Is X = X considered to be knowledge?

Question

Is X = X considered to be knowledge? I am wondering if a circular knowledge that doesn't state anything but state a circular fact or a redundant fact is considered to be knowledge? Is it, and if it is not why? I am thinking some people would define it as knowledge and some people wouldn't depending on what their definition is.

Answer 1

Yes, it is knowledge. "X = X" is not so trivial that it fails to be knowledge. It's actually a little subtle: what does "X = X" actually mean? For this we need to know what "=" means, which is the notion of identity, and that's nontrivial and even context-dependent. Are we talking about equality of numbers, of sets, of functions, of people?

Equality as used in real life is usually actually similitude - we say two things or people are the same if they share all or almost all relevant properties, without necessarily saying that they are exactly equal in the sense that numbers are equal. For instance, we may say that two marbles in a bag are "identical," or that a chair is "the same" chair some historic person sat in, even though there are many physical differences as the chair has changed over time. The Ship of Theseus problem comes to mind. We can even interpret numeric equality in this sense of similitude, as equivalence classes on formulas; "1+2=3" can be interpreted as placing the formula "1+2" in the same equivalence class as the formula "3." The formula "1+2", taken as a sequence of symbols, is not exactly the same formula as the formula "3", but it falls into the same class, which is why we say they are equal.

"X = X" is also useful knowledge. It expresses a fact about equality, which we can use to reason and prove things. If we were writing a computer program to automatically prove theorems, "X = X" would be one of the axioms we'd have to put into the program's knowledge base.

Answer 2

All knowledge is necessarily trivial because obvious. If it isn't obvious, then how do you know it is true to begin with, and if you don't, why say it is knowledge? Sure, we can all still say it, but that in itself doesn't make it true. One example of the triviality of knowledge: I know I am in pain when I am in pain. Well, I certainly expect I would. Another example: I know pain whenever I am in pain. Pain is exactly whatever I feel when I am in pain, and so my knowledge of what I feel defines pain itself, and so I know pain.

Taking X = X to mean "Every thing is identical to itself", which is the Law of Identity, we can certainly say that it is trivial and obvious, although somewhat amusingly you will find people who denies that it is true and sometimes that it is meaningful. Ignoring them, I would certainly say the Law of Identity is trivially true. However, when it comes to expressing verbal arguments and formal reasonings, it is perfectly possible to say "X is not identical to X", or X ≠ X. This is the transgressive "power" of verbal communication that we can make false statements, knowingly or not. Thus, we have decreed the Law of Identity in the same way as we have decreed "Thou Shall Not Kill". Both laws can still be disregarded, but you cannot disregard the law of identity and reason logically, irrespective of what some philosophers and some mathematicians might want to say in this respect. Thus, the law of identity may be regarded as similar to a rule in a game. If you want to play logic with me, you better respect the law of identity or I will myself disregard whatever you say.

The Law of Identify is similar to E = mc². However, you need to know what the symbols E, m and c mean to understand the law. E = mc² is not true if m means the size, the volume or even the colour, rather than the mass at rest. So the Law of Identity just means that you will not be able to reason logically if you assert that some particular thing both has and has not some property. It should be noted that people who disagree with this are invariably people who don't understand what the law means. So the Law is not meant to force people to do something they could not abstain from doing anyway, it is to make clear the rule of the game and legitimise the exclusion of people whose statements somehow disregard the law, and this also justifies calling it a "law".

Answer 3

It seems that you consider knowledge as an equivalent of information. False.

• Knowledge is the model of the world, and in such model, it is important to consider that X = X (see below: Aristotle).

• Information is knowledge obtained by reason or experience. Within such context, X = X provides nothing new. X = X is not information, but it is knowledge.

• X = X is the definition of Aristotle's first law of logic [1]. It is not necessarily circular or redundant, read the definition.

• It might surprise you, but a lot of our knowledge is circular. For example, the dictionary is a set of circular self references, and we consider it a valuable tool. Moreover, all the subjective rules each one follow are based on a tautological structure. Immanuel Kant already noticed that problem: there's no final rule that would validate any other rule. Descartes' cogito ergo sum is an attempt to find such final rule, but Kant associates such cogito with kind of a confusion between reason and existence [2].

Moreover, you can consider X = X as knowledge (also: judgement) analytic a priori: knowledge that is valid per se, without any required experience or perception to make it valid (again: yes, it is knowledge, according to Kant). Read the distinctions between analytic/synthetic and a priori/a posteriori in the introduction of Kant's Critique of Pure Reason.

[1] https://oregonstate.edu/instruct/phl201/modules/Philosophers/Aristotle/aristotle_laws_of_thought.html

[2] https://philitt.fr/2012/10/03/kant-critique-du-cogito-cartesien/