Universal Propositions and “Existential Import”

Introduction. In a note (25, p. 171) in Christopher Shields’ quite admirable overview, Aristotle (2nd edition; London and New York: Routledge, 2014), we find the following passage, setting forth (Ibid., p. 143) the “three simple and, he [Aristotle] thinks, intuitively correct conversion principles” which Aristotle “adduces” as necessary to his syllogistic.

His principles are: (i) from no A is B we may infer that no B is A (if no soprano is a man, then no man is a soprano); (ii) from some A is B we may infer that some B is A (if some woman is a soprano, then some soprano is a woman); and (iii) from all As are Bs we may infer that some A is B (if all sopranos are women, then some soprano is a woman).

The note goes on, however, to say:

Importantly, from the standpoint of modern logic, (iii), however innocuous it may sound, is incorrect. It has, even where it is unwanted, existential import. We might agree that all dragons have wings without wanting to allow that there is some dragon with wings. Rather, what we seem to mean is: If anything is a dragon, it has wings, which does not license us to infer that there is something with wings, namely a winged dragon.

Shields’ view of the matter of existential import is that of the vast majority of contemporary logicians; picking up a copy of the nearest logic textbook to hand, Copi’s Introduction to Logic (7th edition; New York: Macmillan Publishing Company, 1986), I see standard view given expression in Section 5.5, “Existential Import,” of Ch. 5, “Categorical Propositions” (pp. 190-193). But I wish to differ, not by defending some version of, say, the thesis of “two logics” as advanced by Henry Babcock Veatch in works like his Intentional Logic: A Logic Based on Philosophical Realism (Yale University Press, 1952) and Two Logics: the Conflict between Classical and Neo-Analytic Philosophy (Northwestern University Press, 1969), for I understand that there is but one science of logic, of which the logic of Aristotle is but one part, albeit a crucial one.

The intention then of the present post is twofold. First, and immediately, I want to suggest that the view which Shields and modern logic have of existential import, according to which a proposition like “all dragons have wings” should not be understood as having it, represents but half of the truth of the matter. The other half would see that a proposition like “all sopranos are women” should be understood as having existential import. Putting it in summary form, it should be seen as a requirement of logic that a proposition should have existential import if and only if existential import is warranted (I prefer “warranted” and “unwarranted” to “wanted” and “unwanted,” for it has nothing to do with what we want).

Theses of Logic, Theses of Ontology. That done, I would second like to suggest that there is reason for seeing that at least some universal affirmative propositions should be understood as necessarily having existential import. As a first step in so suggesting, I will note that Shields’ rendering of the proposition, “All dragons have wings” as:

If anything is a dragon, it has wings.

is still not quite as “regimented” as it should be to be in accordance with the approach typical of modern logic. Let’s make it a little more so, thus:

For any existent x, if x is a dragon, then x is winged.

or, yet more so:

(x)(Dx –> Wx)

I will next observe that the proposition “If anything is a dragon, it has wings,” “For any existent x, if x is a dragon, then x is winged being,” or “(x)(Dx ® Wx)” is first a proposition about existents and only consequently about dragons and winged beings. As a proposition about existents it is, then, primarily a proposition belonging to ontology, the science of existents, and one only consequently belonging to a theory of dragons. The same is true of the proposition “Some dragon has wings,” “There is something with wings, namely a winged dragon,” or “(Ex)(Dx & Wx).” (I have to interject here my usual apology for and explanation of using “(Ex)” to express the “existential quantifier” in this post; I do so because I do not know how to express it using the standard means within WordPress.)

The Assumption of Existential Import in Universal Affirmative Ontological Propositions. Now, suppose some philosopher, perhaps after a long meditation on the title of the book, The Incredible Lightness of Being, decided that it had to be the case that:

All existents are flying existents.

Let’s so “regiment” that proposition as to have it accord more with the modern mode of expression, thus:

Any existent x is a flying existent.

or, abbreviating it:

(x)(Fx)

We have there three expressions of a universal affirmative proposition. I will now proceed to offer a formal demonstration, if one is needed, that that universal affirmative proposition has existential import. I will do so by means of a reductio ad absurdum argument, one showing that, given the premise, “(x)(Fx),” a contradiction arises with the further assumption of the denial that there is a flying existent, that is, the assumption that:

It is not the case that there is a flying existent.

or:

~(Ex)(Fx)

The proof proceeds as follows:

1. (x)(Fx) {Premise}

2. ~(Ex)(Fx) {Assumption; for a Reductio ad Absurdum}

3. (x)~(Fx) {Logical Equivalence}

4. Fa {1, Universal Instantiation}

5. ~Fa {3, Universal Instantiation}

6. Fa & ~Fa {4, 5, Conjunction}

7. ~~(Ex)(Fx) {2-6, Reductio ad Absurdum }

8. (Ex)(Fx) {7, Double Negation; quod erat demonstrandum. (How I enjoy saying that!)}

The universal affirmative proposition serving as the proof’s premise has, therefore, to be understood to have existential import.

The same, of course, would be true of the universal negative proposition, “(x)(~Fx),” serving as the premise of a similar proof.

By analogy: Let me turn now from ontology, to, say, (philosophical) anthropology. That is, while in the above, I took as the universe of discourse that of existents, without restriction, here I will restrict the universe of discourse to that of human beings. Let me then take the following universal affirmative proposition of philosophical anthropology:

All human beings are rational beings.

and so “regiment” it as to accord more with the modern mode of expression, thus:

Any human being, h, is a rational being.

or, abbreviating:

(h)(Rh)

We have there three expressions of a universal affirmative proposition of philosophical anthropology. I will now, as before, proceed to offer a formal demonstration, if one is needed, that that universal affirmative proposition has existential import. I will do so by means of a reductio ad absurdum argument, one showing that, given the premise, “(h)(Rh),” a contradiction arises with the further assumption of the denial that there is a rational being, that is, the assumption that:

It is not the case that there is a rational being.

or:

~(Eh)(Rh)

The proof proceeds as follows:

1. (h)(Rx) {Premise}

2. ~(Eh)(Rh) {Assumption; for a Reductio ad Absurdum}

3. (h)~(Rh) {Logical Equivalence}

4. Ra {1, Universal Instantiation}

5. ~Ra {3, Universal Instantiation}

6. Ra & ~Ra {4, 5, Conjunction}

7. ~~(Ex)(Rx) {2-6, Reductio ad Absurdum }

8. (Ex)(Rx) {7, Double Negation; quod erat demonstrandum.}

The universal affirmative proposition serving as the proof’s premise has, therefore, to be understood to have existential import.

The same, of course, would be true of the universal negative proposition, “(h)(~Rh),” serving as the premise of a similar proof.

In Posts to Come. What I have presented in this post is a set of suggestions. There is much that I need to do before I can offer it as a theory. I hope, in posts to come, to fill in the many lacunae and to answer the many questions to which the suggestions and the theory give rise. The tasks then I see in front of me include, among others, articulating more fully the identification I have made here of “modern” “logic” with ontology, that of working out the needed understanding of the nature of propositions, and that of developing the epistemology which will allow us to understand how we can have the kind of knowledge which our grasp of the universal propositions figuring above represents.

Until next time.

 

About Rchard E. Hennessey

See above, "About the Author/Editor."
This entry was posted in Aristotelian Logic, Logic, Modern Logic and tagged , , , , , , , , , , , , , , , , . Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s