UofA

Spring 2004

Ling/Phil 596D: Topics in Linguistics and Philosophy

Heidi Harley and Massimo Piattelli-Palmarini

Compositionality

Wednesday April 21

Handout (M. Piattelli-Palmarini)

Von Finkel and Iatridu on conditionals

If and When If -Clauses Can Restrict Quantifiers

Kai von Fintel and Sabine Iatridou

Massachusetts Institute of Technology

Pre-Final Draft, September 17, 2002

Aim of the paper: Show that if -clauses in the scope of nominal quantifiers, while interesting, do not motivate widescale conclusions about non-compositionality. And that the “trivialization” of conditionals is unwarranted. Conditionals are interpreted as conditionals. What’s crucial is to have a good theory of conditionals.

Background 1 (sort of "Logic and Probability 101" on conditionals)

"If A, then B" is a logical connector between the propositions A and B, such that it's false iff A is true and B is false, in all other cases it's true, (or, under the so-called assertibility interpretation, it is simply undefined when A is false). This is the standard "material conditional" interpretation, intuitively corresponding to a (strictly) sufficient condition with no hypotheses at all about its also being (or not being) necessary. The material conditional is logically equivalent to the conditional "if not-B, then not-A" (which generates bizarre results when dealing with the "if...then.." of ordinary language, and which generates the well-known Nicod-Goodman paradoxes of induction in epistemology). It's also logically equivalent to "not-A or B", with equally bizarre implications for the conditionals of ordinary language.

Under these logical equivalences, in fact, (1) should be equivalent to (2) and (3), but it's not.

(1) If you work hard, you'll pass.

(2) Don't work hard or you'll pass. (Except as a joke, I think)

(3) If you will not pass, then you do not work hard. (bizarre, and surely

not equivalent to (1))

What Iatridou and von Fintel (IvF) call “The paradoxes of material implication” come straight from applying these equivalences to the semantics of conditionals.

In terms of probability, the probability of a conditional was first conjectured (by Robert Stalnaker many years ago (1968)) to be the same as the conditional probability of the consequent on the antecedent. But David Lewis (1976, then reprinted many times) produced a rather stunning proof that, under very mild assumptions (and indeed under the most plausible interpretation of conditionals in ordinary language), "If A, then B" just is a cumbersome way of asserting B. That is: p(B/A) = p(B)

This is Lewis's "trivialization theorem". Indeed, it makes the conditional trivial, doesn't it?

What is the assumption? It's just that our interpretation of the conditional be compatible with the following equivalence:

"If A, then, if C, then B" = "If A and C, then B"

(AÉ(CÉB)) = (A&C) É B

This is, indeed, a most plausible assumption, and arguably the garden variety interpretation of conditionals in ordinary language. The following two sentences are equivalent:

If she writes a superb paper, then, if prizes are available, we will suggest

that she applies for one.

If she writes a superb paper and prizes are available, then we will suggest

that she applies for one.

Consequence: According to Lewis's theorem, this is just a baroque way of saying that we will suggest that she applies for a prize. I suggest that we keep this trivialization result in mind, when examining the semantics of conditionals.

In fact, Lewis has suggested that the proper interpretation of conditionals is that of restrictors, in particular, restrictors of higher operators. Whence the passage by Angelica Kratzer cited by IvF:

The history of the conditional is the story of a syntactic mistake. There is no two-place if . . . then connective in the logical forms of natural languages. If-clauses are devices for restricting the domains of various operators. (Kratzer 1986)

This is called Kratzer’s analysis, or “the folkloric analysis”.

Background 2 (sort of Philosophy 101 on Gododman’s lawlike generalizations)

Counterfactuals are truth-functionally evaluable when the connection is a law, they are not otherwise.

Every dime contains silver.

Had this coin been a dime, it would have contained silver. Is OK

Every coin in my right pocket today contains silver.

Had this coin been in my right pocket today, it would have contained silver. Not OK

Whenever the connection conveys something like “Oh, look!” no counterfactual can be built on it.

If one forces a lawlike relation on the interpretation, then the counterfactual may turn into a (moderately) acceptable one. (I am putting coins into my pocket strictly under the condition that they contain silver).

The “folkloric analyis”

Indicating with » semantic equivalence, we have data like the following;

(1) a. Every student will succeed if he studies hard. »

b. Every student who studies hard will succeed.

The equivalence with the relative clause restricting the quantifier justifies the “folkloric interpretation” suggested by Lewis, Kratzer and others.

However, as we saw with Higginbotham 2002 (see my previous handout), there are other cases, pointing in a different direction:

(2) No student will succeed if he goofs off. »

(3) No student who goofs off will succeed.

Under the material interpretation, If A, then B = not-A or B, (2) would be equivalent to

(2b) Every student will goof off and succeed (nevertheless).

A monstrosity!

