Paradox is the product not necessarily of a mistake in reasoning but of a defect in substance. A way of looking at a paradox is a conflict-engendering argument, a piece of deductive reasoning leading to a contradictory result. A paradox sometimes arises in action theory when a person sometimes knowingly and intentionally acts contrary to her own best judgment. This has been termed the problem of incontinence. For example, someone may intend to cut back on sugar consumption but decides to eat a piece of cake anyway. The person has acted incontinently.
The challenge is to show why the person might not have acted incontinently, against one’s own best judgment. Wittgenstein’s views on inconsistency hint at an approach to the problem of incontinence that seems to suggest not how to solve the paradox but how to circumvent the problem altogether. When we arrive at an inconsistent conclusion by a certain proof in mathematics, we should abandon the parts of the proof that lead to the inconsistent conclusion. The aim of this paper is to show that if we apply this procedure to the problem of incontinence, then the inconsistency dissolves for the incontinent person.
Wittgenstein talks of different types of inconsistency in his works, but the one most interesting for this paper is his view that “when [an inconsistency] comes out into the open it can do no harm” (Wittgenstein, Lectures on the Foundations of Mathematics, p. 219). The problem is that if we leave an explicit inconsistency in the system, then it will likely have disastrous practical consequences. For example, should we find and leave explicit inconsistencies in arithmetic, it would be unwise to use this arithmetic to build bridges. Leaving the inconsistencies in the system will lead to the bridge’s collapse, probably killing the people using the bridge. The harm of leaving inconsistencies in a system occurs in trying to apply the formal system to something that will affect people. It is likely that more disasters will occur if a system that contains inconsistencies remains in use to build bridges, skyscrapers, and other buildings. There are at least three possible explanations for a bridge’s collapse. First, the theories engineers used were incorrect or inaccurate. Second, the engineers made a mistake in their calculations. Finally, it could be that the logical system was unsound and led the engineers to make invalid inferences.
The most plausible explanation for the bridge’s collapse is that either the engineers made a mistake in their calculations or used incorrect theories. The focus of this investigation is theories, mathematical or logical, that contain inconsistencies. If a system contains inconsistencies, then it may lead us to make invalid inferences. So, in the example cited, the engineers, if they had known of the system’s flaw, should have used a different consistent system.
Logicians commonly abandon systems that include inconsistencies, particularly contradictions, because the system cannot make substantive claims. If a contradiction implies everything, then there is no way one could be wrong. Everything follows from a contradiction. No answer is false, which seems like a good reason to not take them
seriously. The contradiction makes any consequence of the theory useless.
Without a way to tell whether something is useful, the best we can do is to not use the system that leads to contradictions. This seems to correspond with what Wittgenstein says:
Well then, don’t draw any conclusions from a contradiction; make that a rule. You might put it: There is always time to deal with a contradiction when we get to it. When we get to it, shouldn’t we simply say, “This is no use– and we won’t draw any conclusions from it?” (Wittgenstein, Lectures on the Foundation of Mathematics, p. 209)
When contradictions emerge from a system, we need to get around them somehow. The contradiction plays no important role in the system because it and the consequences that follow have no use.
First, we should notice that the passage suggests we make a rule that when we encounter a contradiction, “don’t draw any conclusions.” We should equip the system with a self-correcting rule. The rule should indicate that nothing follows from a contradiction. If there is no self-correcting rule, then everything will follow from the contradiction rendering the system trivial. The rule circumvents the uselessness of contradictions.
Second, it seems that Wittgenstein denies the usefulness of a contradiction. When we say that a contradiction has no use, we mean that contradictions lack sense. Contradictions fail to function usefully in language by having no substantive meaning.
Logical jamming is neither a failure of the listener nor of the speaker. If it is a logical jamming, then it is a consequence of the way the proposition is built. No action follows logically from the order. Is logical jamming possible? Wittgenstein seems to want to deny the logical jamming account.
What I am driving at is that we can’t say, “So-and-so is the logical reason why the contradiction doesn’t work.” Rather, that we exclude the contradiction and don’t normally give it a meaning, is characteristic of our whole use of language, and of a tendency not to regard, say, a hesitating action, or doubtful behaviour, as standing in the same series of actions as those which fulfill orders of the form “Do this and don’t do that” – that is, of the form ‘p · q’. (Wittgenstein, Lectures on the Foundation of Mathematics, p. 179)
The reason we cannot comply with the order is that we have no way of complying with that order. The order tells the listener to proceed by acting in opposite ways. The use of an order is to compel the listener to comply. The reason we have no use for the proposition is that we cannot apply it to different situations. It is not a matter of the proposition having such-and-such logical form. The logical form is contradictory, but this is not the reason why we have difficulties. The way the proposition is built has nothing to do with why we cannot comply with the order.
