Dear Salvador Escobedo,

as this imop is a very good - though not only your - question,

for sure, it has been answered at several occasions and in several ways.

Closely related is the respective question for (explicit) definitions, and I answered this related question recently here at RG, so you might have a glance at my two comments (especially the second comment) at

https://www.researchgate.net/post/Is_the_distinction_between_real_and_nominal_definitions_still_tenable2

Best

fwg

Theorems are tautological statements, in the sense that proving them amounts to proving their equivalence with the set of axioms that define the framework.

This doesn't mean, however, that new knowledge isn't generated by proving theorems, since the ``space of theorems'' isn't known and proving theorems amounts to exploring this space: It's not possible to say beforehand what are all the theorems that follow from a set of axioms. So the new knowledge that is generated is the knowledge about the space of theorems and how theorems are related to each other.

That's, also, why proving a theorem is necessary, but not sufficient: What is required, upon proving a theorem is stating what are its relations to other theorems and what are the conjectures that it implies can become theorems, in turn.

So the knowledge of the axioms amounts to an ``implicit'' knowledge of the space of theorems; proving theorems amounts to ``explicit'' knowledge of the space of theorems.

``New'' in this context means knowledge about the space of theorems.

There's no problem of logic omniscience, due to Gödel's theorem that, precisely, states that the space of theorems can't be closed: There exist theorems, whose proof can't follow from any given framework of axioms.

Dear Friedrich Wilhelm Grafe

Thanks for your links. I would like to know your opinion related with deduction, since the problem is a bit different when considering definitions rather than theorems.

Dear Stam Nicolis

I agree totally with you. The problem I see is that implicit knowledge does not match any real kind of knowledge. About logic omniscience I don't really see how Göedel's theorem saves the day, because all the theorems in the space of theorems will be known implicitly. Probably you mean: "There exist propositions, whose proof can't follow from any given framework of axioms". because a theorem by definition follows from the set of axioms. In that case I do not see how that would solve the problem of logic omniscience (given that they claim that explicit knowledge is not new knowledge).

Dear Salvador Daniel Escobedo

The statement that ``explicit'' knowledge isn't ``new'' knowledge is, just, wrong. The proof of any theorem, always, involves something new, by construction.

The statement that ``implicit'' knowledge isn't ``real'' knowledge doesn't make sense, unless what ``real knowledge'' is, is defined.

If we define as ``theorem'' a ``proposition'', whose proof can be shown to follow from the axioms, it's true that Gödel's theorem isn't as ``relevant'' as might be thought-that's why appealing to it with respect to certain specific propositions (e.g.Fermat's last theorem, before it was proven or the Riemann Hypothesis) doesn't make sense, since proving that these propositions are, indeed, undecidable is as hard as proving them true (or false). And it must be proved and can be proved-this is the content of Gödel's theorem.

But, once more, ``logic omniscience'' isn't true, simply, because the space of ``all theorems'' isn't known. That there exist propositions whose proof can't be established, is, indeed, not relevant.

Dear Stam Nicolis

Totally agree with you. In my point of view the proposition that deduction generates no new knowledge, is very problematic.

P. D. by "real knowledge" I was referring in a colloquial manner to "knowledge as defined in an acceptable epistemological framework".

... since the problem is a bit different when considering definitions rather than theorems ...

Dear Salvador Daniel Escobedo,

first,

explicit definitions, in the discussion I referred to [ of terms, definable in a deductive theory, or as a conservative extension of a deductive theory] , are simply a special case of theorems of a theory, and, thus far not thus different.

But - secondly -

let's take another point of view, and have a somewhat closer look at a standard example:

The induction axiom is fully expressible in 2nd order logic, so Peano arithmetic in 2nd order logic is known to be a categorical theory, i.e., it determines its (therefore unique) model up to isomorphism.

Now consider e.g. the twin prime conjecture, or the Goldbach conjecture:

nobody will deny, that a proof [deduction] or disproof [e.g. by reductio ad absurdum] of one of these conjectures would be an eminent epistemic gain.

imop, so far, there can be no dispute. Conceptual problems, as indicated with this thread's question, arise only as far, as we take an imop superficial talk on 'tautologies' [inherited from a perhaps undue reception of Wittgenstein's Tractatus statements] as the measure of things. There are - say 'vulgar' - connotations of the word 'tautology' of dubious epistemic value, lighting a smoke grenade in this context.

Dear Friedrich Wilhelm Grafe

You are right about the comment on explicit definitions. I do apologize, I think I've had read the wrong comment in the link.

About your further comments about tautology, I completely agree with you. However I am not sure about the point that there is not dispute on this issue.

The issue is not a new one. W. W. Sawyer wrote in 1967:

>

I would even agree talking about information (in the computational sense), but not about knowledge, as epistemic category. Quine defended this point of view, however he denied that mathematical deduction was a pure deductive process.

However I've recently meet some people that claims that deduction cannot yield new knowledge, and that includes deduction of mathematical theorems (even when talking of relevant proven theorems like Poincaré Hypothesis).

