Three Disjunctions

Consider the following three arguments:

[A]
(1) p [premise]
(2) p v q [1, disjunction introduction]

[B]
(1) p v q [premise]
(2) ~p [premise]
(3) q [1,2 disjunctive syllogism]

[C]
(1) p v q [premise]
(2) p [premise]
(3) ~q [1,2, exclusive disjunctive syllogism]

Each of these three represents a different sort of disjunction. It's pretty clear that [C] is different from [A] and [B]; truth functionally, exclusive disjunction is 0110 (fttf), whereas the other two are inclusive, 1110 (tttf). That is to say, in exclusive disjunction if both are true, the statement is false, whereas with inclusive disjunction if both are true, the statement is true. (As a complete sidenote, can I say I'm actually somewhat impressed with the Wikipedia article on exclusive disjunction, if only for the fact that it follows Barrett and Stenner in recognizing that English does not naturally have an exclusive or. You can, of course, represent it by circumlocutions, but the English 'or' is never exclusive, because English 'or' is never strong enough on its own to require that the truth of both disjuncts makes the disjunction false. This is something that many philosophers never learn. The SEP article on disjunction recognizes this, too, but as it's by Jennings that's unsurprising, since he wrote a very good book, The Genealogy of Disjunction, in which this is discussed at length.*)

But it may seem less obvious that [A] and [B] are different kinds of disjunction. After all, they are truth-functionally the same, and isn't that supposed to be all that you need?

Obviously, I don't think so. My reasoning for treating these as different is this. Even though they have the same truth-functional properties, A2 and B1 cannot be used in argument in the same way. A2 is a weak disjunction for reasoning with, because it arises in the argument solely from disjunction introduction, and the only reason for concluding it is that p is taken is true. And the only thing one can conclude from it is that p is true, unless you also argue that p&q is true. The q has no reason for being there except that you have performed a disjunction introduction on p. If you isolate A2 from its context, and add to it the premise ~p, you cannot conclude q. Your only grounds for accepting p v q was p, and therefore the 'v q' has no independent authority as a disjunct, so to speak; p v q is just a more complicated p.

This is very different in the disjunction syllogism. B1 is a premise; it does not depend on anything. Because of this, it is a very robust disjunction for reasoning, since its existence in the argument, and its truth, does not depend on either the truth of p or the truth of q. Each disjunct stands in its own right.

Thus, I would suggest, contrary to truth-functional orthodoxy, that truth-functional disjunction is rationally ambiguous, i.e., it is ambiguous for the purposes of reasonable argument. We cannot take a disjunctive sentence in propositional logic and know how to understand it for the purposes of actually arguing with it unless we know why that sentence was introduced. Of course, if this is right, it means that every sentence in propositional logic exhibits this ambiguity, because every other operation is related to disjunction by a set of equivalences that are blind to everything except truth values.

What say you?

___

* Jennings, of course, is right in his critique, as were Barrett and Stenner. And yet you still find philosophers saying that "Cream or sugar" on a menu, understood naturally as saying that you can have cream or you can have sugar, but not both, is an instance of the exclusive or. This, of course, is nonsense. The choice really is, as a list:

You may have cream,
you may have sugar,
you may not have both cream and sugar.

And the option, "You may have cream or you may have sugar" is not false if both 'disjuncts' are true. Quite the contrary. For the choice to make any sense at all, the 'disjuncts' must both be true --, i.e., it must be true that you may have cream, and it must be true that you may have sugar, otherwise you've been given no choice at all. The 'or' here actually indicates a conjunction:

You may have cream (to the exclusion of sugar) & you may have sugar (to the exclusion of cream)

and both conjuncts are true.

The Philosophy Job Market

This post by Jon Cogburn on the utter irrationality that is the philosophy job market in North America is all the more remarkable for being quite true. Nothing in this system makes sense; nothing in it has any great probability of improving philosophical work done; nothing in it has any of the nobility, or honor, or, for that matter, good sense that everyone has every right to expect. The whole system is simply a sloppy crazy quilt of practices that no one has ever bothered to think through properly.

That said, I've had a number of very pleasant interactions at APA and campus interviews; I suspect how nice or nasty the interviews are tends to depend a great deal on the type of search and college your AOS tends to get you.

