Overview Of Deduction
| Are we capable of reasoning logically? If so, do we typically reason logically or is this a rarely seen capacity? In order to approach these questions, we must first see whether we can characterize what it means to reason logically and determine whether there are certain capacities that are presupposed. |
| The figure to the right provides my take at succinctly defining deduction in the proof theoretic sense. Deduction presupposes what I refer to as a language, L, that is suitable for expressing statements about a domain. Note that there is not a requirement that this language be universal or "domain independent," although this is usually presumed when we speak of logic and deduction. Secondly, we require some way of forming complex statements from "simple" statements. The logical connectives are used for this purpose. And, finally we include the quantifiers that allow us to make statements over individuals. |
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The language is used to construct statements that are about some domain. In particular, each statement is to have an unambiguous meaning; namely, it is to be either True or False but never both. Now to carry out deduction, we must start with a set of statements about the domain of interest. And, we require that the statements be consistent; that is, we should not be able to derive a contradiction 'p and not p' from the initial set of statements.
Given all this, deduction is simply a procedure that adds statements to S, but in such a way that it is guaranteed that inconsistency not be introduced....said another way, the procedure preserves the truth value assignments of the statements.
What have we required by virtue of this definition? First, note that we have made a distinction between statements about some world and the world that the statements are about. This is often referred to as the distinction between syntax and semantics in logic. This distinction is not unique to logic. A basic assumption about human reasoning is that the "contents" of the mind are about things. For example, I believe that I am typing at this moment. The belief itself is not "me-typing" but a statement that represents "me-typing". Thus, one of the main requisites for reasoning logically is in our possession of a syntax within which to form statements about "things". Let's call this a representational capacity.
A second requirement that arises from our definition is that: 1) statements must be True or False but not both; and 2) the truth value of a statement cannot be changed. Collectively, we can refer to this as the assumption that deduction is truth functional. Consequently, if some statement, s, is True; then the deductive procedure must insure that no statements are ever added that would allow the conclusion 'not-s'.
This is a very strong constraint.It may help you to appreciate this constraint if you think back to the cryptarithmetic problem. The requirements for that problem were quite analogous to this requirement of truth functionality. Each letter had to have one and only one value, and the value couldn't be changed. In that problem, the representation of the problem as a set of equations helped us to see the dependencies that existed among the individual letters. In the present deductive case, we simply have a set of statements. There are no explicit clues in the syntax of the statements to suggest the way in which the truth value of one statement might depend on the truth value of other statements.
There are two ways to deal with this consistency constraint; and, we have seen that both have been considered in the study of deductive reasoning. The first, is to limit the deductive procedure to adding those statements which are permitted under all possible assignments of truth values to the statements that are used in the derivation. This is simply a way of saying that the resulting statement can be derived independently of the particular truth values assigned to the statement in the model. If deductive inference is limited in this fashion we obtain what is termed a monotonic logic; one where no statement is ever withdrawn.
A second way to deal with the consistency constraint is to explicitly check the constraint. Thus, if I "deduce" p; then before adding p to the set S, I see whether I can "deduce" not-p. If I can not, then I allow p to the added to the set S. The adoption of this strategy allows many more statements to be "deduced." The disadvantage is that in the best of circumstances it is very difficult to test whether both p and not-p can be derived from a set S; and, in the worst of circumstances it is impossible. Further, if p is added, and then later q; it may turn out that the addition of q now allows not-p to be "deduced". Thus, although the check for consistency sounds like a "local" property--just check that you can't derive 'p and not-p"; it is really global--that is the check must be made for all p. |
Well, where does that put us with respect to our original question; namely, do we reason logically? On the one hand, it does seem that we possess the basic requirements to reason logically. Our mind uses representations and implicitly understands the distinction between syntax and semantics. And, I suspect that for at least some domains we naturally think of statements about the domain being either True of False but not both; and, therefore recognize that 'p and not-p' makes no sense.
But, what I have tried to show in this little discussion is that deduction turns out to be quite a tricky procedure to realize. And it turns out that deduction in first order logic is only semi-decidable and the problem of deduction is again one of those problems that computer scientists refer to as intractable problems. Consequently, it seems very unlikely that deduction in this technical sense is something that characterizes our reasoning. However, this is not the same thing as saying that our reasoning is illogical or that we do not employ patterns of reasoning that lead us to conclusions. From this point of view, the important question for cognitive psychologists is not whether or not in some special and usually quite simple experimental situations our reasoning arrives at the same conclusion as a formal logic. Rather it is to identify those patterns of reasoning that we do employ to derive new beliefs from our existing beliefs. Clearly, we do add beliefs based on our current beliefs. And, clearly we usually do that under some constraints. We don't usually accept that anything follows from anything. |
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