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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. – Logikal Jun 19 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. – Logikal Jun 19 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. – carina Jun 19 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. – Logikal Jun 19 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. – Yuri Alexandrovich Jul 27 at 1:43
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The following passage that I am quoting at length from Dag Prawitz ("Intuitionistic Logic: A Philosophical Challenge" in Logic and Philosophy edited by G. H. von Wright, Hague, Martinus Nijhoff Publishers, pp. 8-9) may be more illuminating:

Intuitionistic philosophers sometimes use true as synonymous with the truth as known, but this is clearly a strange and unfortunate use. We need a notion of truth where, without falling into absurdities, we may say, e.g., that there are many truths that are not known today. But do we need a notion of truth that allows truths which are even in principle impossible to know?

The non-realistic concept of truth when at all reasonable agrees with the platonistic or realistic concept of truth in the case of sentences that are in principle decidable. Furthermore, the two concepts agree (in contrast to the intuitionistic one mentioned above) in allowing the existence of truths which in fact will never be known. What the above non-realistic principle or truth rules out is the existence of truths that are not even in principle possible to know.

The difference between the two principles boils down to this: on the platonistic principle, a truth condition for a sentence obtains or does not obtain independently of our means of recognizing that it obtains or fails to obtain, and we are then forced to admit that there may be truths that are in principle impossible to recognize (if we are not to assert unwarrantably that all problems are in principle solvable); on the non-realistic principle above, a truth is in principle always possible to recognize, but we must then refrain from asserting that a truth condition either obtains or does not obtain (again, in order not to assert that everything is solvable). Both principles respect the fact that we are not omniscient, but the platonistic principle does this by introducing ideas the need of which are not easily seen.

Martin-Löf and Prawitz seem to claim that, just as when the rules of a game is fixed and definite enough to state that a right move in the game is to be found out by necessity and recognisable as such, a truth in mathematics, according to their conception, (and presumably, other truths under similar conditions) takes on actual existence once a proof is devised, furthermore it has been already there as a potential truth. Hence, it could be inferred that we could describe not the criteria to identify potential truths, but the conditions under which truths could be potentially "awaiting".

Martin-Löf finds a metaphysical ground for their version of verificationism in Aristotle's views and especially, Thomas' reinterpretation of Aristotle's. I do not think that, however, the distinction is as closely related to these views as Martin-Löf puts emphasis on. Sure, the words 'potential' and 'actual' have core meanings that manifest themselves in any context that their employment appears meaningful. Nevertheless, relying almost only on this connection makes a weak argument. The most significant point that Martin-Löf and Thomas Aquinas could be said to share is that the epistemological status of the actual (which is present contra absent in a certain sense) is higher than the potential, and prior to it with regard to dependency relation. An explanation of this point is quite another subject and must be dwelled upon elsewhere. For the present question, I can say that it would have been less misdirecting if Martin-Löf and Prawitz had used 'latent' and 'effective' in place of 'potential' and 'actual'.

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