But do you want to know the think that really shocked me? How much philosophy professors despise the students they teach. Perhaps it's just the questions I ask, or the fact that I tend to design courses that are intellectually very difficult (although I always make it easy for those who participate to pass; the intro course I'm currently teaching is a good example), but it comes up in a surprising number of interviews. Indeed, of all the departments that have interviewed me, of which there have been many, three and only three expressed an explicitly and clearly positive view of the students they teach (Kenyon College in Ohio; Saint Anselm College in New Hampshire; and Saint Vincent College in Pennsylvania). At least three, who shall remain nameless, had interviewers who in one way or another tried to impress upon me the view that their students were in the main stupid and/or selfish. It goes well beyond the concern about whether a set of courses or a particular approach is suitable for undergraduates, which is undeniably a legitimate one given that my approach is always unconventional to the point of eccentricity; it's very different from that, because the contempt for the student shows through. There was one early one where it was so bad that I left the interview with the absolute conviction that I would never teach in such a department, and since then I watch for the warning signs like a hawk.

What I found most fascinating about the 'smoker' is that at any given moment probably a quarter of the people in the room are talking about how bad it is, or how they don't want to be there, or how they'd rather be somewhere else, or how tiresome a responsibility it is. And yet there they all are, milling around between moments of gladhanding so that they can find people to complain to about how there's nothing to do but mill around, and gladhand, and complain.

Various

* With regard to my previous post, Ocham has reminded me of this page at the Logic Museum. For a while (and I hope to get back in the practice after awhile), I did 'LFPP' posts -- Land of Forgotten Philosophy Papers -- which summarized philosophy papers that have become pretty nearly completely forgotten but had worthwhile arguments and valuable insights that were worth remembering. The paper by Land at the link above is certainly one of those. In any case, Land's paper ties in nicely with some thinking I was doing last night. I was reading Keynes's arguments on existential import, which are lovely, it occurred to me that one problem with them is the idea that the matter is really relevant to the doctrine of opposition (at least in the straightforward way he assumes). It's easy to see why one would think that in symbolic logic of the sort he would have known; but I would imagine that one desiderata for a good square of opposition is that it be built up, as entirely as possible, from oppositions of quantity and quality, and nothing else, and the fact that existence keeps popping up as part of the square is a bit worrisome. Land's argument touches on points very close to this issue.

* The Jerome Lejeune Foundation is a medical foundation devoted to funding research and treatment of genetic diseases related to mental handicaps. Jerome Lejeune was the geneticist who discovered trisomy-21, the genetic of cause of Down's syndrome, thus proving that it wasn't (as many people unfortunately had thought) a symptom of 'racial deterioration'; he then went on to link a number of other mental handicaps to chromosomic disorders, doing so much work that he is sometimes called the father of modern genetics. He worked for most of his career toward finding a cure for Down's syndrome; he did not succeed, and died feeling that he had failed his Down's syndrome patients. He died on April 3, 1994; the Catholic Church began its investigation into whether he should be beatified in the spring of this year.

* The upcoming The Dark Is Rising movie looks positively horrid. On the plus side, casting Christopher Eccleston was a good thing.

* Poetry as philosophy at "Just Thomism"

Another Poem Draft

Stars' End

Here
The darkness is only gentler light.

Here
The strands of time and thought grow tangled;
They cross the sky with splendor.

And face to face makes answer
Here.

All roads, each path, leads home.
They each have weight, seek natural place,
The equilibrium of perfectly-there.

Every seeking and all wonder,
Each adventure and every quest,
Returns, after all its wandering,
To the source.

Here
Iron sharpens iron.

Here
Living and knowing are one.

And all stars come to an end
Here.

Puzzles about Existential Import

In this post I may simply show myself to be confused and nothing more (in which case I would be pleased to be enlightened); but I rather suspect that I am confused for the reason that the things confusing me are in a muddled state. I was recently trying to find a good defense of the standard contemporary view of existential import, and failed miserably, because, it would seem, no one has done anything but parrot the same (very bad) arguments for the past fifty years at least. All of the discussions I have been able to find make three major mistakes.

(1) They confuse implicature and implication. When we say "Some X are Y" (with something plugged for X and something else for Y), we usually take this as implying (in a broad, colloquial sense) that X's of some sort exist. But a natural way to understand this is to say that we wouldn't usually have any reason for saying "Some X are Y" if no X's existed at all. And that makes it a matter of implicature, not 'import'. So any appeal to common usage has to make the distinction.

