Kant and Mathematics

This is an excerpt from a paper in which I argue that in the first Critique Kant does not give good reasons for thinking that the truths of mathematics are synthetic a priori.

Mathematical judgments, Kant states, are an example of synthetic a priori judgments (A 10/B 14). Judgments such as “5 + 7 = 12” and “a straight line is the shortest distance between two points” are a priori, since they are both necessary and universal. However, according to Kant they are also synthetic. He states that the concept “twelve” cannot be arrived at solely by examining the concepts of “five,” “seven,” and “addition,” and the concept “shortest distance between two points” cannot be found in the concept “straight line” (B 16). In both cases, Kant argues, we must appeal to an intuition in order to the truth of the judgment. When we add five and seven, we must have recourse to calculating aids such as our fingers, or the visualization of discrete points, etc. (B 15). And in the case of the straight line or other geometrical truths, we must have recourse to intuitions in order to arrive at the correct answer.

But what exactly does Kant mean by appealing to an intuition in this context? As he explains at A 19/B 33, by “intuition” he is specifically referring to the means by which a cognition relates to an object when the mind is presented with that object. In the case of mathematical truths, it is these kinds of intuitions (e.g. the processes of counting or of constructing a geometric figure, cf. Dicker (2004) 27) that allow us to see the truth or falsehood of our judgments, rather than an examination of the concepts involved (Guyer (2006) 61). So an intuition is something over and above the particular concepts involved in a judgment.

In the case of the straight line being the shortest distance between two points, Kant states that the concept of straight “contains nothing of quantity, but only a quality” (B 16), and concludes that this judgment is synthetic. However, a closer examination of the concept “straight line” will reveal the judgment to be analytic. Euclid defines a line as “breadthless length” (Elements I def. 1), and a straight line as “a line which lies evenly with the points on itself (Elements I def. 4); in other words, a straight line is a magnitude in one dimension “which represents the extension equal with (the distances separating) the points on it” (Heath (1956) 166). In addition, Kant’s contemporary Legendre followed Archimedes in simply defined a straight line as the shortest distance between two points (Heath (1956) 169). So the judgment that a straight line is the shortest distance between two points turns out to be analytic after all.

In the case of “5 + 7 = 12,” Kant states that the concept “twelve” cannot be arrived at simply by examining the concepts “five,” “seven,” and “addition,” but requires a process of counting. But it does not seem to me that the process of counting “goes beyond” (B 15) the concepts of the particular numbers and the concept of addition, but that it makes explicit what is already contained in the concepts involved. The concepts “five” and “seven” surely include the concept “magnitude,” and the combination of two particular magnitudes must yield another particular magnitude; in this case, the magnitude that we happen to denote by the concept “twelve.” It seems that e.g. when we represent the concept “five” to ourselves, we unavoidably do so by imagining a certain number of distinct objects (even if those objects are components of another object, as if for instance we represent “five” by visualizing a pie with five slices). I suggest that what Kant calls an “intuition” in this and in similar cases of counting is nothing more than the act of examining the concept of a particular number. That we require this kind of representation in order to perform calculations should be understood as an indication of the genesis of our concepts of numbers (or perhaps an indication of the limits of our particular cognitive capabilities), rather than a proof that mathematical knowledge must be arrived at with the aid of intuitions that go beyond the concepts involved.

Not Getting It

Socrates is my hero. That probably goes for most philosophers. I was a bit surprised at the degree of grief I suffered at reading student essays in which a sizable portion only understand argument as a rhetorical method of persuasion among others. In essay after essay, students failed to get that Socrates is not trying to win in the Apology through rhetoric - he has the option of doing so, but considers it shameful. Maybe it's too much to expect that the students actually read the Apology and noted the importation message of the dialogue that non-rational persuasion is different from persuasion by reason.

Grading is my main interaction with the students (as TA), so I won't have much chance to try to get them to really get Socrates. But it's early in the course and it is (mostly) their first course in philosophy, so after reading more Plato, maybe they'll get it. But if nothing else, I've gotten one of the student errors I want to make sure I address when it's my turn to teach Plato/Socrates to undergraduates.

