Logic

Counterfactuals and Causality

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People have frequently noted some connection between counterfactuals and causation. While it seems backwards to suggest that causation should be defined in terms of counterfactuals, it is reasonable to connect the two concepts, and explains why some counterfactuals are more reasonable than others, as we noted in the last post.

For example, “If I dropped this cup, it would fall to the floor,” is more reasonable than “If I dropped this cup, it would fly up to the moon,” because we are considering the operation of causes like gravity which could cause falling to the floor, and which could not cause (merely by dropping) an object to fly to the moon. In particular, since causes eliminate alternatives, they give us a reason to say “this would have happened rather than that.”

Nonetheless, we cannot get any sort of absolute determination out of this. One would attempt to get a determinate outcome by specifying the counterfactual as clearly as possible: “If I dropped this cup, and everything else was the same.” The “nearest possible world” idea is trying to get at this. However, this is not in fact completely determinate because “everything else” can’t be entirely the same, and what else needs to change is not determinate. In order to drop the cup, there would need to be a course of events that led up to the dropping, and there are many different courses that could have done that. The same thing will happen if you to specify exactly what led to the dropping of the cup; there will need to be something that led to your specification. Thus, at the very least, you will not typically be able to get absolute determination in this way.

Naturally, there is nothing to prevent us from coming up particular examples where we can get complete determination by using something which is always true anyway, or by using logical implication from the counterfactual, e.g. “If I dropped this cup, 2 and 2 would still be 4,” or “If I dropped the cup, it would have been dropped.” But these are not typical cases.

Counterfactuals as Historical Fiction

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Suppose someone reading Anne of Green Gables asks a question about what happened before the story begins. For example, what did Anne have for lunch 37 days before her arrival in Avonlea?

It is easy to see that this question does not have one true answer. There is no such thing as what she really had for lunch, because it is a story, and that meal is not included in it. On the other hand, despite the lack of any absolute truth here, some answers remain more reasonable than others. For example, “She had salad,” is a more sensible answer than “she ate crushed glass that day.” Just as I said in regard to “why” something is the case, one can give a partial answer, in the sense of showing that some options are more intelligible than others, without being able to exclude some options entirely.

These same things will apply to questions about a work of historical fiction, although the intended historical context will provide additional ways to show that some answers are more sensible than others. Thus if a story is set in ancient Rome, the claim that someone had corn for lunch is unreasonable due to the historical context, although not as unreasonable as some other possibilities that you could suggest.

Now consider a counterfactual question about your current situation: “What would you do if it were 120 degrees Fahrenheit in your house?”

There is no fundamental difference between this and the case of historical fiction. In effect, we just created a story about you: “It was 120 degrees in your house. You…”

Like the case of historical fiction, some answers will be more sensible than others, but there is no thing that you really would do in that situation. The story didn’t really take place, but if it did, it would have taken place with a lot more concrete detail, and that concrete detail could determine the specific answer to the question. If Anne of Green Gables were a true story, her concrete situation would have determined what she had for lunch that day. And if it were really 120 degrees in your house, what you would do would depend on how and why things got that way, as well as other factors in your concrete situation.

Some philosophers have spent a lot of time on this kind of counterfactual question, apparently largely from a desire for absolute answers. For example, some suggest that a counterfactual is true if the claim is true in the nearest possible world where the antecedent is true. In a similar way, Molinists argue that in order to be omniscient, God has to know what you would do if it were 120 degrees in your house, and that it must be one specific thing, so that there is one thing that you really would do in that situation. They call this kind of knowledge “middle” knowledge, namely something in between knowledge of what actually is and knowledge of what merely might have been.

All accounts of this kind are wasted effort. The brief account above is sufficient.

Words, Meaning, and Formal Copies

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There is quick way to respond to the implicit questions at the end of the last post. I noted in an earlier discussion of form that form is not only copied into the mind; it is also copied into language itself. Any time you describe something in words, you are to some degree copying its form into your description.

This implies that Aristotle’s objection that a mind using an organ would not be able to know all things could equally be made against the possibility of describing all things in words. There simply are not enough combinations of words to relate them to all possible combinations of things; thus, just as a black and white image cannot imitate every aspect of a colored scene, so words cannot possibly describe every aspect of reality.

Two things are evident from this comparison:

First, the objection fails overall. There is nothing that cannot be described in words because words are flexible. If we don’t have a word for something, then we can make up a name. Similarly, the meaning of a single word depends on context.  The word “this” can refer to pretty much anything, depending on the context in which it is used. Likewise meaning can be affected by the particular situation of the person using the word, or by broader cultural contexts, and so on.

Second, there is some truth in the objection. It is indeed impossible to describe every aspect of reality at the same time and in complete detail, and the objection gives a very good reason for this: there are simply not enough linguistic combinations to represent all possible combinations of things. The fact that language is not prime matter does mean that language cannot express every detail of reality at once: the determination that is already there does exclude this possibility. But the flexibility of language prevents there from being any particular aspect of things that cannot be described.

My claim about the mind is the same. There is nothing that cannot be understood by the mind, despite the fact that the mind uses the brain, because the relationship between the brain, mind, and world is a flexible one. Just as the word “this” can refer to pretty much anything, so also the corresponding thought. But on the other hand, the limitations of the brain do mean that a perfectly detailed knowledge of everything is excluded.

Our Interlocutor Insists

In a sense, the above account is sufficient to respond to the objection. There does not seem to be a reason to hold Aristotle’s account of the immateriality of the mind, unless there is also a reason to hold that language cannot be used to describe some things, and this does not seem like a reasonable position. Nonetheless, this response will give rise to a new and more detailed objection.

