# David Lewis, conditionals and restrictive quantification

In a nutshell:

When are conditionals containing adverbial quantifiers true according to the David Lewis account? In particular, how are they to be judged if the situation in the antecedent never occurs?

The problem:

In Adverbs of Quantification (1975), David Lewis proposed that adverbials such as always, mostly, rarely, sometimes, never and so forth are quantifiers that quantify over cases (as opposed to just time, for example).

So a sentence such as:

• I mostly go to the gym in my pyjamas

would be true if in most cases that I go to the gym I go in my pyjamas.

He then further proposes that when such adverbials appear in conditional sentences, the antecedent of the conditional functions as a restrictor of the adverbial quantifier. So in a sentence such as:

• Mostly, if on a Saturday morning I go to the gym, I go in a suit and tie

then we have a quantifier mostly that quantifies over cases restricted to cases in which I go to the gym on a Saturday. We can model the sentence like this:

• [Mostlycases : I go to the gym on a Saturday morningcases] I wear a suit and tie.

Assuming that I ever go to the gym on a Saturday morning this sentence is true if in a majority of such cases I wear a suit and tie. So far, so good (I hope).

Now I am a supremely lazy individual, and you are more likely to see me in Sainsbury's (a British supermarket) in my birthday suit on a Saturday morning than you are to see me at the gym. I assure you that neither has ever happened. What I want to know, therefore, is how David Lewis would regard my sentence:

• Mostly, if on a Saturday morning I go to the gym, I go in a suit and tie.

If this was a normal conditional without the word mostly, then this conditional would be true according to Lewis, because he subscribed to a material implication account of natural language conditionals. According to the material implication account of conditionals If P, Q is true whenever P is false or Q is true. So given that I never go to the gym on a Saturday morning the sentence above would be true according to such an account. However, Lewis rejects such an account when it comes to conditionals which have adverbials such as always, mostly, never and so forth. The reason is this:

• Mostly if students cheat in their exams, they get a B+.

In a sentence such as the one above, if we take the conditional to be a material implication then all that is needed for the sentence to be true is for the conditional to be true in a majority of cases. All that is needed for the conditional to be true in a majority of cases if the sentence is a material implication, is for most students not to cheat in their exams. If most students don't cheat in their exams but every single student that cheats gets an A+, the sentence would be true. So this is clearly not what we want.

For this reason, therefore, Lewis adopts his restricted quantifier account for such conditionals.

My problem is that I have been unable to find out (maybe because I'm missing something) how Lewis would treat conditionals like my Saturday-morning-at-the-gym one, given that I never go to the gym on a Saturday morning. Does anybody know?

• You may want to edit this a bit (condense explanation, highlight part that is your question), but I think it's a great question. – virmaior Apr 6 '16 at 11:24

Answer: He could say true or false. (That's what I came up with after thinking it through below)

I'll give this a low effort shot. I haven't read anything by Lewis where he says one way or the other, but I'm steeped in Lewisian tradition, so maybe something Lewisian will emerge: As has been pointed out before, Lewis could avail himself to possible worlds to settle this. But you seem to have good textual evidence to think that that isn't the move he would make, so we can take that off the table.

The other options are that it is true, false, or undefined/some other truth value. As far as I recall Lewis is not a fan of trivalent or otherwise nonstandard logics. That might have been a nice solution here, but it would really mess with Lewis's ontology. Given that propositions are sets of world you'd get such nasty things as sets of which some world is indeterminately a member. Definitely not an option for him.

Hence, I assume that he would say that the sentence is either true or false, and that this is a matter entirely of how this world is like, not some other possible world.

I think the final part of your explanation might give you a clue. From what you say it looks like what Lewis is rejecting is a reading of 'mostly' taking a wide scope and the whole sentence coming out true if most of the instances under the scope come out true. I.e. For most x, if Fx then Gx would be true if 'if Fx then Gx' is true, given the standard semantics, for most x. But that is just to reject a certain reading of how the adverbial modifier operates, not the semantics for conditionals altogether.

Given your account so far I think the best thing to say is that it is open for Lewis to go either way. Nothing in his semantics is committing him one way or the other. The best thing to consider is this: What is, intuitively, the right semantics for those weird sentences. Then, check if that answer conflicts with anything in Lewis's system.

Suppose that no students cheat. Then you get: For most of no cases, the students get a B+.

We could call this trivially true or trivially false (I think). It does not conflict with his main account to say it is trivially true, and I don't currently see this conflicting with anything if we say it is trivially false. The question to answer is: What is true in most of zero cases? Is it everything, or nothing?

• +1 Probably the most useful thing you said was that you hadn't read Lewis discuss the problem - which is very useful! – Araucaria - Not here any more. Apr 26 '16 at 12:21

Not a specialist on Lewis, but here's my stab at an answer. I take it your question boils down to something like:

If we buy Lewis's idea that adverbs like this are just disguised instances of quantification over cases, does that change anything about the semantics of the material conditional?