On Authenticity

A former classmate of mine, Matthew Anderson, recently written an article which raises the issue of authenticity. I'm going to take a stab at the subject, but I honestly haven't thought in great detail about the subject, and this is perhaps my first stab at it. Hopefully I'll better understand how authenticity, if a virtue of any sort fits into the ordinary network of duties, values, and virtues after further thought. But here goes - Anderson writes:

Fewer words find more frequent use among young evangelicals than “authentic.” Barack Obama and Mike Huckabee resonated with young evangelicals for the same reason: they appeared authentic in their positions and their mindsets... And while authenticity has a political dimension, it is also a social virtue. Young evangelicals frequently decry the inauthenticity of the mega-church and individualistic evangelical tradition, where people put on a happy face and “played church.” Experiencing “real life together” is the pursuit of the new evangelical small group, where “real life” is always “messy.” Authenticity in social settings is frequently an excuse for sharing sins and problems within a group of people who doubtlessly share the same sins and problems.

Anderson has as his target an issue among a certain group of people, and I don't doubt that he's right about it. My experience has been that this is far from an issue among evangelicals alone. It's a common theme in movies and when I've had deeper conversations with undergraduates and peers outside the academy, it's not infrequently something that comes up. If anything, I suspect that it's currency among evangelicals today has to do with osmosis from the rest of our culture. I suppose that there's an interesting history to the concept - I suspect that it's currency derives from trickle-down existentialism, but I'm far from positive about that - and it really deserves study.

I'm often inclined to get a bit restless when talk of "authenticity" or "being real" comes up. One of the reasons is the nebulousness of the term. I'm never quite sure what qualifies as being authentic. However, there does appear to be some genuine value to authenticity liable both to inflation and deflation.

The sense of the term I have in mind is that one is authentic if one acts honestly. One fails to be authentic if one talks about traditional values but belies those values by the hookers one is secretly meeting with on the side. One fails to be authentic if one hides feelings or desires because of how it will be perceived (being closeted is a failure of authenticity). Part of the concept appears to have to do with not realizing one's desires for the sake of appearance and part of it has to do with publicly speaking of high standards, but actually acting according to a set of lower standards. In brief, one doesn't act honestly. One intentionally acts in a way so that you communicate to others that you are a kind of person that you aren't.

Here's where it's important not to deflate the value of authenticity. The complement of authenticity is a kind of social fraud. Fraud undercuts human relationships extremely effectively, and I'm told that non-familial relationships are paramount for my generation and those following it. Given that virtuous friendships are of great value to having a flourishing life and that social fraud seems to undercut that value, there's something pretty important about authenticity. It may be that if there are such things as "intrinsic goods" of practices (in MacIntyre's), that authenticity is an important precondition for realizing whatever goods are intrinsic to the practice of friendship. Dante seemed to realize this in his placing fraud at the very deepest level of hell. Subverting relationships generated the worst sins in Dante's taxonomy.

However, there's also reason to think that the value of authenticity is often overblown. Firstly, authenticity seems to be mainly a precondition for having other values. If I'm socially honest, then I can participate in enriching relationships with others in a way that I might not otherwise have done. However, authenticity in the sense I'm concerned with is without much value if the person doesn't have many virtues. Authenticity is in some sense a bit cheap. After all, I can be fully authentic without many of the other virtues. An authentic rascal is still a rascal. If he's not authentic, then he's a hypocritical rascal, but removing the hypocrisy doesn't make him less than a rascal. And there does seem to be a bit of a problem here for people like me who are moral realists of some sort or another with authenticity. Authenticity is sometimes the only virtue that undergrads appear to think isn't relative or subjective. In such cases, I get the impression that one can be taken to be socially virtuous so long as one is honest about how deeply bereft one is of other virtues (paradoxically, I suppose that there's a kind of inauthenticity in this).

If the other virtues are whittled away so that only authenticity remains, then one is poor indeed. But, it's also true that adding authenticity to other valuable traits is, as Dante seemed aware of, a precondition for certain sorts of social moral goods.

Ancient Greek Paleontology

This blog is not dead, only dormant.

