Knowledge, Certainty, and Exaggeration

As I explained the other day, I am inclined to accept Butchvarov's view of knowledge as the impossibility of error. If I know that p, then it is not enough that I have a justified true belief that p; I must have a true belief whose justification rules out the possibility of error. Anything short of this is just not knowledge. But then what are we to say about the knowledge claims that people routinely make, claims that that don't come near satisfying this exacting requirement? We won't say that they are mere beliefs, for many of them will be rationally held beliefs. For example, an air traveler who claims to know that he will be in New York tomorrow has a rational belief that will in all probability turn out to be true; but by Butchvarov's lights, a true belief for which one has reasons does not amount to knowledge unless the reasons entail the belief's truth. Since the air traveler's reasons for believing he will be in New York tomorrow do not entail his being there tomorrow, his belief, though rational, is not a case of knowledge. How then do we explain his use of the word 'know'? Should we say that there is a weak sense of 'know' as rational true belief short of certainty?

One idea, also from Butchvarov (The Concept of Knowledge, pp. 54-61), is that the various loose claims of knowledge can be understood as cases of exaggeration. But I'll try to develop this idea in my own way.

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Transitive Sets and the Distinctness of Sets From Their Members

Vlastimil asked for examples of transitive sets.  A transitive set is a set every element of which is a subset of it.  (Hrbacek and Jech, Introduction to Set Theory, p. 50) There is no lack of examples.  The null set vacuously satisfies the condition 'if x is an element of S, then x is a subset of S.'  The set consisting of the null set — {{ }} — is also transitive:  it has exactly one element, the null set, and that element is a subset of it because the null set is a subset of every set.

Now consider the set consisting of the foregoing two sets, the null set and the set consisting of the null set:  {{ }, {{ }}}.  This set has two elements and both are subsets of it.  The null set is a subset of every set, and the set consisting of the null set is also a subset of it in virtue of the fact that the null set is an element of it.

If we identify 0 with the null set, and 1 with the set consisting of the null set, and 2 with the set consisting of the null set and the set consisting of the null set, then 3 will be the set whose elements are the elements of 0, 1, and 2 which is:  {{ }, {{ }}, {{ }, {{ }}}}.  This last set has three elements and each is a subset of it.  One can continue like this and generate as many transitive sets as one likes.  For each natural number there is a corresponding transitive set.

Now how does all this bear upon my assertion that a (mathematical) set is an entity 'over and above' its members (elements)?  That sets are treated in set theory as single items 'over and above' their members can be seen from the fact that some sets have sets as members without having their members as members.   The power set of {Socrates, Plato} has {Socrates} and {Plato} as members, but it does not have Socrates and Plato as members. Therefore, {Socrates} is distinct from Socrates, and {Plato} from Plato.  For if these singletons were identical to their members, then the power set would have Socrates and Plato as members.

Vlastimil seems to think that the existence of transitive sets is somehow at odds with the claim that sets are distinct from their members.  Or perhaps he thinks that some sets are distinct from their members and some are not.  So consider {{ }, {{ }}}.  This is a transitive set since every member of it is a subset of it, which is equivalent to saying that every member of a member of it is a member of it.  Thus { } is a member of {{ }}, which is a member of {{ }, {{ }}}.  But although every member of the set in question is a subset of it, this does not alter the fact that the set is distinct from its members.

So I'm not sure what Vlastimil is driving at.

Note that if every member of a set is a subset of it, this is not to say that every subset of it is a member of it.  {{ }, {{ }}} has itself and {{{ }}} as subsets but not as elements.  Only if there were a set all of whose members are subsets of it and all of whose subsets are members of it could one argue that there are sets for which the membership and subset relations collapse, and with it the distinction between a set and its members.

Butchvarov: Knowledge as Requiring Certainty

Butch We begin with an example from Panayot Butchvarov's The Concept of Knowledge, Northwestern University Press, 1970, p. 47. [CK is the red volume on the topmost visible shelf.  Immediately to its right is Butch's Being Qua Being.  Is Butch showing without saying that epistemology is prior to metaphysics?] There is a bag containing 99 white marbles and one black marble. I put my hand in the bag and without looking select a marble. Of course, I believe sight unseen that the marble I have selected is white. Suppose it is. Then I have a justified true belief that a white marble has been selected. My belief is justified because of the fact that only one of the 100 marbles is black.  My belief is true because I happened to pick a white marble.  But surely I don't know that I have selected a white marble.  The justification, though very good, is not good enough for knowledge. I have justified true belief but not knowledge.