And you can indeed have a good discussion of existential import that distinguishes between the claim that "Some X are Y" implies that X's exist and the claim that "Some X are Y" implicates X's existence. John Venn does it, and he manages to do it more than thirty years before Grice was even born. He doesn't, of course, use the term 'implicature'; instead he talks about "understandings" and "implications" (the latter being used in quotations to indicate that it is being taken in a broad sense). If Venn can distinguish between implication and implicature before anyone had even done serious work on implicature, contemporary logicians have no excuse.

(2) They confuse existence in a particular sense and existence taken generally. Copi and Cohen, for instance, take 'existence' to mean actual existence, and even go so far as to try to wave away existence claims about literary and mythological characters as rare and not really existence claims at all. Because of this, several of their arguments trade on the assumption that the existence indicated by a proposition with existential import must be physical existence in the real world. But as Venn, Carroll, and Keynes recognized in the nineteenth century, this is an assumption that is both implausible and needless; the existence that is relevant here is membership in the domain of discourse established by the terms of the proposition, and that could include dreams or Wonderland. [Of course, this raises an interesting question, since domains of discourse are usually ignored in contemporary logic: Are modern logicians like Copi and Cohen even talking about the same problem as the nineteenth-century logicians?]

(3) They confuse I propositions with existential propositions. I mentioned this in the previous post. As far as I can tell, there have been no good arguments for taking I propositions to have existential import since the nineteenth century. In the twentieth century it is taken as unassailable dogma, which is remarkable. Why must we assume that I propositions have existential import? If I say, "Some impossible things are geometrical impossibilities," am I really implying the existence of impossible things? Perhaps, although it raises the question of how 'existence' would be understood in such a case (it's not a problem if impossible things can be members of the universe of discourse; but this requires that we not confuse 'existence' here with actual existence in the real world). The closest anyone seems to come to doing so appeals to common usage -- and not only does this virtually always commit error (1), it ignores the salutary distinction noted by Carroll between what is logically required and what is convenient for practical purposes. At least when Keynes, the nineteenth century logician who perhaps has the view on existential import closest to the contemporary view (he gets it by generalizing comments made by Venn with regard to symbolic logic to all of logic), appeals to common usage, he doesn't leave it at that, but argues that taking particular propositions to have existential import has a number of logical advantages.

So on this point, at least, logic in the twentieth century seems to have degenerated: points that were once taken as a matter for serious debate are now taken as undeniable, and the arguments put forward are merely cases of uncritical repitition. The nineteenth century logicians seem to do a much better job of handling the matter than anything more recent. In any case, if anyone's come across a better recent defense, let me know.

Here's a puzzle, too. The contemporary view of existential import is usually understood to cause problems for subalternation and conversion per accidens. In Copi and Cohen we find the following argument put forward as an example of the 'existential fallacy':

(1) No mathematician is one who has squared the circle.
(2) No one who has squared the circle is a mathematician.
(3) All who have squared the circle are nonmathematicians.
(4) Some nonmathematician is one who has squared the circle.

We get (2) from (1) by conversion; (3) from (2) by obversion; and (4) from (3) by conversion per accidens. The move from (3) to (4), says Copi (and Cohen), commits the fallacy of assuming that there is someone who has squared the circle. Of course, as already noted, this assumes that (4) is an existential proposition, which should be open to discussion. One could just as easily argue two other positions:

(A) That the real fallacy is committed only if we move from (4) to the claim "Some nonmathematician who has squared the circle exists." But as (4) is obtained from (3) by means of a conversion by limitation, we should really read (4) as simply a more limited (3), telling us that some nonmathematicians (in this case those who are nonmathematicians in the sense that, unlike real mathematicians, they don't exist) are circle-squarers. That is, it doesn't assume existence, but merely that we can take the terms partwise, so to speak.

(B) For that matter, if we went so far as to defend common usage, we might well begin to think if there is an existential fallacy it has to be attributed to the obversion rather than the conversion per accidens. If there is any proposition in the above argument that would suggest to most people in most circumstances that there are people who have squared the circle, it's (3), because we would naturally take it as saying that, Of those who have squared the circle, all are nonmathematicians. And that really would usually suggest that there exist some people who have squared the circle.

I hold to the first, although I think students are owed a non-question-begging explanation why the move from (2) to (3) doesn't commit the existential fallacy.

