Question

Intuitionistic logic is a form of logic that doesn't have the law of the excluded middle, or LEM. The LEM says basically that a proposition that is not true is false, and a proposition that is not false is true. Classical logic has LEM but intuitionistic logic does not.

A 3-value logic is a logic where a proposition can be not merely True or False, but also something else, typically Unknown. It can be used to handle various propositions that 2-value logic cannot, propositions formed such that there is no consistent way to assign either true or false to them. Naturally, LEM also does not apply to a 3-value logic. I tend to view LEM as a device for defining what counts as a proposition. In a logic with LEM, for example, you can't allow a proposition like the Liar, "This sentence is false", so it just can't be formulated as a proposition, but in 3-valued logic it would be possible to allow it.

It occurs to me that Intuitionistic logic is a sort of stub 3-value logic, one where there is no way to deal with the third value, but where the possibility of a 3rd value is still allowed for. That is, without LEM, there is no way to prove that a proposition can only have one of two values.

Are there any developments of this idea in the literature? For example, is there any work where intuitionistic logic is used as a base for either classical logic or 3-value logic just by adding axioms or other features?

Answer 1

Intuitionistic logic is a form of logic that does not have the law of the excluded middle. This means that there may be statements in intuitionistic logic for which it is not possible to determine whether they are true or false.

Step-by-step explanation:

The law of the excluded middle is a logical law that states that for any statement, either that statement or its negation is true. It is related to the law of noncontradiction, which states that a statement and its negation cannot both be true at the same time. Intuitionistic logic is a form of logic that does not have the law of the excluded middle. This means that in intuitionistic logic, there may be statements for which it is not possible to determine whether they are true or false. Intuitionistic logic can also be seen as a sort of stub 3-value logic, where there is no way to deal with the third value, but where the possibility of a third value is still allowed for.