Actual and potential truth for neo-verificationists

Question

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!

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PS: I guess it's not a good question for the maths panel but as a student of maths, I feel really confused when I came across the distinction in Martin-Löf and also his pupils' papers. I seek a precise definition of this distinction (in either model-theoretical or proof-theoretical terms) in order to have a better understanding of Martin-Löf's type theory and its descendants.

Answer 1

Long comment

See The proof-theoretic account of logical consequence:

On the proof-centered approach to logical consequence, the validity of an argument amounts to there being a proof of the conclusions from the premises. [...] The proof-centered approach highlights epistemic aspects of logical consequence. A proof does not merely attest to the validity of the argument: it provides the steps by which we can establish this validity. And so, if a reasoner has grounds for the premises of an argument, and they infer the conclusion via a series of applications of valid inference rules, they thereby obtain grounds for the conclusion (see D.Prawitz, “The Epistemic Significance of Valid Inference”, Synthese,2012).

The condition of necessity on logical consequence obtains a new interpretation in the proof-centered approach. The condition can be reformulated thus: in a valid argument, the truth of the conclusion follows from the truth of the premises by necessity of thought (See D.Prawitz, “Logical Consequence from a Constructivist Point of View”, 2005). Let us parse this formulation. Truth is understood constructively: sentences are true in virtue of potential evidence for them, and the facts described by true sentences are thus conceived as constructed in terms of potential evidence. (Note that one can completely forgo reference to truth, and instead speak of assertibility or acceptance of sentences.)

And see Heinrich Wansing (editor), Dag Prawitz on Proofs and Meaning (Springer, 2015), page 25:

For Prawitz, a sentence is by definition true just in case it is provable, and provability is, in his opinion, a tenseless and objective notion. According to Prawitz, identifying truth with the actual existence, in the sense of possession, of a proof is a “fatal flaw”. In his reply to (M.Dummett, "Truth from the constructive standpoint", Theoria, 1998), Prawitz points out that

a sentence is provable is here to mean simply that there is a proof of it. It is not required that we have actually constructed the proof or that we have a method for constructing it, only that there exists a proof in an abstract, tenseless sense of exists. [...] [T]ruth is something objective: it is in no way we who make a sentence true. To ask whether a mathematical sentence is true is to make an objective question, whose answer, if it has an answer, is independent of time. However, since the theory of meaning that I have in mind is constructive, it does not follow that every such question has an answer; we have indeed no reason to assume that.

See also Proof-Theoretic Semanticsand see Dag Prawitz, Truth and Proof in Intuitionism, Ch.3 of Sten Lindström & Göran Sundholm etc (editors) Epistemology versus Ontology : Essays on the Philosophy and Foundations of Mathematics in Honour of Per Martin-Löf (Springer, 2012).