Axioms are only 'true' within the system that they define. If they don't define a system then they are no more than an opinion. An axiomatic system also needs to define the elements to which it applies.RCSaunders wrote: ↑Wed Jul 10, 2019 8:21 pmI totally agree. Except there is no claim the statement was about a physical fact. It is about the nature of reason itself.A_Seagull wrote: ↑Wed Jul 10, 2019 5:17 amSo I will just conclude with this: There are some truths of the world such as 'Then Earth goes around the Sun in an elliptical orbit' and 'e**ip=-1', and these truths can be demonstrated from a few simple assumptions and a fairly complex sequence of inferences; but to claim that some banal statement like your 'either some fictional aeroplane crashes or it doesn't' should fit alongside them as a truth of the world is just pathetic.
Just as 2 plus 2 equals 4 is not a statement about any actual earthly fact, it is said to be true because two of anything added to two more is four, without specifying any actual things.
Euler's identity is not about any physical fact either. The mathematical elements, e, i, and π are all purely concepetual and have no physical existence. e and π are both irrational which means there is no commensurate (physical) unit of measure in which they can be expressed. i does not exist at all except as a concept.
In logic, the law of excluded middle states that for any proposition, either that proposition is true or its negation is true. It is one of the so called three laws of thought, along with the law of noncontradiction, and the law of identity. The law of excluded middle is the basis of the discussion you objected to.
The law of excluded middle is not based on an assumption, it is based on an axiom, that a thing is what it is and cannot be anything else. An axiom (not the nonsense proposed by modern day philosophers) is true because to deny it is self-contradictory. If you were not who and what you are you would be something and someone else. You could not be that and be you too. An assumption is something simply posited without conclusive evidence or reason. If you are satisfied with assumptions, fine, I am not.
I have no idea why you are uncomfortable with this reasoning, but if you are, then just ignore what I've written, and believe what you wish, but understand I will do the same.
If you want you can set up an axiomatic system with the 'laws' of logic as axioms. But then what does such a system produce? If it doesn't produce some useful theorems then the system can be labelled as 'useless'.
Mathematics is a good example of an axiomatic system. It produces useful and interesting theorems. However before it can be shown to be 'useful' there needs to be a mapping between one or more of its elements and the world of ides (or if you believe in naïve reality between the elements of mathematics and objects in the real world.) In this respect the mathematical elements of 'i' and '2' are equally abstract, it is just that '2' has a simpler mapping to the real world than 'i'. (That said 'i' is useful for both electronics and quantum mechanics).
To ignore this mapping process and claim that there is a direct and unambiguous link between elements of a logical system and the real world and that axioms such as the 'laws' of logic have a direct correspondence to the real world is naïve.
To claim that axioms are objectively true because there are no obvious counter examples is like claiming that 'all sheep are white' and then waiting for someone to find a black sheep.
If the 'law' of excluded middle was really logical it would not just be either A or not-A, but also incorporate the possibilities of both A and not-A and the possibility of neither A nor not-A.
An example for where such a 'law' does not apply in the real world is in the famous 2-slit experiment. It is less than useful to claim that the buckyball (a molecule of some 50 atoms or more) either passes or does not pass through slit A.