**﷽**

A proposition is a statement affirming a positive or negative relation between two concepts. Propositions are therefore either true or false. Such that if a statement can be neither true nor false, then this statement is not a proposition.

For example: “God exists” is a proposition relating existence to God.

**Composition**

Propositions^{[1]} are composed of three parts: the subject (*Al-Mawdu’*), the predicate (*Al-Mahmul*), and the relation (*Al-Nisba*).

**Subject**: that which the predicate is related to.

**Predicate**: that which is related to the subject.

**Relation**: the positive or negative relationship between the subject and predicate.

For example: if we break the proposition “God exists” into the three parts outlined above, we get:

God as the subject, existence as the predicate, and the relating of existence to God as a positive relation^{[2]} .

**The Law of Non-Contradiction**

The law of non-contradiction (*Qanun Imtina’ Al-Tanaqud*) dictates that a proposition and its negation^{[3]} are mutually exclusive^{[4]} and collectively exhaustive^{[5]}. This is because a true proposition is one that is not-false. And a false proposition is one that is not-true.

Thus, if a proposition is true, then its negation is false. And if a proposition is false, then its negation is true.

For example: since the proposition “even numbers are divisible by two” is true, its negation (“even numbers are not divisible by two”) is false by law of non-contradiction.

[1] We deal with Categorical propositions in this submission, those that come in the form “A is B” (e.g. “God is existent” or “God exists” in short).

There are two other formally defined proposition types. Conjunctive conditionals, that come in the form “if A then B” (e.g. “if the sun has risen, then it is daytime”). And Disjunctive conditionals, that come in the form “either A or B” (e.g. “either the world is beginningless or the world is emergent”). All proposition types will be dealt with in future submissions, God willing.

[2] Positive relation because this proposition affirms existence to God.

Inversely, the proposition “God does not exist” is a proposition negating existence from God. It is therefore a proposition consisting of a negative relation between the subject and the predicate.

[3] The negation of a given proposition, is another statement identical to it in all respects, except that its relation is inverted. In other words, the relation of the negation is positive if the original proposition’s relation is negative, and negative if the original’s relation is positive. Negations can often be expressed by simply prefixing the predicate of the original proposition with a “not”.

For example: the negation of the proposition “even numbers are divisible by two” is “even numbers are not divisible by two”.

[4] “**Mutually Exclusive**” meaning:** **both propositions cannot be true.

[5] “**Collectively** **Exhaustive**” meaning:** **both propositions cannot be false.

How do we know that true or false are the only 2 values that could be assigned to a proposition? This may seem unreasonable but is debated in Quantum logic.

Your conceptualization of something includes a conceptualization of what distinguishes it from everything else that is known to you. For if concepts could not be distinguished from one another, then they wouldn’t have been knowable.

For example: if you could not distinguish between the meanings of “pen”, “paper”, and “eraser”, then you would not know what those terms mean.

Hence, when you conceptualize what “eraser” means, you also conceptualize what being “not-eraser” means. And you know that whatever you conceptualize can either be an eraser or not an eraser, because “not eraser” includes all things sans erasers. So being an eraser and being not-eraser are collectively exhaustive (i.e. it is impossible for something to be neither an eraser nor not-eraser).

Once the above is established for all concepts in general, we can apply it to propositions specifically by saying: when you conceptualize what it means for a proposition to match reality, you also conceptualize what it means for a proposition not to match reality. Thus, either a proposition matches reality or it doesn’t. If the first, then this is all what we mean when we say that the proposition is true. If the second, then this is all what we mean when we say that the proposition is false. Thus, a proposition is either true or false. Hence, truth and falsity are collectively exhaustive with respect to any proposition.

That suffices as an answer to your question. As for multi-valued logic then I have a couple of additional comments:

First:there seems to be a conflation between the proposition and the propositional judgment in Western thought. That is to say: they do not distinguish the relation between the subject and the predicate, from the relation between the judge and the proposition. Hence, a statement whose truth value is unknown to the judge, under classical two-values logic (according to the Westerners, at least based on what I’ve been taught at a Western university) is not a proposition. But this is clearly fallacious. Since being unaware of whether or not something is the case, does not mean that it is neither the case nor not the case.To give an example: the proposition “God exists” either matches reality or it does not (i.e. either God actually exists or He doesn’t). If it does, the proposition is true, if it doesn’t the proposition is false. Whether or not the proposition matches reality in of itself, has to do with the relation between the subject and the predicate. And this proposition is true or false, in of itself, irrespective of what anyone else believes about it. That’s one thing.

Another entirely different issue, is how the various people judging the proposition relate to this claim (i.e. “God exists”). Some people may be certain that it is true. Other people may be certain that it is false. Others may think there is a probability in between the two extremes, so they lean to one end of the spectrum without committing to certainty. Others simply don’t know. Others not only hold that this claim is unknown, but they further claim that the answer is unknowable. And so on and so forth.

This second relation (i.e. how the judges relate to the proposition) has no bearing on the previous one (i.e. between the subject and predicate). It has no bearing on the truth or falsity of the proposition in question. And it also has no bearing on whether or not a particular statement is a proposition. So even if you maintained that the existence of God is unknowable, this would not change the fact that “God exists” is a proposition, that is either true or false, in of itself. Contrary to those who hold that not knowing a statement’s truth, means that the statement is not a proposition.

Perhaps multi-valued logic was invented as a way to combat this problem. So they invented a truth value for “unknown”, and a truth value for probabilities between certain truth and certain falsehood, and so on and so forth. I suppose this returns to how one defines “proposition”.

Second:there is a lot of sophistry in highly theoretical fields, like cosmology and quantum physics. To the point where some of them started believing in complete nonsense, like particles emerging by nothing, or the universe creating itself. And their beliefs are not necessarily false because there are problems with the data acquired through raw observation. The issue is how this data is sophistically interpreted to deliberately oppose rational necessity, when it is not proven that alternative interpretations of the data are impossible. And those who preach this falsehood were only able to get away with this, because those theoretical subjects are difficult, and inaccessible to most people. More Muslims need to engage in such fields in order to expose the fallacies and assumptions employed in those theories.