Wittgenstein contra Dummett on determination
Guns don’t kill people, people kill people - though sometimes they need to use guns to do it. Similarly, Wittgenstein says past examples and statements of rules for usage don’t determine how words are to be applied, rather people determine applications though they sometimes it takes a person and a past examples and statements of rules to do it.
In a certain way this thesis seems like it should be uncontroversial. On the one hand, it’s well known that any finite sequence of examples doesn’t logically determine how to continue in new cases: there are always different rules which agree on all the examples given but diverge at some other point. So it can’t be that past examples of how a word has been used act on their own to determine how it should be used in the future. And on the other hand, everyone knows from Lewis Carroll’s story of what the tortoise said to Achilles that a suitably obstinate person can keep accepting every logical rule someone writes down without accepting the consequence which these rules entail. The mere statements of logical laws which we keep dictating and the tortoise keeps accepting (‘if P then Q, and P, than P’ ‘If, P and ‘if P then Q’ and ‘if P then Q and P, then P’ then P) don’t determine the perverse reptile to accept that P, however explicit we make them. So one might think that everyone would agree that a human element is needed and mere examples and statements of rules on their own can’t determine how a rule is to be followed or word applied.
But in fact, lots of smart people don’t agree with this Wittgenstinean thesis. I think this is mostly because it’s initially hard to see how this account can make sense of the ordinary cases where we do say that a formula determines a particular instance or a convention determines future actions. For example Michael Dummett criticizes Wittgenstein’s philosophy of math, by saying that it’s crazy to think that we have a free choice in accepting the conclusion of a proof. He finds Wittgenstein’s view that “there is nothing in our formulation of the axioms and the rules of inference, and nothing in our minds when we accepted these before the proof was given, which of itself shows whether we shall accept the proof or not; and hence there is nothing which forces us to accept the proof” implausible since, “it appears that the mathematical proof drives us along willy-nilly”.
Now, I claim that the Wittgenstinean thesis just mentioned – that rules and examples don’t determine application on their own- can make sense both of the feeling of determination, and of the distinction we draw between formulations of a rule which do determine how we are to treat a certain case and those which don’t and ultimately of when premises determine a conclusion. Thus I think Dummett is wrong to worry that Wittgenstein’s philosophy of math can’t account for the sense in which a set of axioms can commit us to a conclusion or the feeling that we are not freely choosing but compelled when we accept the conclusion.
So, the story I want to tell about rule-following starts with a person being given a bunch of examples of how to follow a certain command or apply a certain word. Remember that we said that no finite collection of examples logically determines how a rule is to applied in new cases, since there are always logically possible rules which agree on the examples given but disagree about new cases. Nevertheless there are some sets of examples you can show to a person such that afterwards they will have definite ideas about when new objects count as examples of the same thing. And, even more remarkably people (especially people with the same culture and history) tend to ‘get an idea of what it would take to continue’ from the same sets of examples, and tend to continue from these sets of examples in the same way. It’s as though certain patterns come more naturally than others to everyone. So, for example if you wanted someone to continue the pattern 111111111… you might only have to give them the first four places 1111, but if you wanted then to continue the pattern 1111222233334444 you would definitely have to give them more than the initial four places 1111. Now, I think that one way of getting someone to use a word correctly or obey a command is to expose them to show them some of these suggestive sets of examples until a) they feel in inclination to say that new cases either do or don’t follow the rule and b) what they say about these cases agrees with what everyone else says. ‘Ball!’ might be an example of a word we learn to use in this way.
In this initial stage I think we can already see three important things. First off, there’s a (vague) distinction between those sets of examples which give people an idea of what it would take to continue in the same way and those which don’t. When someone shows you 1111 or 1235711, you have an immediate idea of what should come next to continue the pattern but if someone shows you 78992 you don’t. Similarly when examples are given about how to apply (or not apply) a word there’s also a (somewhat vague) distinction between the new cases where all the people who have seen the examples are inclined to agree in applying or withholding the term and cases where people are unsure or different people do different things. In this sense we could say that a certain set of examples does or doesn’t determine what to do with a new example. Secondly, there’s a feeling that what you have already seen determines, or forces you to treat new cases in a particular way insofar as you want to continue the pattern. If someone shows you a bunch of examples of grade A eggs you might feel forced to admit that the eggs your chickens lay at home aren’t like the ones you’ve seen. Finally, there’s the fact that where a given set of examples falls with respect to this distinction between examples which determine how to go on and those that don’t, and creating a feeling of compulsion to continue a series in a certain way depends not only on the examples but also, crucially, on human nature. We feel natural in and perhaps even compelled to continue the sequence 1111 with …1111, but there could be perfectly rational aliens who felt equally compelled to continue it …22223333 or who didn’t feel that it suggested any particular continuation. Thus the examples don’t determining future conduct on their own, but only in consort with a person.
