# Understanding, knowledge, and analytic truth

I read through this SEP article and wondered about the possibility of "understanding logic" vs. epistemic logic. One difference I could see would be that uS wouldn't necessarily be factive, meaning I can understand a sentence S without S being true.

But then I could also imagine cases of some S such that uS → kS. These sound like cases of analytic knowledge/truth. Would this be a good attempt to define such knowledge/truth? ("If I understand that bachelors are unmarried, then I know that they are unmarried"? "I understand that triangles have three sides, and so I know that they have three sides"? By contrast, one cannot understand that the round square is round, assuming a consistency-theoretic gloss of understanding? C.f. understanding-how vs. understanding-that, modulo knowledge-how vs. knowledge-that...)

• Analytic truth has two variations: logical necessary & self contradictory. You mentioned one above about bachelors. It is impossible to be a bachelor & also be married. To say a square is also round would be self contradictory. No squares can be fully rounded as squares must have four sides while round shapes like circles do not have formal sides. So to say a proposition is ANALYTIC means the proposition is either logically necessary or self contradictory. This also implies the proposition is deductive instead of inductive. One other distinction needed is analytic vs synthetic propositions. Commented Jun 3, 2021 at 20:11
• I think the idea is in the right direction, analytic truth is classically described as "truth in virtue of meaning alone", but the material conditional uS → kS seems like a wrong vehicle. Both u and k involve subjective psychological aspects, competence and belief, whereas whether something is an analytic truth is supposed to be objective. Perhaps replacing u with a normative version, like uia (understandable to an idealized agent), and dropping k will work: uiaS → S. Commented Jun 4, 2021 at 21:35

What you are describing has to do with a proposition being 'a priori knowable' rather than 'analytic'. There are many sloppy presentations of these concepts that tend to run them together, but they are distinct concepts and both are distinct from 'necessarily true'.

There are at least four different accounts of analyticity. The first is due to Kant, who coined the term. His idea is that with some propositions, the predicate is already contained within the subject. So, for example, "all bachelors are unmarried" might be said to be analytic because the subject 'bachelor' already contains the property 'unmarried'. The problem with this reliance on the concept of 'containment' is that it is too narrow: it doesn't cover cases of sentences that are not in simple subject-predicate form. Frege proposed instead that a proposition can be considered analytic if it can be derived from a logical truth by substitution of definitions. So, we can start with "all unmarried men are unmarried", which is a logical truth, substitute 'bachelor' for 'unmarried man' and arrive at "all bachelors are unmarried". This is much better, though it depends on the use of a particular (ideally formal) logic to make the concept precise. The logical positivists preferred instead to think of analyticity in terms of meanings, or linguistic conventions. On such an understanding, a proposition is analytic if it is true in virtue of its meaning, or in virtue of the linguistic conventions that govern the words from which it is formed. These accounts are broader, though even less precise, than Frege's.

A priority is concerned with whether a proposition is knowable independently of any empirical evidence, beyond that of merely understanding what the proposition means. Necessity is concerned with whether a proposition is not merely true, but could not be otherwise. Thus, analyticity is a linguistic concept, a priority is an epistemological concept, and necessity is a metaphysical concept.

It is important to keep these separate, because substantive philosophical theories depend on the relationships between them. The logical positivists claimed that all a priori knowable propositions are analytic, that all necessary propositions are analytic, and that analyticity provides the explanation of why some propositions are regarded as a priori or necessary or both. This is a non-trivial claim about the relationships between the concepts.

Kripke's theory of metaphysical necessity cuts across this account by holding that there are propositions that are necessary a posteriori, and also propositions that are contingent a priori, and that neither has anything to do with analyticity. Quine rejects the account on the basis that analytic/synthetic is not properly a well-defined binary distinction. Other philosophers such as Paul Boghossian hold that there is a relationship between analyticity and a priority, but not between analyticity and necessity.

All of which is a long preamble to saying that your proposed account might serve as an explanation of what it is for a proposition to be a priori knowable. If a person understands S, then that is all that is needed to know S to be true. On its own, it doesn't really advance our grasp of a priority much, unless perhaps you can embed it in a more substantial theory of what 'understanding' amounts to. And of course it is subject to all the usual objections as to whether there really is such a thing as a priori knowledge at all.

• Can you elaborate why Frege's rejection of Kant's synthetic a priori classification of math (not physical space and time) using principle of substitution of definitions for a logical tautology are more precise than the positivists' meaningfulness criterion? I think Frege's substitution principle is overkill or not needed at all to refute Kant's synthetic a priori after Hilbert's axiomatization movement. 2+2 under group or PA's definition with their axioms is naturally meant to be 4 linguistically, so I don't see any need for another unnecessary intermediate substitution here. Commented Jun 4, 2021 at 23:51
• Appealing to 'meanings' and, in particular, language-independent identity of meanings, is suspect. Quine rejected the idea entirely. The advantage of Frege's account is that we require only logical truths and definitions, which are more tangible things to work with. Of course, we could still criticise it for placing too much weight on the concept of definition. Commented Jun 5, 2021 at 1:03
• I'm treating understanding as a primitive of the system. Sort of. I now think you can define a constellation of epistemic structures by varying certain conditions. So in classical epistemic logic, they posit that kS → S, factivity; iteration, kS → kkS; belief-association kS → bS; or closure, that if kA and A → B then kB. What if there is an epistemic state for each variation on those conditions? Say, understanding is not necessarily factive or belief-associated, though it is iterable and closed. Commented Jun 6, 2021 at 13:24
• And then wisdom might be defined as factive, open, non-iterable, and belief-associated, etc. To an extent, such namings are offered less as analyses of the concepts than as useful placeholders... Commented Jun 6, 2021 at 13:27

According to your same SEP reference:

A key issue in social epistemology is whether understanding, like ordinary propositional knowledge, can be transmitted via testimony. Thus it seems that I can transmit my knowledge to you that the next train is arriving at 4:15, just by telling you. Transmitting understanding, however, does not seem to work so easily, if it is possible at all. According to Myles Burnyeat, Understanding is not transmissible in the same sense as knowledge is. It is not the case that in normal contexts of communication the expression of understanding imparts understanding to one’s hearer as the expression of knowledge can and often does impact understanding.

So if you correctly understand S (may not be the only correct way), then you'll have uS → kS, otherwise such inference rule may not hold. And in above relations between u and K, S doesn't have to be analytic truth, it can be any type of truths. And it's wild and very subjective if you want to redefine knowledge from claimed understanding.