Since Aristotle there are the law of the excluded middle and the principle of bivalence.
The law states that where there are two contradictory propositions one must be true and the other must be false, where the principle of bivalence states that a proposition is true or false.
These assertions will be falsified whenever there is at least one set of contradictory propositions for which can not be said that one is true and the other is false (the law) or when there is a statement which can never be determined to be true or false (both). There are four possible outcomes for falsification regarding a set of contradictory propositions:
The first two possibilities are logically excluded. A valid set of contradictory propositions produces never two true or two false statements. Only the latter two outcomes might happen – and do happen actually.
I do not consider making use of invalid content as appropriate. That will not lead to a denial of the law, just to the denial of the validness of the example. When I would say for instance 'Socrates is mortal' thereby referring to a man who is already dead for more then 2000 years and using that piece of information to prove the falsification of the law, then it is best to deny the validness of this example.
Any example that will falsify this law must not point to reality. It will have to use only symbolic language. Not formal language, as formal language will not be able to express what is needed. In formal language is this law true: p ∨ ~p (p or not-p). That is by definition. It says that whatever you substitute for p, there is a set of contradictions in which one statement is true and the other logically false. I will present two types of examples, that prove that this set of contradictions does not exist always. If you know one sentence then you do not know immediately what the truth value of the other sentence is. It is enough to find sentences, which are false nor true, because this will show that not every sentence will be true or not and therefore will not be able to be part of a set of contradictions. That will satisfy the proof that this law is not correct.
There are two ways that a sentence can not be said to be true or false. That is when it is not possible to determine the truth value of the sentence or when it changes it's truth value every time it is evaluated. An undeterminable sentence is a proof based on the third outcome and a sentence, which changes it's truth value every time a proof based on the fourth outcome.
A set of contradictory propositions which are impossible to determine are:
'there is no proposition that can counter the law of the excluded middle' and 'there is at least one proposition that can counter the law of the excluded middle.' To prove or disprove one of the two propositions all possible true/false sets of propositions must be evaluated. Only when all statements are evaluated can both propositions get determined. It is similar to the function of 1/x. It has two limits into infinity, one to false and one to true, yet they will never become true or false with complete certainty.
This set of propositions is more abstract than the famous example of Aristotle about the sea battle that will be fought tomorrow. My example doesn't need reality. His proposition is an example of future contingency. His proposition is now not determinable, but it becomes determinable at a certain point in time. Every proof that something does not exist will eventually turn into an undeterminable quest with a limit into infinity. It makes it impossible to give those propositions a true or false value. This is another way to challenge his first premisse of the proof of the existence God by Emanuel Rutten then I was doing in What is not, is not what not is. I challenged his first premisse mainly by pointing out there was another valid perspective to deal with propositions that are necessarily unprovable. I referred to the famous last sentence of Wittgenstein as it was ment to be used for those kind of statements. And when there is another valid perspective to look to propositions that are metafysically impossible to know, then is his first statement a choice, not an undisputable starting point. When it is not the only starting point, then can you not conclude that they must be necessarily false. His first premisse, however, is based on the principle of bivalence as he can only conclude that something is false if there are only two choices, namely being true or false. Yet, if that principle is proven to be incorrect like I am stating in this blog, then is the foundation of his statement wiped away completely. It is not a valid starting point anymore.
Let's take a look at two other propositions that are contradictory to each other and one might be evaluated true and the other therefor false:
'this is true' and 'this is false.'
That one word 'this' has apparently two possible outcomes, namely being true and false. The word 'this' is therefore undeterminably true or false. In both propositions is the verb to be used. I understand that this might be different in other languages, but that does not change the principle imho. The function of to be is to make two concepts equal to each other although they are essentialy different. 'This animal is a donkey' has more meaning than the statement 'this animal is an animal', although the validity of the second proposition is easier to establish than the first one. The function of to be is to add a new characteristic to the subject that enhances our understanding of the subject. But the word 'this' can refer to anything. It is in itself an undeterminable word. Just like 'I' and 'you'. It has no inherent meaning. Therefore do those two propositions This is true. and This is false. contradict and not contradict each other at the same time. Both propositions, although apparently each others contradictory propositions, have no internal value without a context. Both are without any further explanation true nor false. They are undeterminable due to absence of context.
Propositions which change their truth value every time they are evaluated are quite common and able to falsify both the law and the principle. All paradoxes have this property and the shortest example of this is:
'I lie'The contradictory proposition 'I do not lie', which is a true statement as long as this proposition refers to itself. Hence this is already an example where a true proposition has a contraditory proposition which is ever changing between true and false when it refers to itself.
Both this law and principle are not correct. As a matter of fact, this prove that the law is not entirely correct is not a real problem for the law. When the law is restated that it is only applicable to statements p which evaluate determinable to true or false, then there is no more problem. Then again is it by definition true. But that was already what it is: true by definition.
The principle of bivalence however does not hold. There are propositions that can not be determined to be either true or false. These propositions are paradoxes or they have a limit into infinity or they require a proper context.