An Enquiry Concerning the Nature of Generalities

This article will slightly digress from my usual topics... Nevertheless it's something I must share with the world.

The other night, in casually conversing with my roommate, I uttered a proposition which received its usual objection. I will frame the conversation, along with its objection and my counter-objection.

Me: I wonder why it is the case that a good number of our ethnic group's females are so culturally conservative.
Roommate: Isn't that too much of a generalization?

Now here is what I want to address to as many people as possible. A generalization is NOT an argumentative fallacy. As a matter of fact, if a generalization were an epistemic illegality, then all of knowledge as we know it would not exist. I will go so far as to say that all knowledge IS a generality. When testing any argument for validity, or set of premises for consistency, a logician will annotate each premise using generalities in propositional calculus, quantificational logic, or any other first-order languages of logic. In this article, I will limit the scope of demonstrating that any proposition of both propositional calculus and quantificational logic are indeed generalizations. Why only these two first-order systems? Simply put, virtually all other systems of logic reduce to either propositional or quantificational logic. Therefore, what is true of these two systems is true of all systems that entail them.

We start with prop calc...
A law or a theorem, be it a law or theorem of physics, mathematics, or anthropology, is a proposition of the nature 'All A's are followed by B's.' We can also regard such a proposition as 'A implies B,' or 'A entails B.' All books have pages. All humans are mortals. All matter is non-destroyable. We know some of these propositions (the first two) to be true by definition alone, as a book entails some number of pages and all humans indeed have a finite lifespan, and the latter to be true given empirical evidence (Law of Conservation of Matter). (Immanuel Kant was the first to make this distinction between analytic and synthetic truths, but that is outside the scope of our argument.)

The types of propositions mentioned above (laws) can be validly regarded as 100% generalizations. This is simply the nature of propositional calculus. 'If A, then B' is a simpler way of asserting all A's must be followed by B's, or that all A's are indeed also B's. If this conditional is true, translated to its most abstract form, we can think of it as 'any object that is ascribed the property of A must be ascribed the property of B.' This utterance is a generality in its purest form. But let us complicate the matter a bit further. Let us suppose we have run into an urn full of 10 marbles – consisting of 5 red marbles and 5 blue marbles.

Quantificational Logic
Now things have gotten slightly trickier. We have as our domain, let us call ∆, 10 objects – 5 of which have been ascribed the color property of redness, and 5 of which have been ascribed the color blue. Using the formal language of quantificational logic, we can summate this situation as follows:
Suppose...
R = red
B = blue
M = marble, which we will designate as an operator
∃x∃y [M(∆) → (Rx v By)]
Admittedly, the formalized notation above is not required for our analytic purposes. It will suffice to know that the logical proposition above can be translated as 'given some values of x and some values of y, for any marble (object) in set ∆, it is either the case that the color property of red or blue is ascribed. Simply put, we can truthfully say some objects in ∆ are red, and some objects in ∆ are blue. And this IS indeed a true proposition, given our setup. Nevertheless, this is a generality.



We can add a further complication that our fellow sociologists and political scientists will utilize. Given our urn, a more specific proposition may be given as follows:
∃x∃y [0.5(M(∆) → Rx) & 0.5(M(∆) → By)]
This proposition above says almost the same as the previous, but a bit more. Simply put, it ultimately translates to '50% of the marbles in set ∆ are red, and 50% of the marbles in set ∆ are blue.' This type of utterance is not as clear-cut as our previous examples of 'All A's are followed by B's,' where 100% of A's are B's. Nevertheless, it asserts something quite similar in nature: 50% of M's in ∆ are R's, and 50% of M's in ∆ are B's. Once again, we have ourselves a truthful generalization.

So much are generalities essential to the existence and continued perpetuation of knowledge, that logicians have gone so far as to build two quantifiers one of which we have already used (also known in the philosophical community as the generality quantifiers). These are are the existential quantifier (for some objects in a given domain, it is the case that) and the universal quantifier (for all objects in a given domain, it is the case that). Saying that no objects in our urn are green would simply be asserting a universal generalization with a negation of the property greenness, or 'all objects in ∆ are not green.' Nevertheless, this is itself a generality of similar nature.

A case my roommate may have legitimately been able to make is one of the following: I disagree, I believe none of our ethnic group's females are conservative, or I believe all of our ethnic group's females are conservative – either one of which would have been inconsistent with my assertion that 'there exists some females in such and such domain that are conservative.' Given that I did not make an utterance of a specific probabilistic nature, and given that neither one of us had a formal survey to cite, neither one of us were in an epistemic position to disagree on an exact numerical value of conservative females in our domain (or ethnic group). Therefore, my utterance was less exact – of the form 'There exists some females in this specific domain that carry the property of conservative-ness.' And any philosopher of language will agree that some, by definition, constitutes 1+ (one or more) objects of a domain. Therefore, so long as there exists at least one conservative female in the ethnic group in question, my proposition was indeed valid.

Possible Philosophical Counterobjection: An Appeal to Modality
One who is well-read in the philosophical arts may find a possible counter-objection which may initially seem as though it bears some truth. One may make an appeal to necessary truths, those whose denial will necessarily result in a logical self-contradiction. But for one, it must be realized that the existence of such necessary truths in nature (independent of our definitions) is arguable even in the philosophical community. Nevertheless, even if opponents of necessary truths were wrong in every possible way, and such modalities do exist, it can be argued that any necessary truth must follow from a tautological disjunct: Av~A (A or not A) either of which must be the case. Therefore, we can have Av~A implies the 'existence of God' or 'a naval sea battle's occurring at noon tomorrow' or any other proposition which an individual may deem as a necessary truth. This conditional can itself be deemed a generality.

Any other possible counter-objections
I invite any reader to hand me a proposition (a semantically and syntactically valid statement) which I will readily be able to show is a generality.

In conclusion, a specific generality in itself may be deemed true or false. However, a generality does not in itself constitute an argumentative fallacy.