And I think the answer to that question is "No, qualifying over cases doesn't change the semantics of the material conditional."

However, you're bringing up cases involving the adverb "mostly" which make it look like Lewis must be wrong--it looks like "mostly, when i go to the gym, I'm wearing a suit and tie" should be false if I never go to the gym. I think this isn't a problem with vacuous conditions, though, but rather a problem with the semantics of "mostly".

Let me start by saying why I don't think Lewis is changing the semantics of the conditional, then I'll come back to what I think is happening with "mostly."

To see why I don't think Lewis is messing with the standard understanding of the conditional, look at Lewis's contrast between the semantic definitions of the standard "selective" first-order quantifiers defined in his (18) and (19) [page 9] and then his semantic definitions of the "non-selective" quantifiers in (20) and (21) that he thinks underly adverbial constructions like the ones under discussion.

I think that Lewis's subsequent discussion of how conditionals introduce restrictions is just refining this basic idea of the truth conditions for these new quantifiers and NOT altering his understanding of the material conditional from its usual sense in first-order logic.

Therefore, I think Lewis is going to say that vacuous sentences like "always, when I go to the gym, I'm naked" are true, even when there are no cases where I go to the gym.

Now, however, notice that something different seems to be happening if instead of "always" I had said, "mostly, when I go to the gym, I'm naked." I think what's happening here is that "mostly" has the natural reading "in >50% of cases", which is more like an existential quantifier than a universal one. In other words, you can't say that I ate most of the cake, if there wasn't any cake served at all. Nor can I be said to have gone to the gym naked in most cases if there were no cases of my gym going at all.

How's that sound?

• Hmmm. According to Lewis it does though. I agree with your last line, but I don't know what Lewis would say thoguh ;) – Araucaria - Not here any more. Apr 26 '16 at 11:35

The kind of conditional you are referring to is often called a counterfactual, because the antecedent is typically false. Lewis regarded these as quite different kinds of conditional expressions from so-called indicative conditionals. Lewis broadly follows a similar path to Stalnaker and uses possible worlds to explain their meaning. Stalnaker has it that a counterfactual "if A then B" is true if B is true in the most similar possible world in which A is true. Lewis modifies this by allowing that there might be a whole set of relevantly similar possible worlds, without requiring that there is a uniquely most similar one. The term 'similar' here is difficult to cash out in practice, because to get plausible results, it is necessary to give priority to possible worlds with the same laws of nature as our own world, even if this implies that tiny miracles have to be posited to account for the divergence between the possible worlds and the actual world. This in turn leads to problems accounting for how to quantify the size and ordering of such miracles. In his book "Counterfactuals" Lewis develops this idea into a formal logic called VC.

To apply it to your example, which we might naturally express subjunctively as "Mostly, if on a Saturday morning I were to go to the gym, I would wear a suit and tie," would on Lewis' account be: in possible worlds which are similar to the actual world, but in which, because of a small miracle, the counterpart of me has decided to go to the gym, in most such circumstances my counterpart is wearing a suit and tie.

• Thanks for the answer, but I don't think that this is a counterfactual conditional, because it doesn't use subjunctive conditional syntax, it uses indicative syntax. Say for example, that person A knows nothing about me apart from the fact that I always wear a suit and tie on Saturdays mornings without exception. Suppose someone asks her what I wear if I to the gym on a Saturday, she may well say that if I go the gym on a Saturday I wear a suit and tie. The fact that I don't go ... – Araucaria - Not here any more. Apr 7 '16 at 12:58
• ... doesn't make the conditional counterfactual. If it did then every indicative conditional with a false antecedent would not be true on Lewis's material implication account, it would merely be a counterfactual. – Araucaria - Not here any more. Apr 7 '16 at 12:58
• The presence or absence of the subjunctive mood is not a reliable indicator. The whole nomenclature of indicative/subjunctive/counterfactual is misleading, but you may replace counterfactual in my reply with "ontic" if you like. I don't see any natural way to interpret your sentence otherwise. So-called indicative conditionals generally have epistemically possible antecedents, or at least ones that may be considered epistemically possible for the sake of argument, or for the sake of some supposition. – Bumble Apr 7 '16 at 14:44
• Given that you never go to the gym, there is no sense in saying "mostly if I go to the gym I wear a suit and tie" unless you are expressing what you would wear if you did go. Compare it with, "Mostly if this button is pressed, the fire alarm goes off." This is not trivially true just because the button has never been pressed. – Bumble Apr 7 '16 at 14:44
• I don't think Lewis would agree with you about the subjunctive there. He used the term counterfactual exclusively to refer to subjunctive conditionals. I agree with you - mostly - about that conditional being counterfactual in the technical sense if I say it myself, but not if someone else says it. But this is by the by for Lewis' account. Incidentally many other scholars would not agree with you that if I say it myself it would be counterfactual see page 6, The Snodgrass affair in Abbott 2004 here – Araucaria - Not here any more. Apr 7 '16 at 14:55

I think this question can best be answered with algebra to get a logical answer. The case being posed can be generalized in linguistics to establish our variables by saying:

Mostly Y I do X and C.