A black and white scene, it will be said, really and truly copies some aspects of a colored scene, and fails to copy others. Thus right angles in the black and white scene may be identical to right angles in the colored scene. The angles are really copied, and the angles are not. But language seems different: since it is conventional, it does not really copy anything. We just pretend, as it were, that we are copying the thing. “Let the word ‘cat’ stand for a cat,” we say, but there is nothing catlike about the word in reality. The form of the cat is not really copied into the word, or so it will be argued. And since we are not really copying anything, this is why language has the flexibility to be able to describe all things. The meaning of thoughts, however, is presumably not conventional. So it seems that we need to copy things in a real way into the mind, the way we copy aspects of a colored scene into a black and white image. And thus, meaning in the mind should not be flexible in this way, and a particular material medium (such as the brain) would still impede knowing all things, the way the black and white image excludes color.

Formal Copies

The above objection is similar to Hilary Lawson’s argument that words cannot really refer to things. In the post linked above on form and reality, we quoted his argument that cause and effect do not have anything in common. I will reproduce that argument here; for the purpose of the following discussion it might be useful to the reader to refer to the remainder of that post.

For a system of closure to provide a means of intervention in openness and thus to function as a closure machine, it requires a means of converting the flux of openness into an array of particularities. This initial layer of closure will be identified as ‘preliminary closure’. As with closure generally, preliminary closure consists in the realisation of particularity as a consequence of holding that which is different as the same. This is achieved through the realisation of material in response to openness. The most minimal example of a system of closure consists of a single preliminary closure. Such a system requires two discrete states, or at least states that can be held as if they were discrete. It is not difficult to provide mechanical examples of such systems which allow for a single preliminary closure. A mousetrap for example, can be regarded as having two discrete states: it is either set, it is ready, or it has sprung, it has gone off. Many different causes may have led to it being in one state or another: it may have been sprung by a mouse, but it could also have been knocked by someone or something, or someone could have deliberately set it off. In the context of the mechanism all of these variations are of no consequence, it is either set or it has sprung. The diversity of the immediate environment is thereby reduced to single state and its absence: it is either set or it is not set. Any mechanical arrangement that enables a system to alternate between two or more discrete states is thereby capable of providing the basis for preliminary closure. For example, a bell or a gate could function as the basis for preliminary closure. The bell can either ring or not ring, the gate can be closed or not closed. The bell may ring as the result of the wind, or a person or animal shaking it, but the cause of the response is in the context of system of no consequence. The bell either rings or it doesn’t. Similarly, the gate may be in one state or another because it has been deliberately moved, or because something or someone has dislodged it accidentally, but these variations are not relevant in the context of the state of system, which in this case is the position of the gate. In either case the cause of the bell ringing or the gate closing is infinitely varied, but in the context of the system the variety of inputs is not accessible to the system and thus of no consequence.

Lawson’s basic argument is that any particular effect could result from any of an infinite number of different causes, and the cause and effect might be entirely different: the effect might be ringing of a bell, but the cause was not bell-like at all, and did not have a ringing sound. So the effect, he says, tells you nothing at all about the cause. In a similar way, he claims, our thoughts cause our words, but our words and our thoughts have nothing in common, and thus our words tell us nothing about our thoughts; and in that sense they do not refer to anything, not even to our thoughts. Likewise, he says, the world causes our thoughts, but since the cause and effect have nothing in common, our thoughts tell us nothing about the world, and do not even refer to it.

As I responded at the time, this account is mistaken from the very first step. Cause and effect always have something in common, namely the cause-effect relationship, although they each have different ends of that relationship. They will also have other things in common depending on the particular nature of the cause and effect in question. Similarly, the causes that are supposedly utterly diverse, in Lawson’s account, have something in common themselves: every situation that rings the bell has “aptness to ring the bell” in common. And when the bell is rung, it “refers” to these situations by the implication that we are in a situation that has aptness to ring the bell, rather than in one of the other situations.

It is not accidental here that “refer” and “relate” are taken from forms of the same verb. Lawson’s claim that words do not “refer” to things is basically the same as the claim that they are not really related to things. And the real problem is that he is looking at matter (in this case the bell) without considering form (in this case the bell’s relationship with the world.)

In a similar way, to say that the word “cat” is not catlike is to look at the sound or at the text as matter, without considering its form, namely the relationship it has with the surrounding context which causes that word to be used. But that relationship is real; the fact that the word is conventional does not prevent it from being true that human experience of cats is the cause of thoughts of cats, and that thoughts of cats are concretely the cause of the usage of the word “cat,” even if they could in some other situation have caused some other word to be used.

I argued in the post on the nature of form (following the one with the discussion of Lawson) that form is a network of relationships apt to make something one. Insofar as an effect really receives form from a cause in the above way, words really receive meaning from the context that gives rise to their use. And in this way, it is not true that form in language is unlike form in a black and white scene, such that one could say that form in the scene is “real” and form in language is not. Both are real.

Thus the objection fails. Nonetheless, it is true that it is easier to see why it is possible to describe anything in words, than it is to see why anything can be known. And this happens simply because “anything is describable in words” precisely because “anything can be known.” So the fact that anything can be known is the more remote cause, and thus harder to know.

 

Fair and Unfair Logic

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St. Thomas discusses cases in which one should not follow the law:

As stated above (Article 4), every law is directed to the common weal of men, and derives the force and nature of law accordingly. Hence the jurist says [Pandect. Justin. lib. i, ff., tit. 3, De Leg. et Senat.]: “By no reason of law, or favor of equity, is it allowable for us to interpret harshly, and render burdensome, those useful measures which have been enacted for the welfare of man.” Now it happens often that the observance of some point of law conduces to the common weal in the majority of instances, and yet, in some cases, is very hurtful. Since then the lawgiver cannot have in view every single case, he shapes the law according to what happens most frequently, by directing his attention to the common good. Wherefore if a case arise wherein the observance of that law would be hurtful to the general welfare, it should not be observed. For instance, suppose that in a besieged city it be an established law that the gates of the city are to be kept closed, this is good for public welfare as a general rule: but, it were to happen that the enemy are in pursuit of certain citizens, who are defenders of the city, it would be a great loss to the city, if the gates were not opened to them: and so in that case the gates ought to be opened, contrary to the letter of the law, in order to maintain the common weal, which the lawgiver had in view.