Knowledge, says Butchvarov, entails the impossibility of mistake. This seems right. The mere fact that people will use the word 'know' in a case like the one described cuts no ice.  Ordinary usage proves nothing.  People say the damndest things.  They are exaggerating, as a subsequent post may show.  'Know' can be used in non-epistemic ways — think of carnal knowledge for example — but used epistemically it can be used correctly in only one way: to mean absolute impossibility of mistake.  Or as least that is Butchvarov's view, a view I find attractive.

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Would Schopenhauer Allow Comments?

Schopenhauer If Schopenhauer were a blogger, would he allow comments on his weblog, The Scowl of Minerva?

I say no, and adduce as evidence the following passage that concludes his Art of Controversy, a delightful essay found in his Nachlass, but left untitled by the master:

As a sharpening of wits, controversy is often, indeed, of mutual advantage, in order to correct one's thoughts and awaken new views. But in learning and in mental power both disputants must be tolerably equal: If one of them lacks learning, he will fail to understand the other, as he is not on the same level with his antagonist. If he lacks mental power, he will be embittered, and led into dishonest tricks, and end by being rude.

The only safe rule, therefore, is that which Aristotle mentions in the last chapter of his Topica: not to dispute with the first person you meet, but only with those of your acquaintance of whom you know that they possess sufficient intelligence and self-respect not to advance absurdities; to appeal to reason and not to authority, and to listen to reason and yield to it; and, finally, to cherish truth, to be willing to accept reason even from an opponent, and to be just enough to bear being proved to be in the wrong, should truth lie with him. From this it follows that scarcely one man in a hundred is worth your disputing with him. You may let the remainder say what they please, for every one is at liberty to be a fool – desipere est jus gentium. Remember what Voltaire says: La paix vaut encore mieux que la verite. Remember also an Arabian proverb which tells us that on the tree of silence there hangs its fruit, which is peace.

Here is the same passage in the German original:

Das Disputieren ist als Reibung der Köpfe allerdings oft von gegenseitigem Nutzen, zur Berichtigung der eignen Gedanken und auch zur Erzeugung neuer Ansichten. Allein beide Disputanten müssen an Gelehrsamkeit und an Geist ziemlich gleichstehn. Fehlt es Einem an der ersten, so versteht er nicht Alles, ist nicht au niveau. Fehlt es ihm am zweiten, so wird die dadurch herbeigeführte Erbitterung ihn zu Unredlichkeiten und Kniffen [oder] zu Grobheit verleiten.

Die einzig sichere Gegenregel ist daher die, welche schon Aristoteles im letzten Kapitel der Topica gibt: Nicht mit dem Ersten dem Besten zu disputieren; sondern allein mit solchen, die man kennt, und von denen man weiß, daß sie Verstand genug haben, nicht gar zu Absurdes vorzubringen und dadurch beschämt werden zu müssen; und um mit Gründen zu disputieren und nicht mit Machtsprüchen, und um auf Gründe zu hören und darauf einzugehn; und endlich, daß sie die Wahrheit schätzen, gute Gründe gern hören, auch aus dem Munde des Gegners, und Billigkeit genug haben, um es ertragen zu können, Unrecht zu behalten, wenn die Wahrheit auf der andern Seite liegt. Daraus folgt, daß unter Hundert kaum Einer ist, der wert ist, daß man mit ihm disputiert. Die Übrigen lasse man reden, was sie wollen, denn desipere est juris gentium, und man bedenke, was Voltaire sagt: La paix vaut encore mieux que la vérité; und ein arabischer Spruch ist: »Am Baume des Schweigens hängt seine Frucht der Friede.«

Why Not Stick to Pure Philosophy?

I ask myself this question.

Why not stick to one's stoa and cultivate one's specialist garden in peace and quiet, neither involving oneself in, nor forming opinions about, the wider world of politics and strife? Why risk one's ataraxia in the noxious arena of contention? Why not remain within the serene precincts of theoria? For those of us of a certain age the chances are good that death will arrive before the barbarians do.

Those in the arena may be admired for their courage, but doubts arise as to their wisdom.

So why bother one's head with the issues of the day? We will collapse before the culture that sustains us does. The answer is that the gardens of tranquillity and the spaces of reason are worth defending, with blood and iron if need be, against the barbarians and their leftist enablers. Others have fought and bled so that we can live this life of solitude and beatitude. And so though we are not warriors of the body, we can and should do our tiny bit as warriors of the mind to preserve for future generations this culture which allows us to pursue otium liberale in peace, quiet, and safety.