The proper equivalence for (2) would rather be:

(4) No student goofs off and succeeds.

Higginbotham’s earlier theory: Conditionals weaken the claim made via a quantifier. Therefore, they mean “if” under upward monotone quantifiers, but mean “and” under downward monotone quantifiers. (See my previous handout on Higginbotham 2002).

Some have interpreted this vagary as a blow to compositionality, and Higginbotham 1986 has been quoted as saying this (but see his more recent papers).

Difficulties with most and few (examples borrowed from Heim and Kratzer)

(5) Most letters are answered if they are shorter than 5 pages.

(6) Few people like New York if they didn’t grow up there.

We have the following real equivalences (respectively)

(7) » Most letters that are shorter than 5 pages are answered.

(8) » Few people that didn’t grow up there like New York.

But the material interpretation of (5) would suggest that (5) is true in the case in which most letters are longer than 5 pages, regardless of how many (if any) are answered. Another monstrosity. So, does the folkloric interpretation really win?

Glitches with the “restriction” interpretation

The interchangeability between conditionals and restrictive relative clauses should go both ways, but it does not.

(9) a. I invited the woman who runs the store downstairs. Not »

b. I invited the woman if she runs the store downstairs.

Nor do we have an equivalence between (11) a and b. (Suppose all stores are run by men. Then the woman in b. refers to no one, while in a. it refers to the woman, whether or not “she” runs a store)

(11) a. I will invite the woman if she runs a store.

b. I will invite the woman who runs a store.

Perhaps the and some are anomalous (non-quantificational) quantifiers. But the glitch extends beyond them.

Restrictive if-clauses force lawlike readings

The restriction contributed by if (even conceding that this is the correct interpretation of conditional clauses) is not all-purpose. It can only restrict nominal quantifiers that are modal, that is, that range over possible states of affairs, not just real ones.

(19) a. Every book that I needed for the seminar happened to be on the table.

b. #Every book happened to be on the table if I needed it for the seminar.

The clear sense of “lucky coincidence” (happenstance) conveyed by (19a) is lost in (19b). The meaning of (19b) is one of “necessity”. Books were on the table under a certain specified condition.

(21) a. No paper that is longer than 50 pages is on this table. Clearly happenstance

b. No paper is on this table that is longer than 50 pages. ??

c. No paper is on this table if it is longer than 50 pages. Clearly a policy of mine

The case is even worse (for the merely restrictive interpretation)

The conditional (if-) clause is not actually inside the restriction of the quantifier. The existential presuppositions do not come out right.

(27) a. Nine of the students will succeed if they work hard.

b. Nine of the students who work hard will succeed.

(28) a. Few of the problems will be solved if we don’t use a computer on

them.

b. Few of the problems that we don’t use a computer on will be

solved.

Clearly, in (27b), there are more than nine students who work hard, out of which, only nine will succeed. This is not the case for (27a).

In (28a), clearly, few problems will be solved overall, regardless of method. In (28b) only few of the computer-less-treated problems will be solved.

The existential presupposition includes the specification from the restrictive relative clause, but not the one from the conditional. The conditional is not inside the restriction of the quantifier.

“Iffiness” lost or found

The (avowedly “nebulous”) intuition, here, is that there is a contrast between clauses meaning that something is the case, period, versus sentences meaning that something is the case, but might well not have been the case. The second is the “iffy” case, indeed conveyed by the if-clauses. Several examples are offered. The best one is from Lewis 1975

(29) a. When Caesar woke up, he usually had tea.

b. (?) If Caesar woke up, he usually had tea.

The reason why (29b) is infelicitous is that it conveys the idea that he might not have woken up at all. This is why the insertion of an adverb like “early” takes all oddness away

(29) c. When Caesar woke up early, he usually had tea.

d. If Caesar woke up early, he usually had tea.

Let’s combine this with quantifiers:

(32) a. Every congressman who is from Florida is a Republican.

b. #? Every congressman is a Republican if he is from Florida.

(32a) expresses a lawlike regularity about voters from a certain State, while (32b) is a bit odd, and not equivalent to (32a). Explanation (?): For a given congressman, it cannot be “iffy” whether he is from Florida or not.

MPP: My explanation for the oddness of (32a) is that the universal quantification over all congressmen is then restricted drastically and anomalously: By a condition that it is not for them to do anything to satisfy (unlike students who work hard). It’s restricted by a state of affairs true only of a tiny minority. It’s like saying

? Every American is rich, if he is Bill Gates.

But it would be equally odd to say

? Every American who is Bill Gates is rich.

I am not persuaded that “iffiness” is the factor here.

IvF punch-line: Conditionals really are conditionals, not merely restrictors.

The meaning of conditionals.