He calls the attitude that leads one to set aside the law in such cases “epikeia,” or “equity,” which in this context means something like fairness or moderation:

As stated above (I-II:96:6), when we were treating of laws, since human actions, with which laws are concerned, are composed of contingent singulars and are innumerable in their diversity, it was not possible to lay down rules of law that would apply to every single case. Legislators in framing laws attend to what commonly happens: although if the law be applied to certain cases it will frustrate the equality of justice and be injurious to the common good, which the law has in view. Thus the law requires deposits to be restored, because in the majority of cases this is just. Yet it happens sometimes to be injurious—for instance, if a madman were to put his sword in deposit, and demand its delivery while in a state of madness, or if a man were to seek the return of his deposit in order to fight against his country. On these and like cases it is bad to follow the law, and it is good to set aside the letter of the law and to follow the dictates of justice and the common good. This is the object of “epikeia” which we call equity. Therefore it is evident that “epikeia” is a virtue.

“Fairness” is probably a good translation here, since someone who rigidly demands the application of the law in such a situation would often be called unfair in relation to the people involved.

Someone might object that much of the benefit of having a law directly depends on following it consistently, without making exceptions based on minute analysis of particular situations, as we saw in the last post. This is correct as far as it goes, but St. Thomas is not talking about analyzing each situation in detail and making an exception whenever there appears to be a benefit, but rather talking about situations which are extremely different from the situations considered by the law. Thus he says in the reply to the second objection:

He who follows the intention of the lawgiver, does not interpret the law simply; but in a case in which it is evident, by reason of the manifest harm, that the lawgiver intended otherwise. For if it be a matter of doubt, he must either act according to the letter of the law, or consult those in power.

To the degree that “laws of logic” can be analogously interpreted as rules for sensible thought and speech, telling one to behave in some ways and not in others, similar principles will apply. Thus, for example, an atheist confronted with the argument of Alexander Pruss for the existence of God based on the indeterminacy of language might not only be inclined to call it sophistical, but to add that it is an unfair way to argue. And indeed it is, precisely in the sense that it applies the rule “either say that A is B or say that A is not B” to situations for which it was not intended, namely situations where B is simply too vague to say. The rule is intended to make people think and speak sensibly, but Pruss is abusing the rule with the opposite result: that he does not speak and think sensibly.

Someone might agree that this is reasonable insofar as we are considering these laws as rules of behavior, but another issue comes up. Human laws are really intended to exclude some kinds of behavior that are really possible. And likewise, rules of logic are really intended to exclude some kinds of behavior that are really possible, e.g. making arguments like:

A: You always say I am wrong.

B: I said you were right about X.

A: See, you just said I was wrong again. You always say I am wrong!

I know from experience that this behavior is possible, and it does violate the laws of logic considered as rules of behavior. But someone might add that the laws of logic are also based on the nature of reality itself, and for this very reason we said that they are not conventions, but could not have been otherwise. So it seems to follow that it should be possible to expound the laws of logic in a form in which they are truly exceptionless, by expressing reality as it truly is.

There is some truth here, but there is also a problem analogous to a similar objection about human law. Consider the third objection and reply in the above article from St. Thomas:

Objection 3. Further, every wise man knows how to explain his intention by words. But those who framed the laws should be reckoned wise: for Wisdom says (Proverbs 8:15): “By Me kings reign, and lawgivers decree just things.” Therefore we should not judge of the intention of the lawgiver otherwise than by the words of the law.

…

Reply to Objection 3. No man is so wise as to be able to take account of every single case; wherefore he is not able sufficiently to express in words all those things that are suitable for the end he has in view. And even if a lawgiver were able to take all the cases into consideration, he ought not to mention them all, in order to avoid confusion: but should frame the law according to that which is of most common occurrence.

The objection here is similar. If there are cases where it wouldn’t be good to apply the law, the lawgiver ought to have enumerated those cases. St. Thomas replies that in reality you will not foresee every case, and that even if you could, enumerating them would simply cause confusion.

A similar thing applies if we consider the laws of logic. You can say, “If you say that A is B in an infinitely precise sense, and that B is C in an infinitely precise sense, you should also say that A is C,” and your claim might be exceptionless. The problem is that your claim has no cases: no one ever says anything in an infinitely precise sense.

And on the other hand, if you try to make your claim include some actual cases, you will not be able to avoid the possibility of exceptions, just as the human lawgiver does not foresee all cases. And as in the case of human law, if you attempt to enumerate all cases, you will simply cause confusion. Thus, for example, someone might say that the problem in the case of Queen Elizabeth is that we simply don’t have a precise enough definition for “old,” and they might then attempt to give a precise definition. But this would have several results:

1. First, the new word “old” would not have the same meaning as the original word, because the very fact that the original word is vague is part of what the word is. It is not accidental; it is not meant to have a precise cut-off.

2. Someone might attempt to remedy the above flaw by enumerating various circumstances, rather than giving a precise cut-off. “If you are less then 10 years old and you say that someone is ‘old,’ it signifies someone who is at least 15.” “If you are in your 30s and you say that someone is ‘old’, it signifies that they are at least 67.” And so on. But attempting to fix the first problem, you have simply compounded it. The new word still does not have the same meaning as the original word, because the original word was meant to be flexible; even your new rules have too much rigidity.

You could attempt to remedy the above problems by listing all the situations where people in fact use the word “old,” but that is not a definition: it is just an indefinitely long list. What St. Thomas said about human law, that it “ought not to mention them all,” is equally true about this situation. The point of defining “old” is to provide an explanation which is both general and flexible. Someone might argue that we should provide a list of all possible circumstances and what should be done in those circumstances, in order to avoid the flexibility of “epikeia,” but such an attempt would be absurd, and harmful to a good life. And it is equally absurd when we attempt to apply the same process to logic or to definitions, and harmful to sensible thought and speech.

