# Does rejecting the law of the excluded middle mean rejecting it for all propositions or only for those one cannot derive?

In logic, the law of excluded middle (or the principle of excluded middle) states that for any proposition, either that proposition is true or its negation is true.

A negation of that statement would suggest to me only that there exists a proposition such that it is not the case that it is true or its negation is true.

This is how Frederic Fitch seems to be rejecting the law of the excluded middle. For Fitch the law still applies to any "definite" proposition that can be derived, but it does not apply to the "indefinite" propositions, such as, "This sentence is false." (page 8)

2.12. Furthermore, we shall assume that there are some propositions which are not to be asserted as true or false.

Is that how the rejection of the law of excluded middle is treated in other logics that deny the law of the excluded middle or do they go further rejecting it as a rule for even definite propositions they can derive?

The reason I ask is that I see this further in the same Wikipedia article:

Many modern logic systems replace the law of excluded middle with the concept of negation as failure. Instead of a proposition's being either true or false, a proposition is either true or not able to be proved true. These two dichotomies only differ in logical systems that are not complete. The principle of negation as failure is used as a foundation for autoepistemic logic, and is widely used in logic programming. In these systems, the programmer is free to assert the law of excluded middle as a true fact, but it is not built-in a priori into these systems.

I wonder if this differs or not from what Fitch is doing.

Fitch, F. B. (1952). Symbolic logic.

Wikipedia, "Law of excluded middle" https://en.wikipedia.org/wiki/Law_of_excluded_middle

• Classical propositional logic is a very very simplified model of natural language. Of course, the language of our everyday experience is full of statements to which LEM is not easily applicable. See e.g. Vagueness. – Mauro ALLEGRANZA Oct 20 '18 at 16:34
• Obviously, we have Intuitionistic Logic : The rejection of LEM is grounded on a semantic where the concept of TRUE is replaced by that of PROVED. – Mauro ALLEGRANZA Oct 20 '18 at 16:38
• @MauroALLEGRANZA Based on your link and Fitch's 10.17 (p.58) intuitionistic logic and his system both lack LEM, but they treat negation introduction differently, Fitch has a "restricted" version requiring "p v ~p" to be an hypothesis or derived first while Heyting's development does not. I haven't seen Heyting's logic yet. I agree that there are many natural language sentences that do not fit LEM. I guess I am really wondering if these other logics divide propositions into something like "definite" and "indefinite" as Fitch does. – Frank Hubeny Oct 20 '18 at 17:11

Rejecting the law of excluded middle is already a somewhat strong committment for constructivists - I think the generally agreed upon is only to not accept it. [If you are confused because you do not see the distinction, you are using the law of excluded middle!] This means that while one does not assume that the law of excluded middle holds, one also does not assume that there is a counterexample.

I am not aware of approach where the law of excluded middle would be rejected for all propositions. To give a trivial example, (TRUE or FALSE) would be considered TRUE in any system that I know of. One reason for rejection the law of excluded middle is the desire to interpret "or" to include the information of which case holds - and while for arbitrary propositions we cannot obtain that, for derivable ones it does work.

• Fitch rejects the law of the excluded middle, however, that mainly allows him to include indefinite propositions such as "this sentence is false". If he is able to derive "p" then by disjunction introduction he can also derive "p v ~p" and from there work with the law of excluded middle for "p". Would you have reference to a text that shows inference rules without LEM? – Frank Hubeny Oct 20 '18 at 17:28

It depends on the the formalism you are working within. For instance, if you are working within a Boolean framework, then any proposition is either true or false. That is, (P->F)->(~P->T). There is no possibility for any other value for a proposition.

For instance, if you are working with fuzzy logic, then you will not deal with binary truth functors. Thus, in this formalism a proposition need not be either true or false. It could have any real valued r between 0 and 1 as your outcome. Therefore, one can see from this that there need not be exclusion of the middle.

As for Fitch, it seems to me he is not doing what is implied by the Wikipedia article. That is, the wiki is saying like I said in the case of fuzzy logic, not restricting truth functors to either T or F will reduce the need to appeal for law of the excluded middle; however, Fitch seems to imply that the law of excluded middle is not necessarily useless, especially in the cases where r, does in reality, equal to either 0 or 1. Moreover, Fitch seems to be working within the former system of logic, that is, Boolean, except where epistemic logic is employed.

• Fuzzy logic looks like a possibility. Do you have a reference for this? Thanks! – Frank Hubeny Oct 20 '18 at 22:35
• @FrankHubeny Hi, a stanford article explaining fuzzy logic has been added. – Bertrand Wittgenstein's Ghost Oct 20 '18 at 22:49