No, a proof is sufficient but NOT necessary. A true proposition is true regardless of whether humans ever construct a proof for it. One more time: a proof pertains to justification, not truth — aletheist

What is justification if not proof?

Why do we need justification if not to establish truth?

:smile: :up:

What is justification if not proof? — TheMadFool

"Proof" has the connotation of rigorous demonstration. We believe all kinds of things for which we do not have "proof" in this strict sense, but we nevertheless are justified in believing them.

Why do we need justification if not to establish truth? — TheMadFool

There are reasons why the standard modern philosophical definition of knowledge is "justified true belief," rather than merely "true belief." Moreover, we are fallible knowers; some of our justified beliefs will turn out to be false.

What is justification if not proof? — TheMadFool

All proofs are justifications, but not all justifications are proofs.

Proof means that a statement necessarily follows from the construction logic of its universe. We do not know the construction logic of the real, physical world. So, we cannot prove anything about it. Therefore, real-world knowledge rests on weaker justification. For example, science rests on the lack of counterexamples ("falsificationism") as justification (on the condition that you really tried to look for them).

Why do we need justification if not to establish truth? — TheMadFool

No, justification is a goal in itself. It turns the belief into a justified belief, i.e. knowledge.

There is no knowledge that is correspondence-theory true. If there is proof, then it is not about the real, physical world. If it is about the real, physical world, then there could still be counterexamples waiting to be discovered.

Hence, knowledge of the truth does not exist.

This does not mean, however, that other, unknown mental faculties cannot be privy to the truth. Still, we are not able to justify knowledge about them. Hence, that question is fundamentally undecidable.

:ok: :up:

You mentioned something that I've been thinking about - that justification, by itself, doesn't guarantee truth. A justification gives us a reason to believe something but there are so many contingencies that prevent it from becoming 100% effective, reducing their evidential force.

Proof, on the other hand is complete, 100%, justification - it's impossible to deny the truth that a proof supports.

Do you agree with me? In other words I'm saying that justifications are not actually 100% sufficient to establish truth but a proof is 100% sufficient to do that.

All proofs are justifications, but not all justifications are proofs. — alcontali

Can you read my reply to altetheist above regarding the difference between justification and proof? Thanks

What is justification if not proof? — TheMadFool

If only proof were justification, then we would have no justification at all for empirical propositions.

Proof, on the other hand is complete, 100%, justification - it's impossible to deny the truth that a proof supports. — TheMadFool

Yes, but proof never provides a truth about the real, physical world.

A proof is 100% justification for a proposition about an abstract, Platonic world by demonstrating that it necessarily follows from its construction logic. To do that for the real, physical world, we would need to know its construction logic, which we don't.

Anything empirical will have to be justified using substantially less strict justification than formal proof, such as the absence of counterexamples.

Do you agree with me? In other words I'm saying that justifications are not actually 100% sufficient to establish truth but a proof is 100% sufficient to do that. — TheMadFool

Yes, agreed, if it were possible to do that, but it isn't. Since proof cannot be provided for statements about the real, physical world, there is no proof for correspondence-theory truth. Hence, proof does not lead to a real-world truth.

:up: :ok:

Wittgenstein briefly entertained a somewhat similar idea in the blue book, when he said "apparently it didn't occur to Socrates to enumerate everything we call 'knowledge' ", as part of an argument for empiricism.

However, to think of knowledge in terms of axioms is misleading, because

i) we can only write down a finite number of axioms, even though our knowledge production faculties can produce an indefinite number of axioms, without an a priori knowable upper-bound.

ii) Knowledge equally consists in the use of axioms and their creation; yet these processes cannot be specified as additional axioms, because we then enter an infinite regress (see Quine's 'Truth by Convention' and Lewis Carroll's Paradox).

iii) we do not always know what we know; furthermore our belief states are invariably inconsistent.

Therefore it is misleading to think of knowledge as an axiomatic system.

There can be no flat assertions. Put differently every statement must be justified or should be an argued position.

Therefore:

1. We go on forever, justifying each claim with another and that with yet another, etc.

or

2. We go for circularity where after a certain number of steps we make a loop where the conclusion supports the first premise

or

3. We settle on, hopefully, self-evident truths or axioms and begin our process of argumentation and truth seeking.

Option 1 is impossible. Option 2 is obviously going to prove anything we like - contradictions included.