What about reality itself? Isn’t it an exceptionless reality that a thing is what it is? Indeed. But this is neither a rule of behavior nor of speech. Nor is it a rule making something be some way; reality does not need something else to make sure that it turns out to be reality rather than something else. There is simply nothing else to be. Parmenides was right at least to this degree.

Laws of Logic

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In the last post, we quoted Carin Robinson’s claim:

For instance, where we use the laws of logic, let us remember that there are no known/knowable facts about logic. These laws are therefore, to the best of our knowledge, conventions not dissimilar to the rules of a game.

Law

I intend to discuss Robinson’s claim in a bit more detail shortly, but first consider the meaning of a law in its plainest sense. In the USA there is a law that you must pay your taxes for the previous year by mid April. What does this law do? Presumably the purpose of the law is to get people to pay their taxes by that time. Without the law, they would likely not pay by then, and if there were no rule that you have to pay taxes at all, people presumably would not pay taxes. So the law is meant to make something happen, namely the payment of taxes by a certain date, something that otherwise might not happen.

Rules of a game

What about the rules of a game? Consider the game of hide and seek. Wikipedia describes it in some detail:

Hide-and-seek, or hide-and-go-seek, is a popular children’s game in which any number of players (ideally at least three) conceal themselves in a set environment, to be found by one or more seekers. The game is played by one player chosen (designated as being “it”) closing their eyes and counting to a predetermined number while the other players hide. For example, count to 100 in units of 5 or count to 20, one two three and keep counting up till it reaches twenty. After reaching this number, the player who is “it” calls “Ready or not, here I come!” and then attempts to locate all concealed players.

This is partly a factual description, but it is also attempting to give the rules. It seems to be a rule that the players who are hiding have some amount of time to hide, and it would seem to be a violation of the rules if the seeker simply starts the game by announcing, “I see everyone here, so I’ve found everyone,” without there being any time to hide.

What do these rules do? Are they like the law?

Yes and no, in different respects. You can certainly imagine a player breaking the rules in the above manner. So the rules, like the law, are meant to make something happen, namely the players act in a certain manner, and they are meant to exclude what might happen without the rule, just like the law.

There is a difference, however. If a player did the above, they would not be playing the game at all. It is possible to go about your life and not pay any taxes; but it is not possible to play hide-and-seek without there being a space or time for people to hide. In this sense, the law excludes some possibilities for life, but the rule of the game does not exclude some possibilities for that game; it simply describes what the game is. It does exclude possibilities that would be rules for other games. So it excludes some possibilities; but not possibilities for the game of hide and seek.

Facts

Why does Robinson say that there are no “facts” about logic? The English word “fact” is taken from the Latin factum, which means “done” or “made.” This is not accidental to the claim here. There is nothing making things follow the rules of logic, and for this reason Robinson asserts that there are no facts, i.e. nothing made to be the case in the realm of logic. Precisely for this reason, you don’t have to go out and look at the “facts”, i.e. things that are made to be the case in the world, to determine whether or not a statement of logic or mathematics is correct or not.

Laws of Logic

Robinson argues that since the laws of logic don’t make anything be the case in the world, they must be conventional, like “rules of a game”. But in our discussion of the rules of a game, we saw that such rules do exclude certain types of possibility, while they constitute the game itself, and therefore do not exclude any possibilities for the game. How would this work if the rules of logic were rules of a game? What sorts of possibility are excluded by the rules, and what game is constituted by the rules?

As we said, it is possible to break the rules of a game, although when you do, you often stop playing the game by definition. It it similarly possible to break the laws of logic?

If we take the game to be a certain sort of speaking, yes, it is. It is possible for someone to say the words, “Blue things are not blue.” It is possible for someone to say the words, “All cats are mammals. Alvin is a cat. Therefore Alvin is not a mammal.” Someone doing this, however, is not playing the particular game in question. What is that game? I suggest we call it “speaking sensibly about reality.” Someone who breaks the laws of logic, by that very fact, fails to speak sensibly about reality, just as someone who breaks the rules of hide-and-seek fails to play the game.

The rules of hide-and-seek are conventional, in the sense that you could have other rules. But if you did have other rules, you would be playing a different game. In the same way, if you had rules other than the laws of logic for your speaking game, you would be doing something entirely different. You would not be doing what we are normally trying to do when we speak, namely speaking sensibly about reality.

Up to this point, we have actually succeeded in making a certain sort of sense out of Robinson’s claim. But does it follow, as supposed, that logic tells us nothing about reality? We pointed out in the previous post that this is not true. But why is it not, if the laws of logic are conventions about how to speak?

Do the rules of hide-and-seek tell us something about the game of hide-and-seek? Clearly they do, despite the fact that they are conventional. They tell us most of what there is to know about the game. They tell us what the game is, in fact. Likewise, the laws of logic tell us how to speak sensibly about reality. Do they also tell us about reality itself, or just about how to speak about it?

They do, in the way that considering the effect reveals the cause. Reality is what it is, and therefore certain ways of speaking are sensible and others are not. So to tell someone how to speak sensibly is to tell them something about reality. However, there is another difference between the laws of logic and the rules of a game. The rules of a game are conventional in the sense that we could have different rules and different games. And similarly, if we didn’t want to follow the “conventions” of logic, we could speak nonsensically instead of trying to speak sensibly about reality. But there is not some possible alternate reality which could be spoken of sensibly by using different “conventions.” In this sense, you can call the laws of logic rules of a game, if you wish. But they are the rules of the game of understanding, and there is only such game, not only in practice but in principle, and the rules could not have been otherwise.

Tautologies Not Trivial

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In mathematics and logic, one sometimes speaks of a “trivial truth” or “trivial theorem”, referring to a tautology. Thus for example in this Quora question, Daniil Kozhemiachenko gives this example:

The fact that all groups of order 2 are isomorphic to one another and commutative entails that there are no non-Abelian groups of order 2.

This statement is a tautology because “Abelian group” here just means one that is commutative: the statement is like the customary example of asserting that “all bachelors are unmarried.”

