Laws of Logic

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

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.