We have to choose option 3. axioms. Hence the axiomatic system of most bodies of knowledge. Sometimes these axioms are a simple matter of consensus e.g. mathematics. Other times axioms are discovered through experimentation like in the sciences.

Deductive logic is what every person in the world wants to do and in it we never conclude more than what's in the axioms. Axioms are like boundaries and all truths that can be derived from them are already present within these boundaries. We can never know more than what's in the axioms. So, in effect, knowing the axioms amounts to knowing everything that follows from them.

I know some people who felt relieved that the key they lost was in the house. Knowing the lost key is in the house (axioms) counts as knowing the location of the key (the truths that follow from these axioms).

I agree with , in the sense that a deductive proof can only ever provide certainty about a hypothetical state of affairs. Whether any given hypothesis matches up with reality can only be evaluated inductively, which means that we can never achieve certainty. Instead, the validity of the method stems from its self-correcting nature, such that the falsity of a hypothesis would become apparent in the long run. Most of our current beliefs are true, which is why we are generally able to get around successfully in the world, but we cannot know for sure that any one in particular is true.

No. 1 expresses the limitless discursive activity of rational analysis, which can sometimes be represented compactly via an infinite loop like in No. 2, except where iterative deduction leads to intermediate conclusions that are not-identical to their premises.

Yet the ultimate inability of rational analysis to defend any given proposition doesn't mean the proposition is false, with Zeno's paradox being the paradigmatic example. For we cannot give a logical proof of motion, and yet we still 'know' of motion because we are nevertheless able to literally construct it; this is a vivid demonstration of why knowledge cannot be represented solely in terms of axioms, and why any account of knowledge must distinguish the activity of practical synthesis from the activity of discursive analysis.

Nevertheless, one way to study such paradoxes is by way of epistemic logic, in which some axioms directly represent beliefs or knowledge, while other axioms represent higher-order beliefs or knowledge such as 'one's knowledge of one's own knowledge' , 'one's beliefs about one's own knowledge', 'one's beliefs about one's own beliefs' etc. etc.


By the way, circular reasoning where the conclusion is considered to be identical to the premise, and hence where no deduction has actually taken place, is characteristic of normative speech acts like "Tidy your room!" whose justification in response to a child's scepticism might consist of the reply "Because I said so!". In my opinion, Metaphysics from the perspective of cognitive psychology, is the study of a particular class of self-reinforcing speech-acts that influence language, motivation and perception.

Most of our current beliefs are true, which is why we are generally able to get around successfully in the world, but we cannot know for sure that any one in particular is true. — aletheist

In my opinion, all of our current beliefs about the real, physical world are resilient Platonic-cave shadows. They are really good at resisting falsification by experimental testing. However, they have no definite connection with the unknown Theory of Everything (ToE), which is the holy grail, i.e. the unknown true belief. If we knew that connection, then we would also know the ToE. A resilient Platonic-cave shadow is eminently suitable as justification but it is still not (correspondence-theory) true. It is merely an excellent heuristic.

Induction is a weaker system than deduction and what is justification in the former is proof in the latter. However note that the utility of inductiom lies in deduction being applicable to knowledge so gained.

Yet the ultimate inability of rational analysis to defend any given proposition doesn't mean the proposition is false, with Zeno's paradox being the paradigmatic example — sime

Good point. However, notice that the issue is:

1. Zeno's argument. Ergo motion isn't possible
2. We can see it move. Ergo motion is possible

In both cases evidence/proof was necessary. The problem is the conclusions contradict.

Induction is a weaker system than deduction and what is justification in the former is proof in the latter. — TheMadFool

There is no "proof" in induction, only evidence.

However note that the utility of inductiom lies in deduction being applicable to knowledge so gained. — TheMadFool

Induction is really the last step in inquiry, although it is ultimately cyclical. First is retroduction, the formulation of a plausible hypothesis. Next is deduction, the explication of what follows necessarily from that hypothesis in order to make predictions. Then comes induction, the testing of the hypothesis to see whether the resulting predictions are corroborated or falsified.

First is retroduction, the formulation of a plausible hypothesis. — aletheist

Ha, but there is no mechanical procedure for discovering a testworthy pattern. That is where other, unknown mental faculties kick in. What procedure would Einstein have followed to discover his 1905 paper on special relativity? I don't think that even Einstein knew. Rationality seems to be the output product of something that cannot be described in language, which is a prerequisite for justification, which is itself the core characteristic of rationality. Hence, rationality is an output product of some process that is not rational at all.