# Actual and potential truth for neo-verificationists

Neo-verificationists such as Martin-Löf and Prawitz make a distinction between actual and potential truth of a proposition, roughly defined as follows:

... that a proposition A is actually true means that A has been proved, that is, that a proof of A has been constructed, which we can also express by saying that A is known to be true, whereas to say that A is potentially true is to say that A can be proved, that is, that a proof of A can be constructed, which is the same as to say, in usual terminology, simply that A is true. (Martin-Löf 1991:142)

This distinction seems closely related to similar distinctions made by Aristotle and Aquinas. But while it is certainly clear when one is entitled to judge/assert that A is actually true (that is, when one constructs a proof for A), it is not so clear what amounts to the conditions for one to be entitled to judge that A is potentially true, or alternatively, what amounts to the conditions for judging that A can be proved. Could anyone help me with this? Thanks!

PS: I am a student of mathematics but I don't have an intuitive idea about the distinction made by these neo-verificationists. I would appreciate if someone explains it in formal terms (e.g., Kripke semantics or proof theory, ...). But I also appreciate it a lot if someone explains the distinction in only ordinary language. Thanks again!

• You must be told straight forward that the terminology may look & sound the same between mathematics & philosophy but the context are NOT identical. In philosophy a proposition is stated in an objective truth context, whereas in Mathematical logic they tend to go with scientific evidence. By scientific evidence I mean sense verification. There are no PROOFS that are not sense verifiable. A proposition is true or false. If you are unaware of the proposition that doesn't remove the truth value. It is STILL either true or false regardless of your awareness. Jun 19, 2020 at 16:01
• The definition presented in your post expresses what a theorem is not a proposition. The potential truth stuff is not found in philosophy. Objective truths for example must remain true forever given the correct details. If I state the proposition " the NY Mets will win three straight world series beginning in 2023." This claim is true or false objectively eventhough it is in the future we become aware of the value. Mathematical logic might not allow that. So how Mathematical logic defines a proposition is not identical to philosophy. Many terms seem to be the same but are different in context. Jun 19, 2020 at 16:12
• @Logikal Hi! Thanks for your comment. I guess you endorse a realist perspective as you are in support of the axiomatization of the Law of Excluded Middle (that a proposition is either true or false). But that's not the case for constructivist or verificationist philosophers, as they would say that it's inappropriate to take LEM for granted. Jun 19, 2020 at 16:14
• The difference is the context which cannot be under stated. Those other so called philosophers are VERY LIKELY to be using a DIFFERENT CONTEXTUAL definition of the LEM. You have people to this day literally reading LEM which is wrong. Propositions express an idea. They are not literally declarative sentences. The LEM also is based on OBJECTIVE TRUTH. That is x can't be true today and false 2 days from now. If you have a discrepancy with the LEM you need to check if the context is used differently in the conversation. In most cases there is a shift in context. Jun 19, 2020 at 16:24
• @Logikal "you need to check if the context" -- exactly. In math and logic, we free to define TRUTH however we want. We cannot do it in the real world, because that is what objective truth IS by definition -- something being objectively true means it is real. Most people struggle with objective truth because the concept -- and the context -- of the objective reality is simply beyond their grasp. Jul 27, 2020 at 1:43