Weakness Does Not Justify

Might does not make right, but neither does impotence or relative weakness. That weakness does not justify strikes me as an important principle, but I have never seen it articulated. The power I have to kill you does not morally justify my killing you. In a slogan: Ability does not imply permissibility.  My ability to kill, rape, pillage & plunder does not confer moral justification on my doing these things.  But if you attack me with deadly force of magnitude M and I reply with deadly force of magnitude 10 x M, your relative weakness does not supply one iota of moral justification for your attack, nor does it subtract one iota of moral justification from my defensive response.  If I am justified in using deadly force against you as aggressor, then the fact that my deadly force is greater than yours does not (a) diminish my justification in employing deadly force, nor does it (b) confer any justification on your aggression.

Suppose a knife-wielding thug commits a home invasion and attacks a man and his family. The man grabs a semi-automatic pistol and manages to plant several rounds in the assailant, killing him. It would surely be absurd to argue that the disparity in lethality of the weapons involved diminishes the right of the pater familias to defend himself and his family.

The principle that weakness does not justify can be applied to the Israeli-Hezbollah conflict from the summer of 2006 as well as to the current Israeli defensive operations against the terrorist entity, Hamas.  The principle ought to be borne in mind when one hears leftists, those knee-jerk supporters of any and every 'underdog,' start spouting off about 'asymmetry of power' and 'disproportionality.'

Which Is More Certain: God or My Hand?

A reader inquires, " I'm curious, if someone asked you what you were more certain of, your hand or belief in the existence of God, how would you respond?"

The first thing a philosopher does when asked a question is examine the question.  (Would that ordinary folk, including TV pundits, would do likewise before launching into gaseous answers to ill-formed questions.)  Now what exactly am I being asked?  The question seems ambiguous as between:

Q1. Are you more certain of the existence of your hand or of the existence of God?

Q2. Are you more certain of the existence of your hand or of your belief in the existence of God? 

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Rod Serling Rules: Twilight Time Again

Serling The semi-annual Twilight Zone marathon is under way at the Sci Fi channel and will continue through New Year's Day and into the wee hours of January 2nd. Here is your chance to view some of the episodes you may have missed.  The best of them are phenomenally good and bristling with philosophical content. I have just given you my analysis of "The Lonely" which aired in November, 1959.  I just now viewed the The Dummy for the nth time, and I note that the ascriptivist theory of personhood I mentioned in my analysis of "The Lonely" also figures in "The Dummy."

The original series ran from 1959 to 1964. In those days it was not uncommon to hear TV condemned as a vast wasteland. Rod Serling's work was a sterling counterexample.

The hard-driving Serling lived a short but intense life. Born in 1924, he was dead at age 50 in 1975. His four pack a day cigarette habit destroyed his heart. Imagine smoking 80 Lucky Strikes a day! Assuming 16 hours of smoking time per day, that averages to one cigarette every twelve minutes.  He died on the operating table during an attempted bypass procedure.

But who is to say that a long, healthy life is better than a short, intense one fueled by the stimulants one enjoys? That is a question for the individual, not Hillary, to decide.

Philosophy From the Twilight Zone: “The Lonely”

Alicia and corry Rod Serling's Twilight Zone was an outstanding TV series that ran from 1959-1964. The episode "The Lonely" aired in November, 1959. I have seen it several times, thanks to the semi-annual Sci Fi channel TZ marathons. There is one in progress as I write.  One can extract quite a bit of philosophical juice from "The Lonely" as from most of the other TZ episodes. I'll begin with a synopsis.

Synopsis.James A. Corry is serving a 50 year term of solitary confinement on an asteroid nine million miles from earth. Supplies are flown in every three months. Captain Allenby, unlike the other two of the supply ship's crew members, feels pity for Corry, and on one of his supply runs brings him a female robot named 'Alicia' to alleviate his terrible loneliness. The robot is to all outer appearances a human female. At first, Corry rejects her as a mere robot, a machine, and thus "a lie." He feels he is being mocked. "Why didn't they build you to look like a machine?" But gradually Corry comes to ascribe personhood to Alicia. His loneliness vanishes. They play chess with a set he has constructed out of nuts and bolts. She takes delight in a Knight move, and Corry shares her delight. They beam at each other.

