During the past century, the focus of work in logic has principally been on the development of logical calculi that bear little apparent relation to the structure of natural language. That is, the syntactic forms postulated for purposes of logical reasoning (e.g. the logical forms of the propositional and predicate calculi) show little relation to the forms postulated within current grammatical theory. This apparent mismatch between logical form and grammatical form is ordinarily taken as an unsurprising consequence of the fact that, since Frege and Russell, one of the central goals of logic has been to help clarify (if not establish) the foundations of mathematics. Mathematics aside, however, it has also been held that natural language is a rather poor medium in which to couch logical reasoning, since natural language is ambiguous, vague, etc.
Yet if we step back a bit and view the development of logic over the last two millennia, we find a rather different picture. Up until the beginning of this century logic was very much concerned with representing the inferences that are made in natural language. So, as everyone knows, classical Aristotelian logic attempted to characterize the valid syllogistic arguments, and the strategy was to elucidate the natural language forms that underwrite valid inferences. It is less well known is that in the 2000+ years after Aristotle a number of successful efforts were made to expand the scope of classical logic as well as to find ways of generalizing and simplifying the rules of inference to a few core cases. Let's use the phrase 'natural logic' to cover this broad research program.
Among the interesting insights of the medieval logicians was the idea that the valid arguments of the syllogism could be generalized to fall within two basic inference paradigms, which they called the dictum de omni and the dictum de nullo, and much of the effort expended in natural logic concerned the characterization of the grammatical environments that corresponded to these two dicta. More recently, it has been observed by Hoeksema (1986) that the dicta de omni et nullo correspond to the upward entailing and downward entailing environments widely discussed in linguistics literature (e.g. in Ladusaw 1979). In effect, the medievals were looking for the syntactic reflexes of directional entailingness.
Can we find the holy grail of natural logic?; can there be syntactic accounts of directional entailingness? Sánchez (1991, 1995) and Dowty (1994) have offered proposals that introduce "monotonicity markings" into the Lambek calculus and categorial grammar respectively, but these efforts, or so I shall argue, have an ad hoc character. Taking an alternative tack, I will describe a formal language L* from Law and Ludlow (1985) in which directional entailingness can be defined in terms of independently motivated features of the syntax of L*. In particular, as proved in Ludlow (1995) a term in L* is in a downward entailing environment if all its occurrences are in the scope of negation and an upward entailing environment if it has no occurrences in the scope of negation. I will then argue that the LF representations in current linguistic theory can easily reflect all the relevant syntactic properties of L*, and I will show how the relevant LF representations can be derived utilizing only "off the shelf" syntactic resources (e.g. functional heads for polarity, number, and conjunction, a copy theory of movement, basic ideas about feature checking and chain formation, and unselectively bound free variables).
Along the way I will have some speculative remarks about the syntactic environments which license negative polarity items, and will make a plea for austerity within semantic theory.
Dowty, D., 1994. "The Role of Negative Polarity and Concord Marking in Natural Language Reasoning." Proceedings of SALT IV. Hoeksema, J. 1986. "Monotonicity Phenomena in Natural Language." Linguistic Analysis 16, 235-250.
Ladusaw, W. 1979. Polarity Sensitivity as Inherent Scope Relations. Doctoral dissertation, University of Texas at Austin. Law, D. and P. Ludlow. 1985. "Quantification without Cardinality." In Berman, Choe, and McDonough (eds.) Proceedings of NELS XV, Amherst: GLSA.
Ludlow, P., 1995. "The Logical Form of Determiners," The Journal of Philosophical Logic 24, 47-69.
Sánchez Valencia, V., 1991. Studies on Natural Logic and Categorial Grammar. Doctoral Dissertation, University of Amsterdam.
Sánchez Valencia,V., 1994. "Monotonicity in Medieval Logic," In A. de Boer et al (eds), Language and Cognition 4, Yearbook 1994 of the Research Group for Theoretical and Experimental Linguistics at the University of Groningen.
Sánchez Valencia,V., 1995. "Natural Logic: Parsing-driven Inference." Linguistic Analysis 25, 258-285.