Denying a logical jamming of contradictions seems to deny the ordinary rules of logic. Contradictory propositions are necessarily illogical. There are two possible interpretations of “necessarily illogical” Wittgenstein may try to avoid. First, the phrase could mean that we could not possibly have avoided the problem. We must get into trouble if we encounter a contradiction. Wittgenstein wants to deny the logical must as much as possible. Second, the phrase could mean that we will necessarily tempt us into trouble. If a proposition necessarily tempts us into trouble, then everything counts as a contradiction. In both interpretations, the way the proposition is built does not necessarily get the listener into trouble. We only need to realize that there is a contradiction present. Realizing a contradiction is present permits us to figure a way around it.
Other places in which contradictions arise include mathematics. They usually arise either as a part of the rules or in matters of configuration. It makes no sense to say that contradictions arise in matters of configuration. Configurations can be manipulated in a number of different ways without being inconsistent. If we decide to start the game of Monopoly at FREE PARKING instead of GO, there is nothing inconsistent about this at all. We just set a new configuration. One may say that we cannot uphold both configurations of Monopoly without falling into contradiction, but this is not what the traditional conception of logical inconsistency. Logical and mathematical inconsistencies imply that two propositions or axioms of the system, which we supposed to be self-evident, support a contradictory conclusion.
Contradictions cannot arise as a matter of configuration in games or mathematics. Matters of configuration are merely creations. Creations fail to have the logical force because they could have been different than they actually are. Logical inconsistency is much stronger and seems as if it could not have been any other way.
Contradictions more likely arise in the rules of a game. When two rules contradict one another, this resembles logical inconsistency. The two rules do not seem to fit in some logical way. For instance, suppose that there is a rule where one type of player cannot do A, but can do B and there is another rule where one type of player can do A, but cannot do B. We could find a player who falls under both categories of player. We would have a logical inconsistency. It seems that playing the game would be impossible should we try to play it while upholding the two rules.
In the later writings on mathematics, Wittgenstein attends to the application of mathematics in order to see the kind of meaning mathematical propositions have. Mathematical proofs are supposed to show how to use the theorem. Proofs follow rules. Without the use – the application – the theorem is just a string of symbols without any mathematical content at all. Before we have the proof, the theorem does not really mean anything. If the theorem does not mean anything, then we do not have enough to understand the proof. The use is the most important aspect of Wittgenstein’s conception of mathematics. Without a use, the proof cannot be applied. If the proof cannot be applied, then that bit of mathematics is empty.
Without the application of mathematics, descriptive propositions, i.e. rules, fail to have any use. We could say that without a use the theorem is useless. Wittgenstein may be implying that if a rule has no use, it shows us its uselessness. The use of the theorem, then, is the fact that it is useless. Such a use may be contrary to fact, but it still has a use and constitutes a mathematical proposition. The mathematical proposition describes a situation in which the proposition is useless. By so describing the situation, we see the proposition’s use. Regardless, we need a use for the proof to be applied.
Another way of seeing the importance of use is to attend to whether mathematical propositions are about something. The realm with which mathematics is concerned has nothing to do with a reality as such; it deals with a Platonic realm of forms. Mathematical propositions are not about something, like tables and chairs. If mathematical propositions
were about something analogous to the way we reference chairs and tables, then we presume that mathematical entities exist. Wittgenstein says:
“20 apples + 30 apples = 50 apples” may not be a proposition about apples.
Whether it is depends on its use. It may be a proposition of arithmetic – and in this case we could call it a proposition about numbers.
You’d expect, if it is about numbers, that you’ve made a discovery in a new realm. But it is not in a new realm at all. You have made something entirely different. (Wittgenstein, Lectures on the Foundation of Mathematics, p. 113-114)
Wittgenstein wants to avoid saying something about a new realm of entities. Numbers in arithmetic cannot be about numbers in the same way saying ‘apple’ is about an apple. In order for numbers to be about something, they must have a use. If there is no use, then the numbers dissolve, so to speak.