Some extend this usage of “trivial” to refer to all statements that are true in virtue of the meaning of the terms, sometimes called “analytic.” The effect of this is to say that all statements that are logically necessary are trivial truths. An example of this usage can be seen in this paper by Carin Robinson. Robinson says at the end of the summary:

Firstly, I do not ask us to abandon any of the linguistic practises discussed; merely to adopt the correct attitude towards them. For instance, where we use the laws of logic, let us remember that there are no known/knowable facts about logic. These laws are therefore, to the best of our knowledge, conventions not dissimilar to the rules of a game. And, secondly, once we pass sentence on knowing, a priori, anything but trivial truths we shall have at our disposal the sharpest of philosophical tools. A tool which can only proffer a better brand of empiricism.

While the word “trivial” does have a corresponding Latin form that means ordinary or commonplace, the English word seems to be taken mainly from the “trivium” of grammar, rhetoric, and logic. This would seem to make some sense of calling logical necessities “trivial,” in the sense that they pertain to logic. Still, even here something is missing, since Robinson wants to include the truths of mathematics as trivial, and classically these did not pertain to the aforesaid trivium.

Nonetheless, overall Robinson’s intention, and presumably that of others who speak this way, is to suggest that such things are trivial in the English sense of “unimportant.” That is, they may be important tools, but they are not important for understanding. This is clear at least in our example: Robinson calls them trivial because “there are no known/knowable facts about logic.” Logical necessities tell us nothing about reality, and therefore they provide us with no knowledge. They are true by the meaning of the words, and therefore they cannot be true by reason of facts about reality.

Things that are logically necessary are not trivial in this sense. They are important, both in a practical way and directly for understanding the world.

Consider the failure of the Mars Climate Orbiter:

On November 10, 1999, the Mars Climate Orbiter Mishap Investigation Board released a Phase I report, detailing the suspected issues encountered with the loss of the spacecraft. Previously, on September 8, 1999, Trajectory Correction Maneuver-4 was computed and then executed on September 15, 1999. It was intended to place the spacecraft at an optimal position for an orbital insertion maneuver that would bring the spacecraft around Mars at an altitude of 226 km (140 mi) on September 23, 1999. However, during the week between TCM-4 and the orbital insertion maneuver, the navigation team indicated the altitude may be much lower than intended at 150 to 170 km (93 to 106 mi). Twenty-four hours prior to orbital insertion, calculations placed the orbiter at an altitude of 110 kilometers; 80 kilometers is the minimum altitude that Mars Climate Orbiter was thought to be capable of surviving during this maneuver. Post-failure calculations showed that the spacecraft was on a trajectory that would have taken the orbiter within 57 kilometers of the surface, where the spacecraft likely skipped violently on the uppermost atmosphere and was either destroyed in the atmosphere or re-entered heliocentric space.[1]

The primary cause of this discrepancy was that one piece of ground software supplied by Lockheed Martin produced results in a United States customary unit, contrary to its Software Interface Specification (SIS), while a second system, supplied by NASA, expected those results to be in SI units, in accordance with the SIS. Specifically, software that calculated the total impulse produced by thruster firings produced results in pound-force seconds. The trajectory calculation software then used these results – expected to be in newton seconds – to update the predicted position of the spacecraft.

It is presumably an analytic truth that the units defined in one way are unequal to the units defined in the other. But it was ignoring this analytic truth that was the primary cause of the space probe’s failure. So it is evident that analytic truths can be extremely important for practical purposes.

Such truths can also be important for understanding reality. In fact, they are typically more important for understanding than other truths. The argument against this is that if something is necessary in virtue of the meaning of the words, it cannot be telling us something about reality. But this argument is wrong for one simple reason: words and meaning themselves are both elements of reality, and so they do tell us something about reality, even when the truth is fully determinate given the meaning.

If one accepts the mistaken argument, in fact, sometimes one is led even further. Logically necessary truths cannot tell us anything important for understanding reality, since they are simply facts about the meaning of words. On the other hand, anything which is not logically necessary is in some sense accidental: it might have been otherwise. But accidental things that might have been otherwise cannot help us to understand reality in any deep way: it tells us nothing deep about reality to note that there is a tree outside my window at this moment, when this merely happens to be the case, and could easily have been otherwise. Therefore, since neither logically necessary things, nor logically contingent things, can help us to understand reality in any deep or important way, such understanding must be impossible.

It is fairly rare to make such an argument explicitly, but it is a common implication of many arguments that are actually made or suggested, or it at least influences the way people feel about arguments and understanding.  For example, consider this comment on an earlier post. Timocrates suggests that (1) if you have a first cause, it would have to be a brute fact, since it doesn’t have any other cause, and (2) describing reality can’t tell us any reasons but is “simply another description of how things are.” The suggestion behind these objections is that the very idea of understanding is incoherent. As I said there in response, it is true that every true statement is in some sense “just a description of how things are,” but that was what a true statement was meant to be in any case. It surely was not meant to be a description of how things are not.

That “analytic” or “tautologous” statements can indeed provide a non-trivial understanding of reality can also easily be seen by example. Some examples from this blog:

Good and being. The convertibility of being and goodness is “analytic,” in the sense that carefully thinking about the meaning of desire and the good reveals that a universe where existence as such was bad, or even failed to be good, is logically impossible. In particular, it would require a universe where there is no tendency to exist, and this is impossible given that it is posited that something exists.

Natural selection. One of the most important elements of Darwin’s theory of evolution is the following logically necessary statement: the things that have survived are more likely to be the things that were more likely to survive, and less likely to be the things that were less likely to survive.

Limits of discursive knowledge. Knowledge that uses distinct thoughts and concepts is necessarily limited by issues relating to self-reference. It is clear that this is both logically necessary, and tells us important things about our understanding and its limits.

Knowledge and being. Kant rightly recognized a sense in which it is logically impossible to “know things as they are in themselves,” as explained in this post. But as I said elsewhere, the logically impossible assertion that knowledge demands an identity between the mode of knowing and the mode of being is the basis for virtually every sort of philosophical error. So a grasp on the opposite “tautology” is extremely useful for understanding.