This can be boiled down to Y being something that is a given circumstance that we pose. X is a variable that's end result is changed by the condition C.

Similarly you can have the case where C is removed from the equation and the sentence still holds and can be in the real world. Use simple laws to subtract variables and

Y = X+C

When C is not true you simply subtract it from y and y becomes a case that is less in value(less likely) but still possible and given the word mostly this is implied in our initial reasoning anyway so:

Y-C=X

X is unchanged Y is a function that gives the likeliness of you to exist on a Saturday at the gym without a suit and tie on correct? This part was mainly to establish the similarity between the two sentences. Simply put the function here would state something about the world similar to "When you go to the gym Y times

Now the question is can mostly be used in a case when you actually never Do the thing in question and mathematically the answer would come from a y function with a negative condition. Saying that you only want to count the times when you do said thing under said condition. So if you want a binary answer in the real world this would give a "False" result. On a graphic representation you do still get a line though so the function is "defined" which is the goal for algebra so along those lines you can HAVE the sentence. That sentence CAN exist. You are simply lying. Its a situation where inevitably X is 0 and C is always 0 or negative. Now You could make this a true statement by adding a new actor that would be similar to an "absolute value bracket" in English this would look something like:

"If I went to the gym on a Saturday, I'd mostly only ever go in a suit and tie."

Notice that just like in the mathematic situation you're forced to only apply the absolute situation after the effect of "in a suit in tie" separate from the "on Saturday" or it makes Saturday a part of the conditional and skews the situation. This function would look like:

|y = X - C|

and it would be solved like

y-C=|X|

This is to say if you apply the absolute value to the "-C" it immediately makes the C posative and creates a different function. These results clearly state imaginary situations or intent. The answers you get are not real they are IMAGINARY numbers just as the situation can exist in conversation and still be a functioning situation but in a context that you say you already do this rather than you would it becomes a "False" to life statement. This is different than being "Impossible" in conversation or "Undefined".

As a wrap up answer I don't know the book just a little googling and what you have said here but I would say:

I see no reason that the statement cannot hold it just exists within the realm of imagination or "In a world that you currently do not inhabit." Not to say it cannot in the future. Meaning that it is not currently a fact but rather a possible future or possible outcome.

(I understand that this is not an answer to the question per se, since it is not particularly about Lewis. But it is to the point, because there is a good reason to reject the material conditional interpretation altogether. It is a toy model for teaching logic, and not a realistic model of actual language use.)

It seems mathematically, by this account, they are true when some statistic applied to all the cases falls in the right range of values, where that statistic and range depend upon the actual word.

Something is mostly the case if more than half of the cases make it true, no? Something never happens when the count of events is zero, no? And it always happens when it never fails to happen.

If it never happens, then zero divided by two is zero and zero is not greater than zero. Something that never happens is both always and never whatever you want. In fact, these are the easy cases. The fact of the matter is that it is not this degenerate case that causes most of the problem with this notion.

In a lot of situations where the referents are abstract or potential, there is no metric that works. So computing this sort of statistic is pointless.

Most of a merely potential set that clearly allows for infinitely many elements simply cannot be computed on a case-by-case basis. We feel like the integers are mostly not prime. But I can put the primes in one-to-one correspondence with the remainder, so clearly just half of them are prime... At the same time, I can clearly produce two non-primes for every prime. And if imaginary kittens are mostly not purple, I can suddenly imagine a whole lot of purple ones and skew the odds.

Even when the cases seem to be finite and bound to reality, there is generally not a metric. I can "usually wear clothes when I go out", but then I can live homeless, in the woods, naked for fifty years, but only once. Are the cases measured by instants of time, or potential instances of observation, or the actual number of times I exit the door?

You need a metric space in order to do this kind of computation, and there is no reasonable metric on 'cases' in general. So this is not a workable theory for any realistic grammar.

You can fall back on context, and claim that the context explicitly or implicity imposes a metric on cases by the framing. But even in the most circumscribed reality there is not enough context to imply one.

This account of language simply tries to hard to move away from dynamics and falls back too hard on mathematics, probability and set theory, without really taking them seriously.

From a framing like that of Wittgenstein's description of language as network of games, or Lacan's account of the dynamics of domains of reference, or even Lawvere's perspective in Category Theory, reference is an active event, not a pre-computed static object. In that context, such parts of language are about controlling assumptions within the context of a narrative, and not about describing some frozen image of reality.

In context, "I mostly do something" is meant to convey that you are to assume when I talk about doing it, that the forward trajectory of the story will work best if you assume it took place. If I don't then involve doing it, it is pointless, and it simply does not have an effect on the presumed narrative, and therefore does not have a meaning, beyond embedding that pending trigger in the context for use later.