TimeSeeker wrote: ↑Sat Nov 03, 2018 1:50 pm
Averroes wrote: ↑Sat Nov 03, 2018 5:41 am
Either way, you have not demonstrated the possibility of a thought that cannot be expressed in language!

You have set yourself up for disappointment. You have contrived an

**impossible challenge**.

There are some inaccuracies in the above quoted statement you made. Firstly, the challenge was not from me but it is the title of this thread!! Secondly, it is you yourself who made a statement such that there are thoughts which cannot be expressed through language thus:

TimeSeeker wrote: ↑Tue Oct 30, 2018 8:49 pm
And yet is precisely

**the thoughts which I can't express THROUGH any medium** are the ones which you require evidence for...

How might one convince you that such thoughts exist?

As you who made a statement implying that there can be thoughts which cannot be expressed through language, so I had asked you to give me evidence for such thoughts! And it seems you have finally recognized that it is “impossible.” I agree with you that it is indeed impossible!

_______________________

TimeSeeker wrote: ↑Sat Nov 03, 2018 1:50 pm
You are asking me to prove a negative

Indeed, I have asked you to give evidence for your statement claiming the existence of the alleged thoughts that cannot be expressed in language!

TimeSeeker wrote: ↑Sat Nov 03, 2018 1:50 pm
The only way I know how to prove a negative is to abandon Aristotelian logic and embrace constructive/intuitionistic logic.

So, now you are claiming that you can prove the existence of thoughts that cannot be expressed in language in intuitionistic logic!

Alright! You can use intuitionistic logic if you want to prove your statement that there are thoughts which cannot be expressed in language. It’s much better for me actually in intuitionistic logic!

From Wikipedia:

Wikipedia wrote:**To an intuitionist, the claim that an object with certain properties exists is a claim that an object with those properties can be constructed.** Any mathematical object is considered to be a product of a construction of a mind, and therefore,** the existence of an object is equivalent to the possibility of its construction**. This contrasts with the classical approach, which states that the existence of an entity can be proved by refuting its non-existence. For the intuitionist, this is not valid; the refutation of the non-existence does not mean that it is possible to find a construction for the putative object, as is required in order to assert its existence. Site: https://en.wikipedia.org/wiki/Intuitionism

The interesting thing in all this is that I have already imposed a necessary condition for the existence of a thought that it be expressible in language,

**exactly** complying with the demands of intuitionistic logic. I am already done in intuitionistic logic!

Now your turn! You now have to show, in intuitionistic logic, the existence of a thought that cannot be expressed in language that it can be constructed, i.e., you have to show the possibility of constructing something that cannot be constructed!!!

TimeSeeker wrote: ↑Sat Nov 03, 2018 1:50 pm
Which necessarily means abandoning the laws of excluded middle AND the laws of non-contradiction!

Absolutely not! In intuitionistic logic, only the law of excluded middle and double negation are dropped, but not the law of non-contradiction. There is no logical framework without the law of non-contradiction!

__________________________

The law of non-contradiction and proof-by-contradiction are very different things! In intuitionistic logic, proof-by-contradiction is a

**proof technique** that is rejected because the latter makes use of the law of excluded middle and double negation (which are not accepted in that framework) but not because of the law of non-contradiction is rejected (as it is not rejected)!

Wikipedia wrote:**Proof by contradiction also depends on the law of the excluded middle**, also first formulated by Aristotle. This states that either an assertion or its negation must be true

(For all propositions P, either P or not-P is true)

(...)

The law of the excluded middle is accepted in virtually all formal logics; however, **some intuitionist mathematicians do not accept it, and thus reject proof by contradiction as a proof technique.** https://en.wikipedia.org/wiki/Proof_by_contradiction

As I already said, there is no logic framework without the law of non-contradiction. In Intuitionistic logic you can still use the principle of explosion (EFQ), i.e., in other words: from a contradiction everything follows, even though that results in trivialities.

Wikipedia wrote:The principle of explosion (Latin: ex falso (sequitur) quodlibet (EFQ), "from falsehood, anything (follows)", or ex contradictione (sequitur) quodlibet (ECQ), "from contradiction, anything (follows)"), or the principle of Pseudo-Scotus,** is the law of classical logic, intuitionistic logic and similar logical systems**, according to which any statement can be proven from a contradiction. That is, once a contradiction has been asserted, any proposition (including their negations) can be inferred from it. This is known as deductive explosion.

Due to the principle of explosion, the existence of a contradiction (inconsistency) in a formal axiomatic system is disastrous; since any statement can be proved true** it trivializes the concepts of truth and falsity**. https://en.wikipedia.org/wiki/Principle_of_explosion

____________________

TimeSeeker wrote: ↑Sat Nov 03, 2018 1:50 pm
So, lets embrace constructive logic and ASSUME that all thought can be expressed in language and see what absurdities/paradoxes this leads to.

Alright, go on and assume whatever you want, and prove in

**intuitionistic logic** the statement: there are thoughts which cannot be expressed in language. Note that you still have to comply with the law of non-contradiction in intuitionistic logic, i.e. you are never warranted to assert a contradiction in intuitionistic logic. There is no getting away from the law of non-contradiction!

I will be waiting for your proof! You can take your time though, I am not in a hurry.

TimeSeeker wrote: ↑Sat Nov 03, 2018 1:50 pm
Of course, now the game is rigged in my favour

Lucky you!! Any way I am glad for you, as this should be an easy proof for you now! I don’t even have to wish you good luck as it seems all the chances are already in your favor! You are nearly there so to speak, but not yet!

__________________________