"If"

Speaking of The Simpsons, (which I did a couple of weeks ago on this blog), let's talk about Principal Skinner's definition of if. This comes from the episode The Boy Who Knew Too Much from Season 5.

In response to a question by Homer, Skinner characterizes if as

A conjunction meaning "in the event that"
or "on condition that".

[This is about 4/5 of the way down the page.]

Now, I don't know how legal dictionaries like Black's define if, because it's the kind of word that search engines will refuse to run searches on. I searched a couple of law dictionaries I happened across in the law library, but found no entry for if. I guess it's not what you'd call a term of art.

I think Seymour Skinner is wrong. Not B. F. Skinner wrong, but wrong. My two disagreements are: in terms of syntactic categories, I don't think if is a conjunction; and semantically, I don't think "in the event that" or "on condition that" are adequate paraphrases. To the extent that ordinary dictionaries adopt a Skinnerian approach - and that extent is significant - it illustrates the shortcomings of relying on dictionaries for the meanings of words. Ya heard that, Justice Breyer?

Is "if" a conjunction?

Here's a definition of conjunction from Dictionary.com:

any member of a small class of words distinguished in many languages by their function as connectors between words, phrases, clauses, or sentences, as and, because, but, however.

If is a connector between words and a member of a closed lexical category, so if may well be a conjunction under this definition. But what did I just say about dictionary definitions?

My objection to calling it a conjunction is that it's not really conjoining constituents the way and, or or but conjoins them, as syntactic equals. I'm no expert on the syntax of conditionals, but I'm not aware of any proposal that groups if with conjuncts like and or or. The analysis that I've seen and believe treats if as syntactically a complementizer, the class that includes that, who, and maybe forms of auxiliaries like be or do, italicized in examples like the following.

I believe that judges are entirely too textualist nowadays.
I like any judge who Oliver-Wendell likes.
Is Easterbrook going to be named to the Supreme Court?
Oh my God! Did Easterbrook really write that?

As the examples show, complementizers are used to subordinate clauses, or else they appear at the left periphery of the main clause, which can be seen as subordinating the main clause to the discourse. If seems to behave similarly. The following sentence is close in meaning and structure to the second sentence in the last set of examples.

I like any judge if Oliver-Wendell likes her.

[Difference: the sentence with who has the property of exhaustivity, which the sentence with if doesn't. The exhaustive sentence tells you that in addition to liking every judge that Oliver-Wendell likes, I don't like any judge who Oliver-Wendell doesn't like].

So I would syntactically categorize if a complementizer - whose function is to subordinate one clause to another - rather than a conjunction, which joins two constituents as equals.

Semantics

But my quibble with Skinner is not just about the syntactic category of if, it's about semantics too. To be sure, he is right that the canonical use of if is in conditional contexts, which makes his definition essentially the canonical one. But notice this: a few lines after he defines the term, Skinner uses if in a sentence where it cannot be paraphrased as "in the event that" or "on condition that." He says:

Justice is not a frivolous thing, Simpson. It has little if
anything to do with a disobedient whale.

It should be clear from applying the substitution test that Skinner's definition will not work for this example:

#Justice is not a frivolous thing, Simpson.
It has little in the event that anything to
do with a disobedient whale.

#Justice is not a frivolous thing, Simpson.
It has little on condition that anything to
do with a disobedient whale.

[Per convention, I'm using crosshatches to indicate semantic infelicity. Since I'm focusing on semantics, I am omitting here the asterisk that is conventionally used to indicate syntactic incorrectness.]

It might be objected that this use of if is a non-conditional use. But in fact Skinner's sentence seems to be shorthand for the following, a clear case of a conditional:

If justice has anything to do with a disobedient whale, it has little to do with it.

So how to define the conditional?

The classic definition is in terms of truth tables. According to this approach, conditional sentences like if A, then B are false when A is true and B is false, and is otherwise true. Equivalently, if A, then B is true exactly when A is false or B is true (or both). The following truth table illustrates:


But there is a recent trend among linguists and philosophers to treat conditionals as quantificational statements, just like sentences with some or every. But instead of quantifying over things denoted by nouns, usually objects, the way these quantifiers do, conditionals are quantificational statements over worlds, events, situations, cases and the like.

Quantification, in turn, is fundamentally about relating two sets to each other. Every Supreme Court justice is wizened means that the set of Supreme Court justices is a subset of the set of people who are wizened. Similarly, if a plaintiff wishes to raise an issue on appeal, she must raise it at trial means [to oversimplify] that the set of situations in which a plaintiff raises an issue on appeal is a subset of the set of situations in which she raised the issue at trial, if we're restricting our consideration to the situations compatible with how the law operates.

Now consider again Skinner's sentence:

Justice is not a frivolous thing, Simpson. It has
little if anything to do with a disobedient whale.

This seems amenable to a quantificational analysis: the set of situations in which justice has anything to do with a disobedient whale is a subset of the set of situations in which justice has little to do with a disobedient whale. Equivalently, there are no situations in which justice has anything to do with a disobedient whale, in which justice doesn't have little to do with a disobedient whale.

Counterfactuals, or: What's a but-for?

If also appears in counterfactual conditionals, which have a different and more complicated semantics than ordinary conditionals, and will not be discussed here.