I you consider that some philosophers even denied the existence of an external physical world, it is not a surprise.

Dear Salvador Daniel Escobedo ,

so we found some imop valuable agreement. let me provisionally conclude

first,

... However I am not sure about the point that there is not dispute on this issue. ...

right and of course, for people enjoying the privilege of living in a free country, opinions are opinions and everybody's got one. So I'd better have said, "for reasons just presented there should be no dispute so far", instead of that "there can be no ... "

Nevertheless, if - but for all that - one engages in such dispute (which I do not recommend), the proponents of the argument from 'tautology' imop were in duty, to specify in some detail as well their concept of tautology as a metalogical concept, covering to some extent 'state of the art procedere' in math logic, and as well the concepts of 'knowledge' and 'information', they refer to.

but, secondly,

and already suggested, my own way - else my advice - here were:

forget about battling against a seemingly mistaken view

( as also in philosophy theories are more likely to be abandoned by loosing followers than by refutation in debates ),

but start elaborating, what instead you think true wrt the question of possible epistemic gain by deduction.

[ But this is an tremendous enterprise, and imop can't really be detailed here - as (again imop) including, beyond others, careful reference to respective views in CPR-Kant and Foundations-Frege as well as reference to the environment of the very start of formal logic in Aristotle, and also including some related review of especially proof theoretical basics of contemporary math logic.]

addendum

just caught on the mastodon/fediverse math philosopher Frédéric Jaêck's recommendation of

Timothy Gowers "what makes mathematicians believe unproved mathematical statements ?"

Already the title indicates a vivid comment to our above discussion.

The download link is to Annals of mathematics and philosophy

https://mxphi.com/wp-content/uploads/2023/01/TG_V2-1.pdf

Dear Friedrich Wilhelm Grafe,

I think that the term 'tautology' does not have an ambiguous meaning in this context, since it is taken as the logical property of a proposition p, that is a tautology if and only if ¬p is a contradiction. So far so good.

The problem arises when they involve knowledge, because it is not a formal-logic concept.

You are complete right about pursuing disputes about rarely defended topics. It is not my intention. What I want is to explore and understand how extended is that opinion among researchers, and what are their arguments.

But yes, the charge of the proof is on them.

A more specific theory about epistemic gain by deduction in the context of epistemic logic would be a interesting line of research, for sure.

Best regards.

Dear Salvador Daniel Escobedo

thx for your reply. In a perhaps last comment of mine in this thread (some excuses for its lengthiness) I'd like to confine to the uses of 'tautology', 'tautologous' in this thread, and expand to some extent, why I hesitate to follow your suggestion wrt 'tautologous'

Anyway thank you for an imop interesting exchange of views.

best

fwg

commenting first on

(1)

\cite{... I think that the term 'tautology' does not has an ambiguous meaning in this context, since it is taken as the logical property of a proposition p, that is a tautology if and only if ¬p is a contradiction. So far so good. ...}

...

ok, at least for two-valued logic systems providing the negation operation,

that's fine, and won't bother us,

but, what does this mention help us to describe

(2)

\cite{ ... The tautologic nature of the deduction ...}

(from your thread starting text)

?

----------------

wrt (1)

let's consider a simple example of an tautology from the logic of relations:

we proof by deduction in 1st order logic:

{ Rxy -> (Ryz -> Rxz), ┐Rxx } imply Rxy -> ┐ Ryx

and this may be done e.g. by showing, that the negation of the conclusion contradicts the conjunction of the premises. [of course, in this case, we get the same result if taking instead the universal closure for each wff]

but one problem with (1) is already, that it is restricted to a finite number of premises, another difficulty, that in less trivial cases one may have to resort to meta-theorems, in order to do a relevant proof,

thus I mean to say:

wrt (2)

things get more complicated, when we have e.g. an infinite set of premises in 1st order logic, as by 1st order compactness, any 1st order deduction is reducible to a deduction from a finite set of premises.

Next let's have some fully fledged (finitely?) axiomatized math theory (example's e.g. in Shoenfield's famous 'Mathematical Logic'), how should we

describe 'the nature of deduction' applied to such theories ?

The general semantical concept of deduction, found at least in every serious textbook on classical 1st order logic, is more or less like:

A wff A follows logically (semantically) from a set of wff's Γ iff any model of Γ is also a model of A

and one standard way, a proof can take, is to start with assuming a model of Γ given and then proceed to show, e.g. by some construction steps (maybe including transfinite induction) and/or via some lemma, it's a model of A too. Another well known way is, to assume the negation of the conclusion -A, and derive from -A, Γ a contradiction, possibly only in the metatheory.

and besides that, what about e.g. classic independency proof cases:

the perhaps most famous 19th century case: the parallel's axiom in geometry ?

the perhaps most famous 20th century case: the continuum hypothesis in [ZF] set theory ?

and what about the Löwenheim/Skolem paradox wrt ZF ?

