find that it seems to have an inevitable need for treating some statement as being both true and false when trying to conduct the implication of "false implies anything" like above. I would like to be clarified whether if such "temporary misuse" of bivalence is allowed when conducting the implication, e.g. when conducting proof by contradiction before we arrive at the contradiction, or if there is some other better explanation on how bivalence is still hold within the arguments like above)

Answer 1

I wonder if a statement could be both true and false within an argument involving contradiction

No; in classical logic, where Bivalence and Excluded Middle hold, a statement cannot be both true and false.

A contradiction is a statment of form $\varphi \land \lnot \varphi$, like e.g. "Every natural number is even and Every natural number os not even".

One of the two conjuncts will be true and the other one, being the negation of the first one, will be false.

Thus, the contradictory conjunction will obviously be false: a contradiction is unsatisfiable, i.e. always false.

Having said that, the Wiki's proof is not a proof that unicorns exist, but a proof of the validity of the Principle of Explosion:

form a contradiction every statement follows.

The principle is the law of classical logic, intuitionistic logic and similar logical systems, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it. This is known as deductive explosion.

As a rule of inference:

$\dfrac { \varphi \qquad \lnot \varphi } {\psi}$,

it is expressed "sintactically": a contradiction is a formula $\varphi \land \lnot \varphi$.

The grounding for the rule is "semantical": the rule is truth-preserving.

The proof exploits the definition of Logical consequence:

we can define logical consequence as preservation of truth over models: an argument is valid if in any model in which the premises are true (or in any interpretation of the premises according to which they are true), the conclusion is true too.

This definition reads: for every situation $s$, if the set of premises $\Gamma$ holds in $s$ (in symbols: $s \vDash \Gamma$), then also the conclusion $\sigma$ holds in $s$. In symbols: $\Gamma \vDash \sigma$.

But for a contradictory formula $\varphi \land \lnot \varphi$ there is no situation where it holds; thus the definition applies vacuously and we can conclude that every statement $\psi$ follows from it.

The conclusion is: inconsistent premises are useless because logical inference applied to them leads to "unreliable" results, i.e. we can "prove" false statement.

This is the reason why consistency is a key-feature of mathematical theories (see your previous post).