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The End of Moderation

Haecker Theodor Haecker, Journal in the Night (Pantheon, 1950, tr. Dru), p. 29:

Many a man thinks to satisfy the great virtue of moderation by using all his shrewdness and bringing all his experience to bear upon limiting his pleasure to his capacity for pleasure. But simply by the fact of setting enjoyment as the end, he has radically violated the virtue.

A penetrating observation.  What is the end or goal of moderation? Haecker is rejecting the notion that the purpose of moderation, conceived as a virtue, is to maximize the intensity and duration of pleasure. Of course, moderation can be used for that end — but then it ceases to be a virtue. For example, if I am immoderate in my use of alcohol and drugs, I will destroy my body, and with it my capacity for pleasure. So I must limit my pleasure to my capacity for pleasure. And the same holds for immoderation in eating and sexual indulgence. The sex monkey can kill you if you let him run loose. And even if one's immoderation does not lead to an early death, it can eventuate in a jadedness at odds with enjoyment. So moderation can be recommended merely on hedonistic grounds. The true hedonist must of necessity be a man of moderation. If so, then the ill-starred John Belushi, who took the 'speedball' (heroin + cocaine) express to Kingdom Come, did not even succeed at being a very good hedonist.

But if enjoyment is the end of moderation, then moderation as a virtue is at an end. Haecker, however, does not tell us what the end of moderation as a virtue is. He would presumably not disagree with the claim that the goal of moderation as a virtue is a freedom from pleasure and pain that allows one to pursue higher goods. He who is enslaved to his lusts his simply not free to pursue a truer and higher life.

For the New Year: Looking Away Shall Be My Only Negation

Nietzsche

One of the elements in my personal liturgy is a reading of the following passage every January 1st. I must have begun the practice in the mid-70s. My copy of The Gay Science was purchased in Boston and is dated 15 September 1974. (You mean to tell me that when you buy books, you do not note where you bought them, and when, and in whose presence?)

Friedrich Nietzsche, The Gay Science, Book Four, #276, tr. Kaufmann:

For the new year. — I still live, I still think: I still have to live, for I still have to think. Sum, ergo cogito: cogito, ergo sum. Today everybody permits himself the expression of his wish and his dearest thought: hence I, too, shall say what it is that I wish from myself today, and what was the first thought to run across my heart this year — what thought shall be for me the reason, warranty, and sweetness of my life henceforth. I want to learn to see more and more as beautiful what is necessary in things; then I shall be one of those who makes things beautiful. Amor fati: let that be my love henceforth! I do not want to wage war against what is ugly. I do not want to accuse. Looking away shall be my only negation. And all and all and on the whole: someday I wish to be only a Yes-sayer.

(Amor fati: love of fate.)

On the Elusive Notion of a Set: Sets as Products of Collectings

In an important article, Max Black writes:

Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should be child's play. ("The Elusiveness of Sets," Review of Metaphysics, June 1971, p. 615)

1. A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set.   A set with two or more members is not identical to one of its members, or to each of its members, or to its members taken together, and so the set  is distinct from its members taken together, though not wholly distinct from them: it is after all composed of them and its very identity and existence depends on them.

In the above quotation, Black is suggesting that mathematical sets are contradictory entities: they are both one and many.  A set is one in that it is a single item 'over and above' its members or elements as I have just explained.  It is many in that it is "wholly constituted" by its members. (We leave out of consideration the null set and singleton sets which present problems of their own.)  The sense in which sets are "wholly constituted" by their members can be explained in terms of the Axiom of Extensionality: two sets are numerically the same iff they have the same members and numerically different otherwise. Obviously, nothing can be both one and many at the same time and in the same respect.  So it seems there is a genuine puzzle here.  How remove it?

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Non-Empty Thoughts About the Empty Set

1. The empty or null set is a strange animal. It is a set, but it has no members. This is of course not a contingent fact about it, but one bound up with its very identity: the null set is essentially null. Intuitively, however, one might have thought that a set is a group of two or more things. Indeed, Georg Cantor famously defines a set (Menge) as "any collection into a whole (Zusammenfassung zu einem Ganzen) of definite and separate objects of our intuition and thought." (Contributions to the Founding of the Theory of Transfinite Numbers, Dover 1955, p. 85) In the case of the null set, however, there are no definite objects that it collects. So in what sense is the null set a set? One might ask a similar question about singletons, sets having exactly one member. But I leave this for later.

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