It's always interesting when something about fossils shows up in antiquity. I was reading some fragments about Xenophanes (b. abt. 570 B.C.) and the following showed up:

He believes that earth is being mixed into the sea and over time it is being dissolved by the moisture saying that he has the following kinds of proofs, that sea shells are found in the middle of the earth and in mountains, and the impressions of a fish and seals have been found at Syracuse in the quarries, and the impression of a laurel leaf in the depth of the stone in Paros, and on Malta flat shapes of all marine life. He says that these things occurred when all things were covered with mud long ago and the impressions were dried in the mud.

- Hippolytus, Refutation (from Cohen, Curd, Reeve, Readings in Ancient Greek Philosophy)

What's interesting here is that Xenophanes knew about fossils, had a vaguely correct notion of what they were (impressions of ancient plants and animals), and used them to formulate theories about prehistory and certain sorts of natural processes. His suppositions were wildly inaccurate, but it was a good early start for paleontology. Additionally, though the "proofs" cited appear to all be hearsay, they do accord with certain kinds of fossils. I wonder whether these were known via rumors passed about in the marketplace or if there were (since it was the dawn of natural science and philosophy in Greece - a self-conscious effort to collect this kind of information among those involved). 

I wonder if there are a) earlier reports of fossils in antiquity and b) if Xenophanes was the first to draw paleontological sorts of conclusions from them. May we rightly call Xenophanes the first paleontologist?

-Bill

Gettiered Development

Once one gets familiar with the Gettier problem, one finds it all over the place. Occasionally, one finds it in popular media, like TV. I happened to find what seems to me a very good case of it in the sitcom Arrested Development.

Unfortunately, if one's not familiar with the show, the clip may require some background. The Bluth family is a wealthy family fallen on hard times. George Bluth is on the lam, while his wife Lucille is living with his twin brother Oscar (distinguishable from his fugitive brother by his head of gray hair). Buster, who is actually the son of an illicit liaison between Lucille and Oscar (but does not know this) has been signed up for the Army by Lucille when challenged by a Michael Moore lookalike to do so, and is about to be deployed to Iraq. George has come to see Buster (since he too believes that Buster is his son), wearing a wig as a disguise so that he won't be arrested on sight. The key things to note is that Oscar is Buster's father, and George appears to be Oscar.

Buster believes, on the word of George, whom he takes to be Oscar, that Oscar is his father. His belief is true. His belief also appears to be justified. He sees a man that looks just like Oscar, testifying to him that he is his father. 

It seems pretty obvious that at this point, Buster doesn't actually know that Oscar's his father, but he has a justified true belief that he is. 

Unconscious Inference

I received internalism about justification in my mother's milk, so to speak. Most of my earliest philosophical teachers were internalists in this way, and I have usually thought that this is about right when it comes to justification, despite my flirtations with proper-functionalism.

Recently, I have been more latitudinarian in my internalism - favoring what's been termed "mentalism" by Conee and Feldman in Evidentialism. Mentalism is a kind of bare internalism, it holds that what's justified for me to believe supervenes on my mental states. Thus, a brain-in-a-vat can have all the same justified beliefs that I have if he has all of the same mental states I do. That sounds about right to me (pace externalism about content worries). The upside of this form of internalism is this: my grounding for certain beliefs may not be easily accessible to me in the way that classical internalism envisions.

Philosophers often think (or it seems so) of the thinker as an intellectual, and perhaps even a caricature of an intellectual: a person who comes to all of their beliefs through conscious deliberation. But really, if I survey my own mind, and others seem to avow this, most of my beliefs are not formed this way: I have flashes of insight about theoretical matters when I'm working on something else (maybe after awaking from sleep, in the shower, while I'm working on an unrelated topic), I seem to make inductive inferences without going through any sort of explicit process (I see the groceries on the floor and immediately believe that my wife is home), I act in a way that shows that I have some sort of pro-attitude toward some proposition without it ever crossing my mind that I do (especially in skilled activity - like driving - I evade a road hazard without even thinking about it), etc. 