Mathematical propositions have meaning by virtue of whether they can be applied practically. For mathematics, there are certain rules of the proof that show how to use the theorem in question. We could say the same thing for practical reasoning. Practical reasoning may not be about anything, but our actions may depend on the use of rules to perform or not to perform certain actions. Understanding propositions of mathematics in terms of their use clarifies the philosophical confusion we have when we think that mathematical propositions of practical reason. We must understand them in terms of their use. We should abolish thinking about morality in terms of talking about moral facts.
The dependence on use in mathematics and practical reasoning makes the similarities between them apparent. Primarily, we are interested in the way we use propositions. No matter where we find ourselves, in mathematics or practical reasoning, whether there is a use of the proposition is a primary concern.
According to Wittgenstein, it seems that mathematics may be intimately associated with practical reasoning. If this is true, then the way we solve inconsistencies in mathematics may be applicable to practical reasoning as well. We have see that the way Wittgenstein suggests solving inconsistencies is to circumvent the problem. We may not understand a part of the proof leading to the contradiction, so we just have to get around it.
Some concrete examples will assist us in seeing how Wittgenstein may have suggested solutions to problems of mathematics and practical reasoning. First, suppose we have a proof that 4 + 1 = 6. Part of the proof for this contradictory conclusion is Russell’s Paradox, the set of all sets that do not contain themselves. We do not understand what to say about the paradox. In fact, the paradox seems to fail to have any use. Since we have already determined that the parts of a proof need a use in order for it to be a part of the proof, we just throw away Russell’s Paradox. The contradictory conclusion is false, and we circumvent the problematic part of the proof. We have extinguished the contradictory conclusion.
Wittgenstein seemingly endorses this suggestion in a similar example. Wittgenstein entertains the idea that 4 = 5. This is a result of dividing both sides of the equation 4 X 0 = 5 X 0 by 0, or using the rule: if a X b = c and d X b = c, then a = d. Wittgenstein says, “if the danger is simply that someone might go this way unawares and get absurd results which we do not want, then the only thing is to show him which way not to proceed from a contradiction” (Wittgenstein, Lectures on the Foundation of Mathematics, p. 222).
In a similar way, we can control the problem of incontinence. There are two examples we can use to show how Wittgenstein would want to circumvent the problem of incontinence. For example, suppose that Jones prefers to cut her consumption of sugar over having a piece of cake after dinner, but she has a piece of cake after dinner anyway. Just before dessert her preferences reversed: she preferred having a piece of cake over reducing her consumption of sugar. We can say that Jones has acted incontinently since the piece of cake exceeds her intended consumption of sugar for the day. In the mathematical proof, we freed it of contradiction by eliminating Russell’s Paradox. In this case, we must do something similar. Jones’s wants to cut back on sugar consumption more than she wants to have a piece of cake, but she eats the cake anyway. By eating the piece of cake, a contradiction has emerged. To rid the situation of a contradiction Jones merely needs to not get into situations where she is likely to desire a piece of cake. Jones
just doesn’t eat the piece of cake.
Another example that Wittgenstein might be able to help us resolve is inspired by an example in Michael Bratman’s “Toxin, Temptation, and the Stability of Intention” (1999). I love a good Café Americano, but I also enjoy getting to sleep before 11:00pm. I know that if I have more than one Café Americano, I will not be able to get to sleep until around 2:00am. Prior to going to Cocoa Caffe Wednesday evening I prefer one Café Americano and getting to sleep before 11:00pm to more than one Café Americano after dinner and getting to sleep around 2:00am. While at Cocoa Caffe, after my first Americano, my preference reverses. I succumb to this desire and have a few more Americanos. After leaving the café, my preference changes back. Throughout my time at Cocoa, I continue to prefer one Café Americano and getting to sleep before 11:00pm. My intentions are inconsistent since while I was at the café I preferred one Café Americano
and getting to sleep before 11:00pm and I preferred more than one Café Americano and getting to sleep around 2:00am. I have acted incontinently. When someone shows me which way to proceed, I just need to admit that I have to get around it another way. As Wittgenstein seems to suggest, I just don’t go to Cocoa.
In the case of incontinence, the only thing we need to not go to Cocoa or not get into the situation where I’m tempted by the cake. We get around the problem of incontinence by abandoning the part of the proof where the contradiction arises. We could make it a part of our system that we just abandon those proofs or arguments that lead to contradiction. The new rule rids the system of contradictions. Thus, incontinence never arises.