Copyright © 1999 Massachusetts Institute of Technology
No matter what "Lake Woebegone" refers to, what time is being talked about, exactly what kind of majority "most" refers to, or what sense of "above average" is meant, this little proof is valid -- as long as the meaning of these terms is held constant. If we replace "most" by the determiner "all," "some," or "at least three," the proof remains valid, but if we replace it by "no" or "few," the proof becomes invalid. As the science of reasoning, logic attempts to understand such phenomena.
It is instructive to compare logic with linguistics, the science of language. Reasoning and using language have a number of properties in common. Both characterize human cognitive abilities. Both exhibit INTENTIONALITY -- they refer to objects, events, and other situations typically outside the skin of the agent. And both involve an interaction of SYNTAX, SEMANTICS, and PRAGMATICS. Given these similarities, one might expect logic and linguistics to occupy similar positions vis-à-vis cognitive science, but while linguistics is usually considered a branch of cognitive science, logic is not. To understand why, we must recognize that, apart from the properties noted above, logic and linguistics are strikingly dissimilar. What constitutes a proper sentence varies from language to language. Linguistics looks for what is common to the world's many languages as a way to say something about the nature of the human capacity for language. By contrast, what constitutes a proper (i.e., valid) piece of reasoning is thought to be universal. Twentieth- century logic holds that no matter what language our sample argument is couched in, it will remain valid, not because of some cognitive property of human beings, but because valid reasoning is independent of how it was discovered, produced, or expressed in natural language.
Since antiquity Euclidean geometry, with its system of postulates and proofs, was taken as the shining example of a logical edifice, having its foundations in what we would now call "cognitive science." Under the influence of KANT, nineteenth-century mathematicians and philosophers had assumed that the truth of Euclid's postulates was built into human perceptual abilities, and that methods of proof embodied laws of thought. For example, the great mathematical logician David Hilbert (1928 p. 475) wrote, "The fundamental idea of my proof theory is none other than to describe the activity of our understanding, to make a protocol of the rules according to which our thinking actually proceeds."
Given this historical relationship between logic, geometry, and cognition, it is not surprising that logic was profoundly influenced by the discovery of non-Euclidean geometries, which challenged the Kantian view, and which brought with them both a deep distrust of psychology and an urgent need to understand the differences between valid and invalid reasoning. What were the valid principles of reasoning? What made a principle of reasoning valid or invalid? Refocusing on such normative questions led logicians, following Gödel and Tarski in the mid-twentieth century, to the semantic aspects of logic -- truth, reference, and meaning -- whose relationships must be honored in valid reasoning. In particular, a valid proof clearly demonstrates that whenever the premises of an argument are true, its conclusion is also true. That is why our sample argument is valid, not because of cognitive abilities of humans, but because, if its premises are true, so is its conclusion.
Cognitive science, on the other hand, is concerned with mechanism, with how humans reason. Why do they make the reasoning errors they do? Systematic reasoning errors are at least as interesting as valid reasoning if one is looking for clues as to how people reason. Why are some inferences harder than others? For example, people seem to take longer on average to process the inference in our original sample argument than they do the one in the variant where "most" is replaced by "some" or "at least three," even though all three versions are valid. (This claim is based on informal surveys; I know of no careful work on this question.)
The logician's distrust of psychological aspects of reasoning has led to a de facto division of labor. The relationship between mind and representation is considered the subject matter of psychology, that between representation and the world the subject matter of logic. (There is no such division of labor involving linguistics since it has long been interested in mechanism.) This division has resulted in a distance between the fields of logic and cognitive science. Still, ideas and results from logic have had a profound influence on cognitive science.
Late-nineteenth- and early-twentieth-century logicians (e.g., Hilbert, FREGE, Russell, and Gentzen, before the shift to semantics noted above), developed FORMAL SYSTEMS, mathematical models of reasoning based on the syntactic manipulation of sentencelike representations. The import of "formal" in this context is that the acceptability of an inference step should be a function solely of the shape or "form" of the representations, independent of what they mean. Within linguistics, this has led to the view that a sentence has an underlying LOGICAL FORM that represents its meaning, and that reasoning involves computations over logical forms.