The basing relation with mentalism points the way toward explaining how we have justified belief (or some other doxastic/quasi-doxastic state) in these cases. Various mental processes, those I don't have direct control over dispose me to form the correct attitude from my evidence without the slightest conscious thought about it. These seem to be operating when I'm thinking about other things, when I'm asleep, when I'm driving, and so on. Often the only indication of these processes are when they seem to 'pop' into my conscious mental landscape, but even then, like in the case of driving, I only seem to be aware of them from the way my mind has directed my behavior.

Anyway, this seems interesting and bears more thought.

Content Externalism

(So, this is musing at this point... I'd love to be corrected by those who understand this issue better, especially if I'm making any simple mistakes due to my ignorance in this area. But blogs are good for this - exploring and figuring out when one has something seriously in error.)

I'm currently reading through Timothy Williamson's Knowledge and Its Limits[1] and one of the crucial issues in the book is Williamson's adoption of an "externalist" theory of mental content. I wish to muse a bit on such theories, since I am undecided about what the correct attitude toward externalism of this sort ought to be. I think Williamson is right in at least so far as he sees that the adoption of externalism about content will require some serious rethinking of traditional epistemology. 

Externalism has been typically argued for via "Twin Earth"-style thought experiments. I'll adapt for my own use one given by Tyler Burge. [2] Al is a fairly resident of Earth. Al has a variety of beliefs about aluminum. "He thinks that aluminum is a light, flexible metal, that many sailboat masts are made out of aluminum, that someone across the street recently bought an aluminum canoe."[3] Suppose that there is a duplicate of Al living on Twin Earth, a planet in every way like our own with one important exception, what plays the functional role of aluminum on Twin Earth. When twin-Al (call him 'Twal') runs his hand over twin-aluminum (call it 'twaluminum') he has just the same sorts of tactile stimuli as when Al runs his hand over aluminum. It weighs the same in each's hand, it reflects light in just the same way. Twal also calls twaluminum 'aluminum'. So, it would seem when Twal thinks of twaluminium, it's hard to see how his internal concept of twaluminum differs from Al's concept of aluminum. This is made more explicit on supervenience theses. If mental states, including intentional propositional attitudes, supervene on brains, then surely the mental state that Twal and Al are in are precisely the same. But if Al asserts, "This is a fine aluminum canoe" and Twal asserts, "This is a fine aluminum canoe" they surely are saying something different. Twal has never encountered aluminum and Al has never encountered twaluminum. But additionally, how could they be saying something different? If their mental act of asserting some proposition with those words supervenes on precisely the same (ex hypothesi) type of brain state, then surely they are making the same kind of assertative (sp?) act.

I think that it's intuitive, even with the rejection of a supervenience thesis about mind, that this problem occurs. I think that one of the central issues in these externalist thought experiments is that Al's and Twal's experiences underdetermine what it is that their intentional states are of. Suppose I don't have any beliefs about what the microstructure of aluminium is. All I know about it are the typical descriptive characteristics that accompany aluminum. When I am introduced to it, suppose that someone says "That's aluminum" and points at it. My experience underdetermines whether or not what I am seeing is aluminum or twaluminum. Every subsequent encounter with aluminum will be similar, because there is nothing that will tell the two metals apart without very close examination (let's say requiring a laboratory). So, how could my mental content about aluminum be different from my counterpart on twin-earth who has just the same sorts of experiences... someone points out to him twaluminum saying "That's twaluminum." He goes on to have the same sorts of experiences I do, but in every case with twaluminum instead of aluminum. What grounds do I have for saying that his concept of twaluminum differs in the least from my concept of aluminum? On the classical sense-datum conception of mental content, there doesn't seem to be any reason to think that there could be any difference between Al's and Twal's concepts, because all of the sensible properties that Al and Twal are aware of are exactly similar phenomenologically. 

The solution that has generally been recommended is externalism about content. That a belief is about a given proposition is not only determined by the thinker, but by the thinker's environment. Putnam seemed to think it was the local linguistic community. It might just as well be determined by causal contact with the things "out there." 