In any case, that wasn't my puzzle. My puzzle is this. How can people straightfacedly reject subalternation as committing the 'existential fallacy' and yet accept universal instantiation, given that the two are logically very similar? (Certainly from a term logic standpoint there is very little difference, and free logics often treat UI as doing more or less what subalternation is accused of doing -- Karel Lambert, for instance, seems to be very explicit that this inconsistency with regard to the treatment of UI and subalternation was one of the motivations for his work in free logic, although certainly not the only one. And it is worth pointing out as a side note, since people forget, that (x)(Fx) is in term logics an A proposition even though it is not in the hypothetical form into which most A propositions are recast when translated into the predicate calculus, and thus yields no hypothetical form when instantiated.) And yet we find this done all the time. At the very least textbooks that criticize subalternation for violating existential import should give reasons why universal instantiation (esp. in combination with existential generalization) should not be treated the same way, but I have found none that do so, despite there being obvious reasons why such a point would need to be addressed. (If you look carefully in more advanced works you find that the reason for distinguishing the two has to do in part with a particular theory of interpretation, one in which a great many propositions apparently of the form "Some x is F" are not really I propositions at all, and many propositions apparently of the form "a is F" are not really singular at all. This is the only reason why "Some things are impossible" can be held to be intelligible if I propositions have existential import, for instance: we have to say that it is not an I proposition at all. We get mired in a great deal of controversial theory if we look at why universal instantiation and subalternation are supposed to be treated differently; and we begin to wonder whether we are dealing with a legitimate logical need (as some say) or a set of epicycles designed to save the phenomena for an approach that has serious weaknesses.)

And, of course, it seems slightly absurd in a broader sense, too. As Venn noted, no one takes the possibility that in a particular case you might divide by zero to be a good argument for never dividing by variables. One might well ask, even if the standard contemporary view is entirely correct, why we would talk as if we should reject subalternation and conversion per accidens altogether rather than recognize them under some set of conditions. (We certainly do that with existential generalization anyway.)

So I find the whole mess to be obscure and puzzling; even if the standard contemporary view is correct, it certainly doesn't seem to be taught properly.

Lewis Carroll on Existential Import

Lewis Carroll has a fascinating argument about the existential import of propositions in the notes to what would have been the second part to the Symbolic Logic. (Only Part I was published in Carroll's lifetime. Part II seems to have gotten as far as galley proofs, but most of these were lost.) First Carroll has some salutary advice against dogmatism on the matter:

...I maintain that any writer of a book is fully authorised in attaching any meaning he likes to any word or phrase he intends to use. If I find an author saying, at the beginning of his book, "Let it be understood that by the word black I shall always mean white, and that by the word white I shall always mean black," I meekly accept his ruling, however injudicious I may think it. (232)

The question of existential import really boils down to two questions: what views may logically be held and what views may conveniently be held. The three types of propositions we are considering are those that typically are indicated by 'some' (I), by 'no' (E), and by 'all' (A). (If you are wondering where O is, Carroll treats "Some x are not y" as the I proposition, "Some x are non-y".) He identifies two views he regards as logically consistent.

(1) I and A have existential import, but E does not.
(2) E and A have existential import, but I does not.

His reasoning is as follows.

(1) Suppose that I asserts the existence of its subject. We must then regard A as having existential import as well, since A necessarily contains a proposition in I. Thus if I has existential import, so does A. Are we forced on this view to a particular view of the status of E? Yes. Assume that E has existential import. Then, if "No xy are z" is true, it follows that some things exist that are x and y. But since I has existential import, the same thing follows from "Some xy are z." "No xy are z" and "Some xy are z", however, are contradictories; whence it follows that if I, A, and E all have existential import that it is a necessary truth that some things exist that are x and y (for any x and y, of course), which is absurd. Thus, if I has existential import, so does A; and if I and A have existential import, E cannot have it.

(2) Let us suppose that I does not assert the existence of its subject. Suppose, given this, that E does have existential import. Then "No x are y" indicates that some x exist and none of them are y, i.e., all x are non-y, which is an A proposition. Further, "All x are non-y" proves "No x are y"; the two are equivalent. Thus every proposition in A is equivalent to one in E, and has existential import. So if I does not have existential import and E does, A must have existential import.

(3) But what of the third option, that neither I nor E has existential import? If neither I nor E has existential import, this implies that A doesn't have it, either. But this view wreaks havoc with logical facts. Take the valid syllogism Darapti:

All m are x; All m are y; Therefore, Some y are x.