Shall we really consider to state,

that these independence proofs, or last mentioned a proof of the uncountability of the Real Numbers in a theory admitting a countable model

are by 'the tautologous nature of deduction' ?

Deduction does not generate "new" knowledge. In fact, it does not generate knowledge at all. Deductive thinking does not and cannot construct new truths. What it does is affirm specific truths implied in a universal or general truth. The more important question should be "How is deductive knowledge generated?" I believe this is the greatest evil in the detail.

A simple answer. IMHO, if by “new” we understand something not known before, then yes, deduction can provide new results. But if we suppose (as Imre Lakatos did in his Proofs and Refutations) that everything we deduce is already implicit in the premises, then of course there will be no new results.

Dear Friedrich Wilhelm Grafe

You indicate very interesting points about the problems with the tautologist point if view.

Regarding to (1) there is actually a problem using those concepts with non two-valued logic systems.

However the mathematical deductions are usually done via two-valued logic (unless your are an intuitionistic mathematician). Anyways, I think that the tautologist do not care if a specific mathematical inference can be performed in a non tautologic deductive way, but rather if such mathematical theorem can be stated in a formal tautologic form.

So, he can argue that a (tautologically) deduced theorem do not increase knowledge, without stating anything about non-tautological cases.

Regarding to (2), Indeed I've said "the tautologic nature of the deduction". However I must say that this was a concession to simplify the discussion. Given the assumption that deduction is tautologic, can we conclude that deductions do not represent new knowledge?

I personally do not think that formal tautology is what characterizes deduction, and you pointed out a lot of problems with this characterization.

I would define deduction as a logic process which has the function to progress from general to (equally or) more specific propositions. However I would have to add a lot of caveats to this definition.

What is, in your opinion, the attribute that characterizes logical deduction?

Dear Jiolito Benitez

You claim that what deduction does is to affirm specific truths implied in a universal or general truth. But implication do not means knowledge. That is in some cases given a set of axioms, the true/falsehood of many wff remain unknown. Think for example in the Riemann Hypothesis (let us suppose that it is decidable in ZFC). It seems clear to me that implication do not equals knowledge, and therefore deduction give rise to a truly progress in knowledge.

Dear Décio Krause

Thanks for your comment.

In my case I find that use of the term 'implicit' very problematic. In my opinion it is case of language abuse, because the term 'implicit' is normally used to refer to thinks that are known, but not said. For example, if I say: "buy a car", it is implicit that you will buy it with money. I don't have to say: "buy a car with money", because it is already known, and no need to be explicit. So the original meaning of 'implicit' is an implied idea that it is not need to be said.

However, used in the way of Lakatos, we would have to say that abc conjecture is implicit in the ZFC axioms, not because it is known at all, but because its true/falsehood is a possible conclusion of a deduction. Which in my opinion is nonsense, and introduce the questionable concept that un-deduced conclusions are already known someway before the deduction is actually performed.

Therefore I do not think that all logically implied conclusions fit into the concept of implicit.

Salvador Daniel Escobedo,

You wrote, "You claim that what deduction does is to affirm specific truths implied in a universal or general truth. But implication do not means knowledge." I did not ever claim that implication means knowledge. If you read back my statement, and let me quote it here, "Deduction does not generate "new" knowledge. In fact, it does not generate knowledge at all." Aristotle's syllogistic reasoning (deductive logic) does generate knowledge.

I have an opinion regarding this interesting topic. Of course, the answer depends on what we call "knowledge". I adopt here the view that to acquire knowledge means to be aware of something that we didn't have noticed before. For instance, when arriving in America, the Europeans knew new things about the people who are living here, so they had new knowledge of several things, despite these things `existed' before their arrival. In this sense, a new mathematical theorem is a new knowledge, despite, from a logical point of view, it is already implicit in the axioms (deductive logic is supposed). Imre Lakatos has argued in this sense in his Proofs and Refutations, saying that an axiomatic system does not involve new knowledge. But in my interpretation, since the theorem was not known before, it constitutes new knowledge.

Almost all new mathematical knowledges are new definitions-concepts and new deductions-theorems. For example, Gödel's incompleteness theorems tells debunked the hope of founding mathematics on an consistent system of axioms. This was totally unexpected and put an end to the Hilbert program in mathematical foundation. Most of the knowledge in mathematics is totally irrelevant to the logical axioms onto which mathematics is currently expressed. A totally new logical foundations could one day substitute the current one and it would not be relevant to the mass of mathematical knowledge. The core of mathematical knowledge originates from abstactions made on different core human trade activities: logical debate, accounting, land measurement, art of building, etc. Mathematics is the systematic organisation of the world of relations originally present in these trade activities and which were already present in our sensory-motor sysyems allowing our bodily interaction with the world. It is why abstracting these relation in the first place and organising them are natural to us.

All knowledge should be confirmed by observation in a sense. I doubt that pure formats can create new knowledge but definitely better assumptions.