What makes this somewhat disturbing for the traditional epistemologist is something like this: I find the notion that epistemology is a first-person enterprise attractive and think that there are some propositions that I am the expert on, probably to the point of infallibility. If I am in intense pain, there is no way that I can be wrong about thinking that I am. If I am thinking about giraffes, there is some part of that thought that is simply not in doubt. But if content externalism is true, this first-person privelege seems somewhat fantastical. I cannot tell, just by inspecting my own thoughts what I am thinking about. I may not even be able to tell what propositional attitude I am in, if any. For instance, belief requires a proposition, at least on a standard interpretation (we do sometimes talk of believing in a person - but let's set those cases aside). Suppose that propositions have their subjects as constituents, so when I mentally token the proposition expressed by "That canoe is made of aluminum", I express a proposition that has a canoe as a constituent. Suppose when I take myself to token that proposition and believe it, I do it in a case where I do not actually see a canoe, but am the victim of an illusion. Thus, we a case similar to a Frege-puzzle. But unless propositions can have gaps (and they might), then there is properly no proposition that I am believing, so though it seems to me that I am believing a proposition, I am NOT. So, it would seem that I'm not in a believing state at all. But that seems batty in the extreme - how could I be mistaken about that kind of thing? 

[1] Williamson, Timothy, Knowledge and Its Limits...

[2] Burge, Tyler, "Two Thought Experiments Reviewed", Notre Dame Journal of Formal Logic, Vol 23.3 July 1982

[3] Burge, 284.

Stoic Views on Value

In EN 1096a2, Aristotle maintains that no one would call a virtuous person happy if she were suffering great misfortunes, except to defend a “philosopher’s paradox.” The Stoics, however, embrace this paradox by insisting that virtue is sufficient for happiness. But unlike Aristo and his followers, they also hold that so-called external goods, while not choiceworthy in themselves, do have a kind of value in that they are to be preferred to their opposites. However, it remains to be seen whether the Stoics can consistently maintain these two views of value without their position collapsing into either the Aristotelian view or the view of Aristo. It is this tension that Cicero highlights in the dilemma he poses to Cato in De Fin. III.10-12. I will argue that Cato’s subsequent attempt to reconcile these two commitments ultimately fails.

Cato states that what is valuable is that which is “either itself in accordance with nature, or brings about something that is” (III.20) and that the only things worth choosing are things that are valuable. He then describes “appropriate actions” as being impulses directed towards selecting what is in accordance with nature: initially as infants we seek to preserve our own existence, but as we become able to exercise our reason we see an overall order and pattern in the world that we come to seek for its own sake. Thus, we discover that acting virtuously, i.e. acting in accordance with the order and pattern of nature, is the only thing that is truly valuable and thus truly choiceworthy. Virtue is necessary and sufficient for eudaimonia, and all else is indifferent. But of the things that are indifferent, some such as health and beauty are in accordance with nature, and are thus to be preferred over their opposites such as sickness. It seems to me as though Cato is saying that virtue is the only thing we need in order to be happy, but if we are presented with the option of either having or not having certain external goods, we will take those things that are in accordance with nature. So, the Stoic sage will not go out of his way to obtain health and money, but if they happen to come his way he will not reject them. This view of virtue, taken with the Stoic belief that virtue is within our power, shows that for the Stoics happiness is something that is completely within our power, and not dependent on things such as external goods that are often beyond our control.

Cato sees the potential difficulty in making a distinction between what is to be chosen and what is to be selected, and at III.22-24 he tries to illustrate how the Stoics can maintain this distinction while still maintaining that virtue is the only thing that is ultimately valuable. His example of archery is rather confused (why else does the archer practice his craft, if not for the sake of hitting the mark?) but his examples of acting and dancing seem to work better, since they accord somewhat with his account of virtue in that the craft is practiced for its own sake, and not for the sake of some external object. But even dancing and acting are not perfect examples, Cato says, because their success can depend partially on what is outside of the dancer’s or actor’s control, such as the audience’s reaction.

However, it seems to me that the distinction between what is choiceworthy and what is selected cannot be consistently maintained. Earlier in III.20 Cato stated that what is in accordance with nature is valuable, and thus choiceworthy. But at III.44, Cato admits that health has some value, although he does not admit that it is a “good.” He seems to be saying that the value of heath and the value of virtue are different in kind, even if his comparisons of them at III.45 seem to imply a difference in degree. But he is faced with a dilemma: either health is in fact in accordance with nature, and thus is valuable and choiceworthy, or else it is not choiceworthy, and thus can be neither valuable nor in accordance with nature, and thus there would be no grounds for preferring it to sickness. As it is, Cato seems to want to say that health is both in accordance with nature, and yet not choiceworthy, which is inconsistent with his own stated system of value.