 

Perfectly Random

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Suppose you have a string of random binary digits such as the following:

00111100010101001100011011001100110110010010100111

This string is 50 digits long, and was the result of a single attempt using the linked generator.

However, something seems distinctly non-random about it: there are exactly 25 zeros and exactly 25 ones. Naturally, this will not always happen, but most of the time the proportion of zeros will be fairly close to half. And evidently this is necessary, since if the proportion was usually much different from half, then the selection could not have been random in the first place.

There are other things about this string that are definitely not random. It contains only zeros and ones, and no other digits, much less items like letters from the alphabet, or items like ‘%’ and ‘$’.

Why do we have these apparently non-random characteristics? Both sorts of characteristics, the approximate and typical proportion, and the more rigid characteristics, are necessary consequences of the way we obtained or defined this number.

It is easy to see that such characteristics are inevitable. Suppose someone wants to choose something random without any non-random characteristics. Let’s suppose they want to avoid the first sort of characteristic, which is perhaps the “easier” task. They can certainly make the proportion of zeros approximately 75% or anything else that they please. But this will still be a non-random characteristic.

They try again. Suppose they succeed in preventing the series of digits from converging to any specific probability. If they do, there is one and only one way to do this. Much as in our discussion of the mathematical laws of nature, the only way to accomplish this will be to go back and forth between longer and longer strings of zeros and ones. But this is an extremely non-random characteristic. So they may have succeeded in avoiding one particular type of non-randomness, but only at the cost of adding something else very non-random.

Again, consider the second kind of characteristic. Here things are even clearer: the only way to avoid the second kind of characteristic is not to attempt any task in the first place. The only way to win is not to play. Once we have said “your task is to do such and such,” we have already specified some non-random characteristics of the second kind; to avoid such characteristics is to avoid the task completely.

“Completely random,” in fact, is an incoherent idea. No such thing can exist anywhere, in the same way that “formless matter” cannot actually exist, but all matter is formed in one way or another.

The same thing applies to David Hume’s supposed problem of induction. I ended that post with the remark that for his argument to work, he must be “absolutely certain that the future will resemble the past in no way.” But this of course is impossible in the first place; the past and the future are both defined as periods of time, and so there is some resemblance in their very definition, in the same way that any material thing must have some form in its definition, and any “random” thing must have something non-random in its definition.

 

Open Future

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Let’s return for a moment to the question at the end of this post. I asked, “What happens if the future is indeterminate? Would not the eternalist position necessarily differ from the presentist one, in that case?”

Why necessarily different? The argument in that post was that eternalism and presentism are different descriptions of the same thing, and that we see the sameness by noting the sameness of relations between the elements of the description. But if the future is open, as Aristotle supposed, it is hard to see how we can maintain this. Aristotle says that the present is open to either having the sea battle tomorrow or not having it. With an eternalist view, the sea battle is “already there” or it is not. So in Aristotle’s view, the present has an open relationship to both possibilities. But the eternalist view seems to be truly open only to the possibility that will actually happen. We no longer have the same set of relationships.

Notice the problem. When I attempted to equate eternalism and presentism, I implicitly assumed that determinism is true. There were only three states of the universe, beginning, middle, and end. If determinism is false, things are different. There might be beginning, middle, and two potential ends. Perhaps there is a sea battle in one of the potential ends, and no sea battle in the other.

This suggests a solution to our conundrum, however. Even the presentist description in that post was inconsistent with an open future. If there is only one possible end, the future is not open, even if we insist that the unique possible end “currently doesn’t exist.” The problem then was not eternalism as such, but the fact that we started out with a determinist description of the universe. This strongly suggests that if my argument about eternalism and presentism was correct, we should be able to formulate eternalist and presentist descriptions of an open future which will be equivalent. But both will need to be different from the fixed “beginning-middle-end” described in that post.

We can simply take Aristotle’s account as the account of presentism with an open future. How can we give an eternalist account of the same thing? The basic requirement will be that the relationship between the present and the future needs to be the same in both accounts. Now in Aristotle’s account, the present has the same relationship to two different possibilities: both of them are equally possible. So to get a corresponding eternalist account, we need the present to be equally related to two futures that correspond to the two possiblities in the presentist account. I do not say “two possible futures,” but “two futures,” precisely because the account is eternalist.

The careful reader will already understand the account from the above, but let us be more explicit. The eternalist account that corresponds to the presentist account with an open future has multiple timelines, all of which “exist”, in the eternalist sense. The reader will no doubt be familiar with the idea of multiple timelines, at least from time travel fiction. In a similar way, the eternalist reworking of Aristotle’s position is that there is a timeline where the sea battle takes place, and another timeline where the sea battle does not take place. In this view, both of them “actually” happen. But even in this view, an observer in the middle location will have to say, “I do not, and cannot, know whether the sea battle will take place or not,” just as in Aristotle’s view. For the observer cannot traverse both timelines at once. From his point of view, he will take only one, but since his relationship to the two possibilities (or actualities) is the same, it is indeterminate which one it will be.

Even if one cannot prove my account of equivalence to be wrong, the reader may worry. Time travel fiction frequently seems incoherent, and this suggests that any view with multiple timelines may also be incoherent. But this potential incoherence supports the equivalence, rather than subtracting from it. For as we noted in the post on Aristotle, there is a definite appearance of incoherence in his position. It is not even clear how his view is logically possible. So it would not be surprising, but quite natural, if views which are intended to be equivalent to his position are also not clearly coherent. Nonetheless, the multiple timelines description does have some logical advantage over Aristotle’s position, in the sense that “the sea battle will take place in timeline A” does not even appear to contradict “the sea battle will not take place in timeline B.”