One might postulate that the logical forms involved in our sample argument are something like
(Early work did not treat determiners such as "most" and "few" at all -- only "every," "some," and others that could be defined in terms of them.) In this view, recognizing the validity of the argument would be a matter of computing the logical forms of the natural language sentences and then recognizing the validity of the inference in terms of these forms (see, for example, Rips 1994).
As models of valid reasoning, formal systems have important uses within mathematical logic and computer science, but as models of human performance, they have been frequently criticized for their poor predictions of successes, errors, and difficulties in human reasoning. Johnson-Laird and Byrne (1991) have argued that postulating more imagelike MENTAL MODELS make better predictions about the way people actually reason. Their proposal, applied to our sample argument, might well help to explain the difference in difficulty in the various inferences mentioned earlier, because it is easier to visualize "some people" and "at least three people" than it is to visualize "most people." Cognitive scientists have recently been exploring computational models of reasoning with diagrams. Logicians, with the notable exceptions of Euler, Venn, and Peirce, have until the past decade paid scant attention to spatial forms of representation, but this is beginning to change (Hammer 1995).
In the 1930s, Alan TURING, a pioneer in computability theory, developed his famous machine model of the way people carry out routine computations using symbols, a model exploited in the design of modern-day digital computers (Turing 1936). Combining Turing's machines and the formal system model of reasoning, cognitive scientists (e.g., Fodor 1975) have proposed formal symbol processing as a metaphor for all cognitive activity: the so-called COMPUTATIONAL THEORY OF MIND. Indeed, some cognitive scientists go so far as to define cognitive science in terms of this metaphor. This suggestion has played a very large role in cognitive science, some would say a defining role, and it is implicit in Turing's original work. Still the idea is highly controversial; connectionists, for example, reject it.
A third contribution of logic to cognitive science arose from research in logic on semantics. Most famously, the logician Montague (1974), borrowing ideas from modern logic, developed the first serious account of the semantics of natural languages; known as "Montague grammar", it has proven quite fruitful. One successful development in this area has been use of generalized QUANTIFIERS to interpret natural language determiners and their associated noun phrases (Barwise and Cooper 1981). The meaning of each determiner is modeled by a binary relation between sets; the relations themselves have very different properties, properties that can be used in accounting for associated logical and processing differences.
Our final application has to do with cognitive interpretations of the first of GÖDEL'S THEOREMS. This theorem, one of the most striking achievements of logic, demonstrates strict limitations on what can be done using formal systems and hence digital computers. Various writers, most famously Penrose (1991), have attempted to use Gödel's first theorem to argue that because there are things people can do that computers in principle cannot do, the formal systems of logic are irrelevant to understanding human cognition, although this argument is very controversial (see, for example, Feferman 1996).
If it is to be the science of full-fledged reasoning, logic still has much to accomplish. What features might this more complete logic have? The logician C. S. Peirce suggested that the relationship between mind, language, and the world was irreducibly ternary, that one could not give an adequate account of the binary relation between mind and language, or between language and the world, without giving an account of the relationship among all three. According to this view, the division of labor depicted above, and with it the divorce of logic from cognition, is misguided. Peirce's thinking has been reincarnated in the situated cognition movement, which argues that any adequate cognitive theory must take account of the agent's physical embeddedness in its environment and its exploitation of regularities in that environment (see SITUATEDNESS/ EMBEDDEDNESS).
Situatedness infects reasoning and logic (Barwise 1987). The ease or difficulty of an inference, for example, depends on the agent's context in many ways. Even the validity of an inference is in a limited sense a situated matter because validity depends not just on the sentences used, but on how they are used by the agent. This arises in our sample argument in the requirement that the meaning of the terms be held constant. The way the agent is situated in the world in part determines whether this caveat is satisfied. For example, if the agent uttered the two premises in different years, the conclusion would not follow.
Logic has had a profound impact on cognitive science, as the above examples show. The impact in the other direction has been less than one might have expected, due to the distrust of cognitive aspects of reasoning by the logic community. One hopes that the synergy between the two fields will be greater in years to come.
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Copyright © 1999 Massachusetts Institute of Technology