Distinction: Law of Bivalence vs Law of Excluded Middle

Alex O'Connor uploaded an interview with Joe (Unsolicited Advice). Joe studied logic in college, and he mentions that he got in trouble for confusing the Law of Bivalence with the Law of Excluded Middle. I totally thought these were the same thing, but apparently they are not. And yet, trying to spell out these laws is surprisingly annoying. Consider:

Law of Bivalence(1): For any proposition P, P is either true or false.

Expressed in logic: ∀P(P∨¬P)

Law of Excluded Middle(1): For any proposition P, P is either true or false.

Expressed in logic: ∀P(P∨¬P)

These are identical. Sometimes both bivalence and excluded middle are described as saying "P is either true or false", causing the confusion. If we define both laws as nothing more than the above, then they are the same law. But LEM is usually taken to be saying something slightly different. Consider:

Law of Excluded Middle(2): For any proposition P, either P is true or its negation is true.

So now we have explicitly added in the idea of negation. But how would we logically express this version of the law? This version is saying that disjunction is always true between a proposition and its negation. So for any situation involving P or ~P, that system is logically true. In other words: ∀P(P∨¬P).

...But that's just the same thing again.

Another source of confusion, for me anyway, is referring to these as laws. But how does a law differ from an axiom, a theorem, or a principle? Are the above laws in the sense that they are basic axioms, or are they conclusions derived from axioms?

Also, if bivalence is a 'semantic principle' and LEM is a 'logical truth', which is another thing I've heard, then how are these different? Maybe this means that when we are talking specifically about "P or ~P" we are talking about LEM, and when we are talking about bivalence we are not using logic sentences at all and are only, in that context, using English. But then that makes LEM nothing more than the logic version of bivalence, which doesn't sound right.

Putting these questions to the side, if we define LEM in conditional terms we get:

Law of Excluded Middle(3): If a proposition is true then its negation is false, and if a proposition is false then its negation is true.

If we take 'false' and 'not true' to be logically equal, then "If a proposition is true then its negation is false" means: P ⟶ ¬¬P (Double Negation Introduction). This is equivalent to: ¬¬P ⟶ P (Double Negation Elimination). Or: P ⟷ ¬¬P.

And "If a proposition is false then its negation is true" means: ¬P ⟶ ¬P. This shows LEM being a tautology when we equate 'P is false' with 'Not-P is true'.

Note: Consider the standard belief terms for God:

(1) God exists.

(2) God does not exist.

Theism = the view that affirms (1) and rejects (2).

Atheism = the view that affirms (2) and rejects (1).

Agnosticism = the view that neither affirms nor rejects (1) or (2).

According to LEM, "God exists or God does not exist" is true. So it might seem like agnosticism is the denial of LEM. But an agnostic can affirm that it will, in the end, either be the case that God exists or does not, but it's just that the agnostic is not able to epistemically affirm one proposition or the other. So one can accept "P or ~P" while at the same time withholding belief in either P or ~P. Compare: I might know that either my lottery ticket is a winner or is not a winner, but I won't actively believe one way or another until I check it.

Another problem: 'Not true' applies both to false propositions and non-propositions. So 'not true' and 'false' are not always equivalent. However, in many contexts not being true entails being false. If it's not true that God exists, then it's true that God does not exist. If it's not true that I want cereal for breakfast, then it's true I want not-cereal for breakfast (either I want nothing for breakfast or something other than cereal, assuming that wanting nothing and not wanting anything are equivalent).

This is related to the issue of "neg raising" and the scope of negativity. Basically, we need to be careful with where we place our 'nots' so that we know exactly what it is we are claiming.

A neat way to do things:

Law of Bivalence(2): There are exactly two truth values, 'true' and 'false.' 

Law of Excluded Middle(4): All propositions take on exactly one truth value.

This makes things much more clear. Bivalence(2) is about the number of truth values (there are exactly two) while LEM(4) is about propositions and the truth values they can take on (they can take on exactly one).

Combining these yields the Law of Non-Contradiction.

Notably, LEM here implies that if something does not have a truth value, then it doesn't count as a proposition.

What does denying bivalence look like?

There are three-valued logics, such as the one proposed by Łukasiewicz, which uses true (1), false (0), and possible (1/2).

Łukasiewicz also proposed what would come to be known as fuzzy logic, which says that truth is not a binary, but a spectrum, and so there are an infinite number of truth values ranging from 0 to 1.

See: https://plato.stanford.edu/entries/lukasiewicz.

A third truth value could be proposed as something like 'undefined' or 'indeterminate' or 'unknown.'

In theory you could deny bivalence by saying there is only one truth value, but I don't know of any reason to do this.

What does denying LEM(4) look like?

You can deny it by saying there are propositions with no truth values, or deny it by saying there are propositions with more than one truth value.

This is where 'gaps' and 'gluts' come into play. A gappy proposition is one that is neither true nor false, and a glutty proposition is one that is both true and false. Gluts are also known as dialetheias, also known as true contradictions. So LEM(4) amounts to saying "There are no gaps or gluts."

See: https://plato.stanford.edu/entries/dialetheism/. 

Note: In that SEP entry, Graham Priest says: "It is a subtle issue . . . whether a gap should count as lacking any truth value, or as having a non-classical value distinct from both truth and falsity, and similarly whether a glut has two truth values, or some third ‘glutty’ value."

So it's not necessarily clear which law is being violated when we posit gaps and gluts. It depends on how we define things.

Consider the Sorites paradox. "This is a heap of sand." You could say that this proposition is a genuine proposition, but is vague, and we cannot determine whether it's true or false. And so you deny bivalence and add a third truth value, 'undetermined,' and apply it to vague propositions. 

Or you could affirm bivalence and say that "This heap of sand" is neither true nor false, denying LEM(4).

So on bivalence(2) and LEM(4), it's clear that the two laws are not logically the same and it's clear that one doesn't imply the other. This way, gaps and gluts are clearly compatible with bivalence and incompatible with excluded middle, and multi-valued logics are clearly compatible with excluded middle and incompatible with bivalence.

This is not standard though. What's more standard is to take the three laws of classical logic...

(1) Law of Identity = ∀x(x=x) (Anything is numerically identical to itself.)

(2) Law of Excluded Middle = ∀P(P∨¬P) (For all propositions, either they are true or false.)

(3) Law of Non-Contradiction = ∀P(~(P∧~P)) (For all propositions, it's not the case that they are true and false.)

...and say that gaps deny (2) but do not deny (3), while gluts deny both (2) and (3). Confusingly, Law of Bivalence is called a law and yet is not counted among the "three laws of classical logic."