Not the material interpretation, nor the “chameleon” interpretation, nor the restriction interpretation. Based on previous work by von Fintel, the interpretation on offer is the following (in essence, see the paper for details): A strict conditional with contextual variability. (Basically, a Stalnaker conditional plus some conditions)

If p, then q

Claims that all p-worlds in some contextually limited domain are q-worlds. The following properties have to be satisfied:

(a) In the relevant limited domain, there are p-worlds. No incompatibility between p and the domain being quantified over.

(b) Conditional Excluded Middle (see also Higginbotham 2002) either “if p, then q” applies, or “if p, then not-q” applies, but not both. It’s a Stalnaker conditional. This warrants that we can decide about the conditional (check its truth-conditions) in all the relevant worlds. (Homogeneity condition)

(c) In the course of a conversation, there will usually be a dynamic of the domain of worlds being quantified over. Condition (b) warrants that issues of monotonicity can be tracked.

Conditionals are now combined with quantifiers (every, no, and most)

We have, in their analysis, the following ingredients (see the paper):

Individuals (indicated by x, used as a suffix)

Domains (indicated by R)

Possible worlds (indicated by w, also used as a suffix).

The formula for every is:

(34) every_x [R_x] [if_w p_w,x, q_w,x]

Every individual x in the domain R is such that in any world w in which p holds of x, q also holds of x.

Iffiness and existential presuppositions follow directly.

(Notice that the if-clause is not – anyway not automatically – part of the restriction of the nominal quantifier). Existential presuppositions are explained by the schema.

Other applications

The case of no

(36) a. No student will succeed if he goofs off. º

b. Every student will fail if he goofs off.

The equivalence derives directly from the Excluded Middle and the Stalnaker interpretation. No individual x in the domain is such that in all of the worlds where p is true of x (i.e. x succeeds), q is true of x (i.e. he goofs off). But since there is the homogeneity presupposition, this is equivalent to: every individual x in the domain is such that in all of the worlds where p is true of x, q is false of x.

The conjunctive character of if under negative operators is also explained. This applies also to cases when the operator is not “really” a quantifier over individuals:

(37) a. I doubt that John will succeed if he goofs off. »

b. I doubt that John will goof off and (still) succeed.

The case of most

(38) a. Most but not all of the students will succeed if they study hard.

b. Most but not all of the students who study hard will succeed.

(38a) says that studying hard will be effective for most, but not for all the students. Suppose that those students for whom studying hard will not be effective realize that, and don’t even try. The ones who study hard all succeed therefore. In that situation, (38a) is true, but (38b) is not. Therefore, even under most the difference between restriction and conditionalization holds. It also holds under counterfactuals. The truth conditions are different.

(39) a. Every one of these students would have succeeded if he had studied

hard.

b. None of these students would have succeeded if he had goofed off.

Epistemic conditionals

Consider the following case

If a student’s light is on, then that student is home.

The light being on is evidence for his being home. But, of course, it’s not a cause. A previous paper by Iatridu and von Fintel (“Epistemic containment”) had established that quantifiers cannot have scope over epistemic operators. Therefore, (40) is ungrammatical

(40) *Every student_x is home if his_x light is on.

But the clausal restriction would be Ok

Every student whose light is on is home.

The direct application to conditionals confirms the bona fide conditional nature of if-clauses, and weakens the “folkloric interpretation”. QED

Comments (MPP):

In essence, their thesis is that the Stalnaker interpretation, combined with plausible additional requirements, offers a good semantics of conditionals (with tentative hints also at counterfactuals). We saw in Higginbotham’s paper (presented at the same conference, then revised) that things are not so neat. There are further conditions that limit the interpretability of a conditional as a Stalnaker conditional. Further machinery is needed (notably for counterfactuals – see my previous handout).

Reminder:

(32) No professor will retire early if not offered a generous pension.

is not equivalent to (33)

(33) No professor not offered a generous pension will retire early.

But (32) does not mean, either, what (38) means

(38) (No x) (x is not offered a generous pension Þ x will retire early).

Counterfactual relevance is crucial here: We need to know whether professor X, who did in fact retire early, would have done so had he not been offered a generous pension.

And the Stalnaker interpretation for the conditional leaves us with the highly problematic equivalence (repeated here from Higginbotham’s paper)

(24) Few students will get A’s if they work hard. »

(40) Most students will not get A’s if they work hard.

Both papers (Higginbotham’s and IvF’s) want to re-establish compositionality, and get rid of a “chameleon” semantics of conditionals. I am sympathetic to the thrust of the IvF’s paper that conditionals do indeed express conditions, and not simply a restriction (their “iffy” character is an important consideration). All depends, however, on finding the right semantics for the conditional. This is a hard task. Maybe the conditionals of ordinary language express several distinct connectors “that just sound alike” (to borrow Higginbotham’s own words in another context). This would save compositionality, provided we identify them all, and explain how speakers and hearers know in each case which one is meant.