A Mereological Argument for Theism

This argument, as far as I know, is novel. It mirrors an argument for the existence of God in Richard Swinburne's The Existence of God, which argues for the existence of God from otherwise inexplicable brain and mental event correlations.

Consider the Special Composition Question: "What necessary and jointly sufficient conditions must any xs satisfy in order for it to be the case that there is an object composed of the xs?"[1]

Ned Markosian has argued that every answer yet suggested for answering this question has failed. He argues that we ought to take composition to be brute: "There is no true, non-trivial, and finitely long answer to SCQ." [2] Let's agree with Markosian that all previous answers to this question have failed and hold off on endorsing his answer to the question.

Suppose however, that none of the views above work, but it is still the case that the some xs are such that the xs compose some y but some zs are such that they do not compose some w. That is to say, some, but not all mereological simples compose something. There should be some fact of the matter about this too. In theory, we should be able to draw up a list of all of the composite objects and their simples with their properties. On brutalism, there is not going to be any explanation for why these simples with these properties compose something, rather than not. The set of correlated xs with their properties and composite ys will be brute, with no explanation.

Brute explanations are only epistemically permissible if there is no uneliminated explanation. [4] I submit that there is an uneliminated explanation. The correlation of xs with their properties and composite ys has a personal cause, a la Swinburne. [5] In fact, whether or not humans are composite physical objects, they will be unable to make it the case that these correlations are true. This is easy to see, there was a time before the first human. It is not plausible to think that the advent of humans started making trees out of xs shaped tree-wise. So, if there is a personal explanation, then it is not a human person. Furthermore, the personal cause must be a mereological simple himself otherwise, he would be insufficient to explain his own composition out of some xs. God fits this description and it is more plausible that the correlations of simples and their properties and the objects they compose are the result of the action of a divine personal cause than that they are brute.

So,

1. There are correlations between simples and composite objects.

2. If there are correlations of this sort, then said correlations are either explained or they are brute.

3. Said correlations are either explained or they are brute. (MP 1, 2)

4. If the correlations are explained, they are either explained by some form of Contact (and all strengthened forms, up to Fusion), Life-ism, Nihilism, Universalism or have a personal explanation.

5. It is false that they are explained by some form of Contact, Life-ism, Nihilism, or Universalism. (Not argued here, between van Inwagen and Markosian every one of these is dispatched). [6]

6. If the correlations are explained, they have a personal explanation.

7. Said correlations either are explained by a personal explanation or they are brute.

8. It is more probable that there is a personal explanation than that they are brute.

9. If there is a personal explanation, then there is a God. (see argument above).

10. It is more probable that there is a God than that the correlations are brute.

Even if this argument works, it has a modest conclusion. As with Swinburne's argument from mental/physical event correlations, it works best as part of a cumulative case.

I'm interested to hear what people think. I have my doubts about premises 5 and 8.

[1] Markosian, Ned, "Brutal Composition" Philosophical Studies 92: 211-249, 1998., 212

[2] Ibid., 214.

[3] Ibid., 223.

[4] I take it that this is plausible. However, I haven't the foggiest idea how to argue for it.

[5] Swinburne, Richard The Existence of God, 2nd Edition(Clarendon Press: Oxford), 2004, 35-51.

[6] van Inwagen, Peter, Material Beings, ( ), 1990.

On the Nature of Numbers

The following definition of the number 2 has been brought to my attention: "[t]he set of all pairs." Against this definition, I raise the following objections:

I. It seems viciously circular: if 2 is the set of all pairs, what are pairs other than a set of two objects? The same objection seems to hold for defining 3 as the set of all triples, etc.

II. Even if this kind of definition can be used for the natural numbers, I am at a loss to understand how numbers such as fractions, irrational roots, pi, imaginary numbers, and negative numbers are to be defined in terms of sets of objects.