Since this view requires that we take all these statements as hypothetical, it is equivalent to the following argument:

If there were any m in existence, all of them would be x;
If there were any m in existence, all of them would be y;
Therefore, if there were any y in existence, some of them would be x.

But this is invalid. For there could be a case consistent with the premises in which the conclusion would be false, namely, if y existed but x and m did not. The same problems arise with the syllogisms Disamis, Datisi, Felapton, and Fresison.

But even if you were willing to sacrifice Darapti, Disamis, Datisi, Felapton, and Fresison together just to save the no-existential-import view, this view has other problems. I is taken by virtually everyone to allow conversion, i.e., from "Some x is y" you can infer "Some y is x" and vice versa. But if we make these hypotheticals, this simply cannot be done.

Thus only (1) and (2) are viable approaches. (2), however, has the problem (which Carroll brings out very humorously) of diverging quite considerably from ordinary language. (It requires us to say that, "Some millionaires are in my club" can be true without implying that there are any millionaires at all; and that "No one convicted seven times of forgery are allowed" implies that there are people who have been convicted seven times of forgery.) This is not a logical problem, but makes translation into abstract form extraordinarily tricky. Thus he prefers (1).

(Carroll does recognize that there are other possible views that do not assert straightforwardly one way or another. For instance, one could say that "Some x are y" merely means that x and y are compatible. But this would run into a great many practical inconveniences as well. Further, one could hold that A propositions sometimes have existential import and sometimes don't, which does confessedly appear to have some support in common usage.)

The sharp eye will note that Carroll assumes that A propositions contain I propositions. (Indeed, he holds that all A propositions are really compound propositions. "All dogs are canines" is really the double proposition "Some canines are dogs and no dogs are noncanines.") In his editorial notes, William Warren Bartley III says that Carroll is here begging the question, "which is precisely whether it is indeed convenient to regard propositions in A as 'necessarily containing' propositions in I" (233n). This, of course, is not true. The question is not whether A contains I but whether A and I have existential import, and they are not the same. In modern logic it is always assumed that I must have existential import. But Carroll makes no such assumption, nor would any of the algebraic logicians of the time have taken it to be true without argument. (There's a lovely little summary of the history of this point at Buckner's Logic Museum). Thus in assuming that A contains I he is not assuming that A has existential import, because he does not assume that I has existential import. He thinks, of course, that it is more convenient to take I as having existential import than not, but you will notice that one of his logically viable options is the view on which I does not assert the existence of its subject. Because modern logicians are willing to throw out conversion per accidens and subalternation,, i.e., the view that A propositions contain I propositions in some way, they can hold that A and E lack existential import while I has it; I's having existential import is the point they take to be non-negotiable. But Carroll takes this to be entirely negotiable; what he takes to be non-negotiable is the containment principle. And thus we have a different view.

****
References are to Lewis Carroll, Symbolic Logic, Bartley, ed. Clarkson N. Potter (New York: 1977).

A Poem Draft

Dragon Psalm

Earth shakes, mountains tremble;
they reel at the flaring of wrath,
as water at the flaring flame.
Smoke rises from his nostrils,
a fire pours from his mouth,
kindling stones like coals pour forth.
He touches mountains and they smoke.
Before him goes a devouring fire;
it whirls about him, a mighty tempest.
Hills and stones melt like wax;
all his foes are consumed.

The mountains saw him and quivered;
the raging waters drew back in fear;
the deep gave forth a groaning voice
and the stars stood still in the heavens.
He crushes the head of the wicked;
his arrows of light shoot forth,
his lightning like a glittering spear.

The heavens are shaken, rent,
darkness is under his feet.
He is borne on wings of wind.
The eternal mountains are shattered;
the fragments of hills pave his way.
Before him goes lightning and splendor;
the earth sees and quakes.
He rains down flame and coals of fire,
gives the wicked a scorching wind. Selah.

Drowning fire precedes; it storms around him.
The bed of the sea is uncovered,
the world's foundations are laid bare,
at the Lord's roar, the storm of his breath.

I was drowning in deep waters.
He drew me out and saved me;
he destroyed the demon of the sea.
His wings are wings of morning;
he sets the heavens glowing, aflame.
The mountains bow down before him
that they may declare his justice:
the Lord of hosts is his name. Hallelujah.