Now, admittedly, this feeling of compulsion and distinction between examples which do or don’t determine how to respond to a new case aren’t what Dummett is talking about. But, on the story I am telling they will turn out to be a sort of primitive ancestors of these notions.
The next step is to suppose that someone is trained in the way just mentioned to apply a bunch of different words. They can then be trained, again by example, in how to apply certain combinations of these words, or what to do when a certain modifier is added to a certain words. Just as originally hearing ‘red’ applied to strawberries and tomatoes gave them an idea about whether it applied to cherries, hearing how ‘not red’ and ‘not orange’ and ‘not striped’ are applied can give them an idea of how to apply ‘not green’. Thus someone can learn to apply a phrase without seeing any direct examples of how it applies but rather by seeing examples of how all the various components and compositional devices within it work. So now our subject’s language includes complex expressions with internal structure. Being the inquisitive creatures they are they may then notice some patterns in their applications of sentences and phrases of the form that everything which they are inclined to apply one phrase to they are also inclined to apply another phrase to. These patterns between how different terms are applied can acquire a kind of motivational force on their own. So that if normally you would apply ‘not not red’ in almost all the cases where you have applied ‘red’ and then you notice this as a pattern you will likely be inclined to modify your future usage so that you now apply these terms to exactly the same things. We feel a sort of compulsion to apply words to new cases in a way that fits with the patterns (either in cases where the word has been applied in the past or in the relation between how one word is applied and how another is) which strike humans as simple and salient.
Now, I realize that giving a theory of logical consequence is a tall order on its own and if we add to this the claim that the theory in question was Wittgenstein’s one arrives at something which is almost certainly hubristic. So I won’t claim that the theory I am now articulating was Wittgenstein’s but I do think that it combines the attractive view of logic and logical necessity in the Tractatus with the ideas about the absence of external determiners in rule-following and the abandonment of deep objective logical structures for thoughts in the Investigations. On this view A being a logical consequence of B is ultimately founded on facts about our use of language: (roughly[1]) that we are always willing to apply B whenever we are willing to apply A. The amazing scope of logical necessity, that it holds for every possibility however remote, is accounted for not by any deep structure of the world, but rather by our own abiding dispositions in using language.
Now, if this view of logical necessity and consequence works, it shows us how the necessitation of a proof’s conclusion by its premises and our feeling of compusion in reading it are ultimately consistent with ‘free choice’ and the absence of determination in the sense of the Wittgenstinean story about rule following. For it’s a contingent special fact about humans that certain patterns come to us naturally and exert a compelling force on our future behavior. Our past usages of language and the premises of the proof on their own don’t determine anything further. An alien who responded to different patterns might continue to ‘use the words of the premises in the same way’ (according to its different sensibilities about how to go on in the same way) while rejecting the conclusion. In this sense accepting the conclusion is a free human choice. But at the same time, there are striking (to us) patterns in the choices which humans make, and noticing these patterns inclines us to conform to them even more. And logical necessity arises from these patterns in how we choose (and feel we must choose). Thus, relationship between what people are inclined to say and logic is reversed with respect to the traditional picture. Rather than logical laws articulating the deep structure of the world and people magically being able to discern this and then conforming our use of words to these laws, on this picture there are first patterns in our use of words which then acquire motivational force for us and get codified in the laws of logic. So when we accept the conclusion of a proof we are at once making a free choice which goes beyond what the premesis and our past usage and state of mind determined and obeying the irresistible laws of logical consequence.
Despite what Dummett thinks, free choice in the sense of rule-following and the logical determination of a proofs conclusion by its premises are no more at odds then freedom and determinacy are in the classic problem of free will. If the laws of logic arise from and articulate patterns in how humans are inclined to freely choose to extend past examples and expression, just as ascriptions of beliefs and desires arise from and articulate patterns in action, there is no more question of how someone’s free choice can be determined by the laws of logic then of how someone’s free actions can be determined by their beliefs and desires.
Slogan: the laws of logic articulate patterns in how we freely choose to ‘go on in the same way’ in applying our terms.
[1] There is, of course, the question of weeding out general firmly held empirical beliefs that A always indicates that B and patterns arising from applications which the agent would under more ideal circumstances recognize as erroneous.