Aristotle on Future Contingents

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In Chapter 9 of On Interpretation, Aristotle argues that at least some statements about the future need to be exempted from the principle of Excluded Middle:

In the case of that which is or which has taken place, propositions, whether positive or negative, must be true or false. Again, in the case of a pair of contradictories, either when the subject is universal and the propositions are of a universal character, or when it is individual, as has been said,’ one of the two must be true and the other false; whereas when the subject is universal, but the propositions are not of a universal character, there is no such necessity. We have discussed this type also in a previous chapter.

When the subject, however, is individual, and that which is predicated of it relates to the future, the case is altered. For if all propositions whether positive or negative are either true or false, then any given predicate must either belong to the subject or not, so that if one man affirms that an event of a given character will take place and another denies it, it is plain that the statement of the one will correspond with reality and that of the other will not. For the predicate cannot both belong and not belong to the subject at one and the same time with regard to the future.

Thus, if it is true to say that a thing is white, it must necessarily be white; if the reverse proposition is true, it will of necessity not be white. Again, if it is white, the proposition stating that it is white was true; if it is not white, the proposition to the opposite effect was true. And if it is not white, the man who states that it is making a false statement; and if the man who states that it is white is making a false statement, it follows that it is not white. It may therefore be argued that it is necessary that affirmations or denials must be either true or false.

Now if this be so, nothing is or takes place fortuitously, either in the present or in the future, and there are no real alternatives; everything takes place of necessity and is fixed. For either he that affirms that it will take place or he that denies this is in correspondence with fact, whereas if things did not take place of necessity, an event might just as easily not happen as happen; for the meaning of the word ‘fortuitous’ with regard to present or future events is that reality is so constituted that it may issue in either of two opposite directions. Again, if a thing is white now, it was true before to say that it would be white, so that of anything that has taken place it was always true to say ‘it is’ or ‘it will be’. But if it was always true to say that a thing is or will be, it is not possible that it should not be or not be about to be, and when a thing cannot not come to be, it is impossible that it should not come to be, and when it is impossible that it should not come to be, it must come to be. All, then, that is about to be must of necessity take place. It results from this that nothing is uncertain or fortuitous, for if it were fortuitous it would not be necessary.

The argument here is that if it is already true, for example, that I will eat breakfast tomorrow, then I will necessarily eat breakfast tomorrow, and there is no option about this and no ability of anything to prevent it. Aristotle is here taking it for granted that some things about the future are uncertain, and is using this as a reductio against the position that such claims can be already true. He goes on to give additional reasons for the same thing:

Again, to say that neither the affirmation nor the denial is true, maintaining, let us say, that an event neither will take place nor will not take place, is to take up a position impossible to defend. In the first place, though facts should prove the one proposition false, the opposite would still be untrue. Secondly, if it was true to say that a thing was both white and large, both these qualities must necessarily belong to it; and if they will belong to it the next day, they must necessarily belong to it the next day. But if an event is neither to take place nor not to take place the next day, the element of chance will be eliminated. For example, it would be necessary that a sea-fight should neither take place nor fail to take place on the next day.

These awkward results and others of the same kind follow, if it is an irrefragable law that of every pair of contradictory propositions, whether they have regard to universals and are stated as universally applicable, or whether they have regard to individuals, one must be true and the other false, and that there are no real alternatives, but that all that is or takes place is the outcome of necessity. There would be no need to deliberate or to take trouble, on the supposition that if we should adopt a certain course, a certain result would follow, while, if we did not, the result would not follow. For a man may predict an event ten thousand years beforehand, and another may predict the reverse; that which was truly predicted at the moment in the past will of necessity take place in the fullness of time.

Further, it makes no difference whether people have or have not actually made the contradictory statements. For it is manifest that the circumstances are not influenced by the fact of an affirmation or denial on the part of anyone. For events will not take place or fail to take place because it was stated that they would or would not take place, nor is this any more the case if the prediction dates back ten thousand years or any other space of time. Wherefore, if through all time the nature of things was so constituted that a prediction about an event was true, then through all time it was necessary that that should find fulfillment; and with regard to all events, circumstances have always been such that their occurrence is a matter of necessity. For that of which someone has said truly that it will be, cannot fail to take place; and of that which takes place, it was always true to say that it would be.

Yet this view leads to an impossible conclusion; for we see that both deliberation and action are causative with regard to the future, and that, to speak more generally, in those things which are not continuously actual there is potentiality in either direction. Such things may either be or not be; events also therefore may either take place or not take place. There are many obvious instances of this. It is possible that this coat may be cut in half, and yet it may not be cut in half, but wear out first. In the same way, it is possible that it should not be cut in half; unless this were so, it would not be possible that it should wear out first. So it is therefore with all other events which possess this kind of potentiality. It is therefore plain that it is not of necessity that everything is or takes place; but in some instances there are real alternatives, in which case the affirmation is no more true and no more false than the denial; while some exhibit a predisposition and general tendency in one direction or the other, and yet can issue in the opposite direction by exception.

Now that which is must needs be when it is, and that which is not must needs not be when it is not. Yet it cannot be said without qualification that all existence and non-existence is the outcome of necessity. For there is a difference between saying that that which is, when it is, must needs be, and simply saying that all that is must needs be, and similarly in the case of that which is not. In the case, also, of two contradictory propositions this holds good. Everything must either be or not be, whether in the present or in the future, but it is not always possible to distinguish and state determinately which of these alternatives must necessarily come about.

Let me illustrate. A sea-fight must either take place to-morrow or not, but it is not necessary that it should take place to-morrow, neither is it necessary that it should not take place, yet it is necessary that it either should or should not take place to-morrow. Since propositions correspond with facts, it is evident that when in future events there is a real alternative, and a potentiality in contrary directions, the corresponding affirmation and denial have the same character.

This is the case with regard to that which is not always existent or not always nonexistent. One of the two propositions in such instances must be true and the other false, but we cannot say determinately that this or that is false, but must leave the alternative undecided. One may indeed be more likely to be true than the other, but it cannot be either actually true or actually false. It is therefore plain that it is not necessary that of an affirmation and a denial one should be true and the other false. For in the case of that which exists potentially, but not actually, the rule which applies to that which exists actually does not hold good. The case is rather as we have indicated.

