Philosophical Validity Proofs: Logic vs. Mathematics

[AllCombinations]

Hello. I am currently studying "Introduction to Logic" 2nd Edition by Harry J. Gensler and I have a question about writing logical proofs. The book's preferred method is one of assuming the opposite of an argument and then taking the original argument apart while looking for a contradiction to arise. If a contradiction arises from assuming the opposite of the original conclusion, then (in a binary system of true and false statements) the original conclusion is proven to follow and the argument is said to be valid.

 

My question is this: I know that in mathematics it is very common - even required - to prove a statement; that is, a claim will be made and then a proof will immediately follow the claim up. But what is common practice in philosophy? Are statements made but then it is left to the reader to work out for themselves if the reasoning is logical (if not sound)? Or is proof required here too? Of course I do understand the difference between validity and soundness, but is it common to prove validity?

 

To illustrate, here is a problem from the book I mentioned: "Some are logicians. Some are not logicians. Therefore, there is more than one being." (problem 2, section 9.2b, pg. 210)

 

Strictly in terms of validity, would a philosophical text simply leave it at this? Or would it be correct to include a proof such as the following?

 

Assume that it is false that more than one being exists. Then it follows that for all beings that do exist it is not the case - for some other being - that one is not the other. Because some being is a logician, let that person be called Adam; and, because some other being is not a logician, let that person be called Eve. Then it is not the case - for some being - that Adam is not that person; that is, for all beings, anyone is Adam. Then Adam is Eve. But because Eve is not a logician and Adam is Eve, it follows that Adam is not a logician, and therefore more than one being exists.

 

That just seems like a mess... I mean, look at it. But I have never read a philosophical text before, so I don't know.

 

In philosophy, what is proven and what isn't? And how does it compare to mathematics?

 

Thanks ahead of time.

[Eise]

Hmmm... Seems a bit complicated, indeed.

 

The conclusion of the argument is 'there is more than one being'. So denying this would be:

 

There is only one, or zero beings

 

The sentences 'Some are logicians. Some are not logicians.' are not quite unambiguous, so let's rewrite them:

 

There exists at least one being who is a logician.

There exists at least one being who is not a logician.

 

No let's split our denial in two:

 

There exists no being.

 

Well both sentences say there is a being so 'There exists no being' is false.

 

Now:

 

There exists exactly one being.

 

Assume this being is a logician. Then the second sentence says there exist a person who is not a logician. Well one person cannot be a logician and not a logician at the same time so this is a contradiction.

 

Of course the same the other way round. So the denial:

 

There is only one, or zero beings

 

is wrong, and therefore the conclusion is correct, which makes the original logical argument valid.

 

I consider pure logic not as philosophy: it is however an essential tool for philosophers. They should be able make sound arguments, and recognise wrong logical derivations.

 

Mathematics I consider as applied logic: to the field of mathematical objects, like numbers, transformations, forms, etc etc. Where in other sciences one can operate at least temporally with concepts that are not rigidly defined, where the logic might be hidden for the moment, mathematics must rigidly comply to logic. If a logical argument to some mathematical statement cannot be found, it cannot be considered true. But in physics one can formulate an empirical law that exactly fits the observable data, but one has no idea how this empirical law can be derived from known physical laws. Examples that come to my mind are Balmer's formula, and Planck's formula. It was only later that Planck could derive his law for black body radiation later from more fundamental principles, but for many years could not believe these principles were true.

Edited October 20, 2016 by Eise

[AllCombinations]

Thank you for your feedback, Eise. However, my inquiry remains unanswered: in philosophy, would one provide a proof as we have done here? It is common practice in mathematics, but would it be done in philosophy? Did Kant do it in his Critique of Pure Reason? (I haven't read it. It was a title that sprung to mind.)

[Eise]

Hi AC,

 

It is not often done so explicitly as in your example, but some philosophers tried it rigidly (e.g. Spinoza, or Wittgenstein in his Tractatus. I would not take Kant as example, even if his work is impressive). Most philosophers do not argue so exact, in my opinion mostly because philosophical concepts are not that clear. A lot of philosophy is just trying to get such clarifications, and some philosophical problems 'evaporate' by providing rigid definitions.

 

 

But there are modern day philosophers who at least partially build up such logical arguments. But never so rigidly as is mathematics.