the week of september 24

The big topics: meanings constraining logic forms

Apart from filling in some formal background, we used this week to work on two very different arguments showing that the logical form of sentences might not always be what we think it is, or what tradition claims it is.

Trying to find a way to avoid both the so-called "Paradoxes of Material Implication" and the unwelcome consequences of Gibbard's proof, we concluded that we had to give up one very basic assumption about if ....then clauses: that there is a two-place connective in natural languages that corresponds to if ... then. If-clauses should probably be analyzed as adjuncts that modify possibly non-overt operators like modal operators or adverbs of quantification. In the best of all possible worlds, then would simply be a species of anaphoric pronoun. Here is some history leading to what is now often called the "restrictor" view of if-clauses. An important contribution on the syntactic side of the story was Michael Geis' dissertation on Adverbial Subordinate Clauses in English (MIT 1970). On the semantic side, evidence accumulated during the sixties and seventies that not all types of conditionals could be analyzed as material implications, and, crucially, not even as material implications under the scope of some modal operator or adverb of quantification: Lewis' 1973 paper "Adverbs of Quantification" played an important role in linguistics, but in philosophy and philosophical logic, it had already become clear that probability conditionals, deontic conditionals, and counterfactuals needed a non-traditional analysis. In my 1978 dissertation and subsequent work, I proposed a unified analysis for all if-clauses as restrictors of overt or non-overt operators. The analysis also explains why indicative conditionals without overt modal operators or adverbs of quantification come in different flavors and can be interpreted as material implications in the limiting case. A summary of some of this work is my short 1985 paper "Conditionals". An important follow-up is von Fintel's 1994 Restrictions on Quantifier Domains. Bhatt and Pancheva's survey article Conditionals represents the state of the art with respect to the syntax of conditionals.

The second big question we investigated was whether all of a verb's arguments are true arguments of the verb root, or whether at least some of a verb's argument structure is syntactically constructed via neo-Davidsonian argument association. We looked at Barry Schein's argument for neo-Davidsonian association of the agent argument. We attempted to formalize a simplified version of Schein's original sentence "Three video games taught every quarterback two new plays" (Schein 1993) and saw that we can't seem to get an adequate formalization without neo-Davidsonian association of the agent argument. We convinced ourselves that Schein's argument carries over to other external arguments, but we couldn't make a similar case for theme arguments and other internal arguments. There may be a deep semantically hard-wired asymmetry between external and internal arguments, as proposed in my Event Argument. This asymmetry may ultimately account for the difference between unaccusatives and unergatives. Unaccusative roots do, but unergative roots do not have a non-event argument.

material implication

You practiced the "quick falsification method" ("reductio ad absurdum", PMW pp.108 -110) with two examples which, at the same time, illustrated some apparent shortcomings of a material implication interpretation of English indicative conditionals. The first example was a version of the three barber example by Lewis Carroll. The second example was Ernest Adams' switch example. I hoped to convince you that the material implication interpretation cannot be the correct interpretation for English indicative conditionals. Noah wrote up his thoughts on this issue and talked about his observations on Friday (access restricted). Something interesting happened in class when we tried to assess the truth of the conditional if Allen is out Brown is in on the Lewis Carroll scenario. You all were tempted to fill in a modal and said things like "it's not necessary that if Allen is out then Brown is in", or "it could be that both Allen and Brown are out and Carr is in". Might this tell us that there are hidden operators in those conditionals that may be non-overt modal operators or adverbs of quantification? Since this proposal came up spontaneously, I promised I would walk you through Gibbard's proof (Allan Gibbard). Suppose we claimed that English sentences of the kind "if A then B" had to be analyzed as material conditionals under the scope of an operator, as in "necessarily (A → B)" or "always ( A → B)". If that proposal amounts to saying that there is a two-place connective "if ...then" such that "if A then B" is to be understood by definition as "necessarily (A → B)" or "always ( A → B)", then Gibbard's proof will get us into serious trouble. The proof shows that, given three very plausible assumptions about that alleged two-place connective, it has to express material implication.