Basically, then, there are two arguments. First there is the argument that if statements about the future are already true, the future is necessary. If a sea battle will take place tomorrow, it will necessarily take place. Second, there is the argument that this excludes deliberation. If a sea battle will take place tomorrow, then it will necessarily take place, and no place remains for deliberation and decision about whether to fight the sea battle. Whether you decide to fight or not, it will necessarily take place.

Unfortunately for Aristotle, both arguments fail. Consider the first argument about necessity. Aristotle’s example is that “if it is true to say that a thing is white, it must necessarily be white.” But this is hypothetical necessity, not absolute necessity. A thing must be white if it is true that is white, but that does not mean that “it must be white, period.” Thus for example I have a handkerchief, and it happens to be white. If it is true that it is white, then it must be white. But it would be false to simply say, “My handkerchief is necessarily white.” Since I can dye it other colors, obviously it is not simply necessary for it to be white.

In a similar way, of course it is true that if a sea battle will take place, it will take place. It does not follow at all that “it will necessarily take place, period.”

Again, consider the second argument, that deliberation would be unnecessary. Aristotle makes the point that deliberation is causative with respect to the future. But gravity is also causative with respect to the future, as for example when gravity causes a cup to fall from a desk. It does not follow either that the cup must be able not to fall, nor that gravity is unnecessary. In a similar way, a sea battle takes place because certain people deliberated and decided to fight. If it was already true that it was going to take place, then it also already true that they were going to decide to fight. It does not follow that their decision was unnecessary.

Consider the application to gravity. It is already true that if the cup is knocked from the desk, it will fall. It does not follow that gravity will not cause the fall: in fact, it is true precisely because gravity will cause the fall. In a similar way, if it true that the battle will take place, it is true because the decision will be made.

This earlier discussion about determinism is relevant to this point. Asserting that there is a definite outcome that our deliberations will arrive at, in each case, goes against our experience in no way. The feeling of “free will,” in any case, has a different explanation, whether or not determinism is true.

On the other hand, there is also no proof that there is such a determinate outcome, even if in some cases there are things that would suggest it. What happens if in fact there is nothing ensuring one outcome rather than another?

Here we could make a third argument on Aristotle’s behalf, although he did not make it himself. If the present is truly open to alternative outcomes, then it seems that nothing exists that could make it be true that “a sea battle will take place,” and false that “a sea battle will not take place.” Presumably if a statement is true, there must be something in reality which is the cause of the statement’s truth. Now there does not seem to be anything in reality, in this scenario, which could be a cause of truth. Therefore it does not seem that either alternative could be true, and Aristotle would seem to be right.

I will not attempt to refute this argument at this point, but I will raise two difficulties. First of all, it is not clear that his claim is even coherent. Aristotle says that “either there will be a sea battle or there will not be,” is true, but that “there will be a sea battle” is not true, and “there will not be a sea battle” is not true. This does not seem to be logically consistent, and it is not clear that we can even understand what is being said. I will not push this objection too hard, however, lest I be accused of throwing stones from a glass house.

Second, the argument that there is nothing in reality that could cause the truth of a statement might apply to the past as well as to the future. There is a tree outside my window right now. What was in that place exactly 100 million years ago to this moment? It is not obvious that there is anything in the present world which could be the cause of the truth of any statement about this. One might object that the past is far more determinate than the future. There are plenty of things in the present world that might be the cause of the truth of the statement, “World War II actually happened.” It is hard to see how you could possibly have arrived at the present world without it, and this “necessity” of World War II in order to arrive at the present world could be the cause of truth. The problem is that there is still no proof that this is universal. Once things are far enough in past, like 100 million years, perhaps minor details become indeterminate. Will Aristotle really want to conclude that some statements about the past are neither true nor false?

I will more or less leave things here without resolving them in this post, although I will give a hint (without proof at this time) regarding the truth of the matter. It turns out that quantum mechanics can be interpreted in two ways. In one way, it is a deterministic theory, and in this way it is basically time reversible. The present fully determines the past, but it equally fully determines the future. Interpreted in another way, it is an indeterministic theory which leaves the future uncertain. But understood in this way, it also leaves the past uncertain.

Necessity, Possibility, and Impossibility

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I spoke here about various kinds of necessity, but did not explain the nature of necessity in general. And in the recent post on Hume’s idea of causality, it was not necessary to explain the nature of necessity, because the actual idea of causality does not include necessity. Thus for example a ball can break a window even if it would have been possible for someone to catch the ball, but the person did not do so.

Sometimes it is asked whether necessity implies possibility: if it is necessary that Tuesday follow Monday, it is possible for Tuesday to follow Monday? I am inclined (and I think most are inclined) to say yes, on the grounds that to say that something is not possible is normally understood to imply that the thing is impossible; thus if it is not possible for Tuesday to follow Monday, it is impossible. But this is largely a verbal question: regardless of how we answer this, the real point is that the necessary is the same kind of thing as the possible, except that possibilities are many while the necessary is one. And likewise, a count of zero for the same things implies impossibility. Thus there is something that we are counting: if we find none of them, we speak of an impossibility. If we find only one, we speak of one necessity. And if we find many, we speak of many possibilities.

What are we counting here? Let’s take an example. Horses can be white, or red, or brown, among other possibilities. So there are many possible colors for a horse. And on the other hand snow is always white (or so let us pretend.) So there is only one possible color for snow, and so snow is “necessarily” white. Meanwhile, air is always colorless (or so let us pretend.) So it is impossible for air to have a color. Based on this example, we propose that what we are counting is the number of forms that are suitable for a given matter. Someone might object that if we analyze the word “suitable” here it might involve some sort of circularity. This may well be the case; this is a common occurrence, as with desire and the good, and with virtue and happiness. Nonetheless, I think we will find it worthwhile to work with this definition, just as in those earlier cases.