III. If “pairs” is understood to mean “pairs of physical objects,” then it would seem that in a possible world with only two physical objects, numbers larger than two could not exist, since e.g. “the set of all triples” would be empty.

IV. Or again, take two possible worlds which each contain only twenty physical objects, with the stipulation that each world contains objects that are entirely different than the objects in the other. In this case, it would seem that “2” in one world would not be identical to “2” in the other, since the set defined as “the set of all pairs” would have different members in each possible world.

Thales' Well

The origin of this blog's name, from Theaetetus:

"[T]hey say Thales was studying the stars... and gazing aloft, when he fell into a well; and a witty and amusing Thracian servant-girl made fun of him because, she said, he was wild to know about what was up in the sky but failed to see what was in front of him and under his feet. The same joke applies to all who spend their lives in philosophy."[1]

This blog will be dedicated to the pursuit of looking aloft and will avoid discussion of what is at our feet.

[1] Plato, Theaetetus, trans. M.J. Levett, rev. Myles F. Burnyeat, from Plato: Complete Works ed. John M. Cooper, Hackett: Indianapolis, IN, 1997., 174a.

Aristotle on Knowledge and Necessity

The determinist’s argument in De Int. 9 (as given by Aristotle and interpreted by Richard Sorabji[1]) runs as follows: it is either true or false that event X will occur. If the statement “X will happen” turns out to be true, then it will have been true in the past. But if the statement “X will happen” was true in the past, then there is no point to deliberating whether or not to bring it about that X, since “X will happen” was true in the past, and the past cannot be altered. Thus, “X will happen” is necessarily true.

As Sorabji points out, Aristotle seems to accept the inference from the premise that any given statement is either true or false to the determinist conclusion. According to what Sorabji calls the traditional interpretation, Aristotle answers the determinist by arguing that predictive statements like “X will happen” are not yet true or false, but become true or false when X actually happens (or becomes inevitable). Sorabji finds this response unsatisfying, because e.g. if a statement’s being true means that it corresponds to a state of affairs in the world (which was a fairly common understanding of “true” for the Greeks) then the statement “there will be a sea battle tomorrow” is true (when uttered) if in fact it corresponds to a state of affairs, viz. the sea battle happening tomorrow; to speak of it as “not yet true” seems odd (Sorabji 101).

Sorabji’s own solution is to deny the inference between the premise that any given statement is either true or false to the determinist conclusion by making reference to one’s power, or lack of power, to bring about a state of affairs. Thus, if the statement “X will happen” turns out to be true, i.e. actually happens, then I do not have it in my power to make it the case that not-X, since X has already happened and I cannot alter the past. But, Sorabji thinks that this scenario is disanalogous to a situation in which I have been told “X will happen,” but X has not yet happened. Sorabji argues that since the event X still lies in the future, we still have the power to bring about not-X, and thus even if the statement “X will happen” does turn out to be true, it is not necessarily true, as the determinist thinks.

However, it seems to me as though something like Sorabji’s reply was addressed by Aristotle. At 19a7-23, Aristotle argues that it is false that all events happen of necessity: before an event X happens, both X and not-X are possible states of affairs. Since not-X is possible, then if X happens, it cannot have happened necessarily since it was in some agent’s power to bring about not-X. This passage seems to mirror Sorabji’s discussion of irrevocability in the first paragraph on p. 102, and Aristotle seems to argue here for the same disanalogy between past events and future events that Sorabji does.

Returning to Sorabji’s objection that it seems as though statements about the future should be said to be actually true or false depending on whether they in fact correspond to a state of affairs, it seems to me that Aristotle could answer that they are not yet true or false simply because they do not yet correspond to a state of affairs, and that only when they can truly be said to correspond, or fail to correspond, to a state of affairs can they be said to be true or false. It seems to me that this is the line Aristotle takes in his conclusion to this section at 19b1. Thus it appears that Aristotle argues both for a qualified principle of bivalence (cf. Sorabji p. 94-95) and against the determinist’s belief that all events happen of necessity.

---

[1] Sorabji, Richard. “Necessity, Cause, and Blame: Perspectives on Aristotle’s Theory.” Duckworth, 1980, pp. 91-103.