getting around gibbard's proof

On Thursday, we went through Gibbard's proof. We were ready to accept the crucial three premises of the proof, but felt uneasy about a hidden premise: is "if ...then" truly a discontinuous two-place connective? What if we had to analyze "if"-clauses as regular adjuncts without any inherent connection to the pronoun "then", as Geis had proposed? Those adjuncts could then restrict possibly non-overt quantificational operators, including modal operators and adverbs of quantification. The impact of the proposal is most easily seen with stacked "if"-clauses. Stacked "if"-clauses are now analyzed like any other stacked adjunct clauses, stacked relative clauses, for example. Like stacked relative clauses, stacked "if"-clauses should be able to constrain one and the same operator, and do no longer have to involve structures where whole conditionals are embedded within other conditionals. The adjunct view of "if"-clauses seems to provide an escape from the devastating verdict of Gibbard's proof, while still allowing us to endorse the hidden operator view suggested by the so-called "Paradoxes of Material Implication". It now becomes possible to maintain that the logical form of "If A, then if B, then C" is "((operator: if A): if B) C", not "operator (A  →  operator ( B →  C) ". The second, but not the first formalization can be parsed as involving a two-place connective satisfying Gibbard's three premises (see the definition of such a connective above). Consequently, the second, but not the first formalization falls under the verdict of Gibbard's proof.

logic background for this week

In PMW, 13.2.3, pp. 343 - 348, you are introduced to a typed lambda calculus that is labelled "TL" for "Type Logic". You are first presented with a syntax that generates all and only the well-formed expressions for a given type (subsection I). In subsection II, you find a compositional semantics that assigns to every expression of a given type a corresponding denotation of the same type. All expressions of type t are assigned a truth-value, for example. Denotations are relativized to a model. What are models used for in logic? They give you a semantic definition of the basic semantic notions like 'tautology','contradiction', 'logical consequence' with respect to the formal languages under consideration. Take the language PL ("Predicate Logic") in chapter 13 of PMW. Given the syntax and semantics of PL, we can say that any formula of PL is a tautology, for example, iff it is true in all possible models for PL. Truth in a model is defined in definition (13-10) on p. 327: A formula of PL is true in a model M just in case its denotation with respect to M and g is 1, for all possible variable assignments g. We now have to know what possible models for PL and possible variable assignments are. You find the definition of a model for languages like PL in chapter 7, p. 143. A possible model for PL is a pair consisting of a domain of individuals D and an interpretation function F that assigns appropriate values to the non-logical constants (individual constants and predicates) of PL, as specified on p. 143. Given a model M, the set of all admissible variable assignments is the set of all total functions from the set of variables of PL to the domain of the model.

A typical activity in mathematical logic consists in giving several formal definitions of intuitive notions like 'tautalogy', 'logical consequence', 'effective computability' and then prove that the definitions are equivalent. For PL, for example, we can give a semantic definition of which formulas of PL are tautologies in terms of truth in all models. We can also give a syntactic definition of what the tautologies of PL are in terms of derivability in an axiomatic system, for example. A completeness proof is then given to show that the two methods of picking out the set of tautologies are equivalent: they characterize the same set of tautologies. See PMW, chapter 8 for more discussion. If you want to reflect on whether the model-theoretic method of characterizing the basic semantic notions is right for natural language semantics check out some of the recommended references about model theory.

background on mereology: sums of individuals & events

When trying to come up with a definition for non-Boolean conjunction, we needed a way to talk about the sum of two events. This need generated a discussion about mereological notions like the part relation and the mereological sum operation. The part relation and the sum operation are interdefinable: The sum of a and b is the smallest individual that has both a and b as parts. Or: a is part of b just in case the sum of a and b is b. I again recommended the article on mereology (by Achille Varzi) in the Stanford Encyclopedia of Philosophy, and distributed the first three pages of David Lewis' 1991 book Parts of Classes (Blackwell Publishers, Oxford). Can you write down all the mereological principles that David Lewis illustrates with cat parts and cat fusions (use Varzi's technical notation from the Encyclopedia article). With all the mereological principles in place, we can now also talk about plural individuals as sums of atomic individuals, which was useful for our formalizations of Schein sentences.