An argument for conditional excluded middle.

Conditional excluded middle is the following schema:

if A, then C; or if A, then not C.

It’s disputed whether everyday conditionals do or should support this schema. Extant formal treatments of conditionals differ on this issue: the material conditional supports CEM; the strict conditional doesn’t; Stalnaker’s logic of conditionals does, Lewis’s logic of conditionals doesn’t.

Here’s one consideration in favour of CEM (inspired by Rosen’s “incompleteness puzzle” for modal fictionalism, which I was chatting to Richard Woodward about at the Lewis graduate conference that was held in Leeds yesterday).

Here’s the quick version:

Fictionalisms in metaphysics should be cashed out via the indicative conditional. But if fictionalism is true about any domain, then it’s true about some domain that suffers from “incompleteness” phenomena. Unless the indicative conditional in general is governed in general by CEM, then there’s no way to resist the claim that we get sentences which are neither hold nor fail to hold according to the fiction. But any such “local” instance of a failure of CEM will lead to a contradiction. So the indicative conditional in general is governed by CEM

Here it is in more detail:

(A) Fictionalism is the right analysis about at least some areas of discourse.

Suppose fictionalism is the right account of blurg-talk. So there is the blurg fiction (call it B). And something like the following is true: when I appear to utter , say “blurgs exist” what I’ve said is correct iff according to B, “blurgs exist”. A natural, though disputable, principle is the following.

(B) If fictionalism is the correct theory of blurg-talk, then the following schema holds for any sentence S within blurg-talk:

“S iff According to B, S”

(NB: read “iff” as material equivalence, in this case).

(C) The right way to understand “according to B, S” (at least in this context) is as the indicative conditional “if B, then S”.

Now suppose we had a failure of CEM for an indicative conditional featuring “B” in the antecedent and a sentence of blurg-talk, S, in the consequent. Then we’d have the following:

(1) ~(B>S)&~(B>~S) (supposition)

By (C), this means we have:

(2) ~(According to B, S) & ~(According to B, ~S).

By (B), ~(According to B, S) is materially equivalent to ~S. Hence we get:

(3) ~S&~~S

Contradiction. This is a reductio of (1), so we conclude that

(intermediate conclusion):
No matter which fictionalism we’re considering, CEM has no counterinstances with the relevant fiction as antecedent and a sentence of the discourse in question as consequent.


(D) the best explanation of (intermediate conclusion) is that CEM holds in general.

Why is this? Well, I can’t think of any other reason we’d get this result. The issue is that fictions are often apparently incomplete. Anna Karenina doesn’t explicitly tell us the exact population of Russia at the moment of Anna’s conception. Plurality of worlds is notoriously silent on what is the upper bound for the number of objects there could possibly be. Zermelo Fraenkel set-theory doesn’t prove or disprove the Generalized Continuum Hypothesis. I’m going to assume:

(E) whatever domain fictionalism is true of, it will suffer from incompleteness phenomena of the kind familiar from fictionalisms about possibilia, arithmetic etc.

Whenever we get such incompleteness phenomena, many have assumed, we get results such as the following:

~(According to AK, the population of Russia at Anna’s conception is n)
&~(According to AK, the population of Russia at Anna’s conception is ~n)

~(According to PW, there at most k many things in a world)
&~(According to PW, there are more than k many things in some world)

~(According to ZF, the GCH holds)
&~(According to ZF, the GCH fails to hold)

The only reason for resisting these very natural claims, especially when “According to” in the relevant cases is understood as an indicative conditional, is to endorse in those instances a general story about putative counterexamples to CEM. That’s why (D) seems true to me.

(The general story is due to Stalnaker; and in the instances at hand it will say that it is indeterminate whether or not e.g. “if PW is true, then there at most k many things in the world” is true; and also indeterminate whether its negation is true (explaining why we are compelled to reject both this sentence and its negation). Familiar logics for indeterminacy allow that p and q being indeterminate is compatible with “p or q” being determinately true. So the indeterminacy of “if B, S” and “if B, ~S” is compatible with the relevant instance of CEM “if B, S or if B, ~S” holding.)

Given (A-E), then, I think inference to the best explanation gives us CEM for the indicative conditional.

[Update: I cross-posted this both at Theories and Things and Metaphysical Values. Comment threads have been active so far at both places; so those interested might want to check out both threads. (Haven’t yet figured out whether this cross-posting is a good idea or not.)]

18 responses to “An argument for conditional excluded middle.

  1. Just a few further thoughts about my premises; specifically (B) and (C).

    (B) is a substantive position on what fictionalism consists in. Other versions would e.g. say that “blurgs exist” was false (after all, there are no blurgs!) and that it’s only assertible because we assume an “according to” operator is tacitly invoked.

    A more interesting way of denying this would be to restrict the equivalence somehow. Daniel Nolan suggests doing this, in response to Rosen, in his Stanford encyclopedia article on modal fictionalism.

    (C) is again substantive. Not everything hangs on the claim that “according to fiction” should quite generally be analyzed as an indicative conditional (Cian Dorr suggests that a fictionalism about mereology should be cashed out in terms of a counterfactual conditional: I don’t think he intends that to be part of a general analysis of fiction. FWIW I think that the indicative analysis does better than Cian’s counterfactual analysis as part of a formulation of mereological fictionalism considered on its own merits.)

    But it’d be nicer if “according to” in general could be given this analysis. And I don’t think it’s crazy to think this is the case. Stefano Predelli gave an interesting talk in Leeds this year, where he pointed to some odd (monstrous!) semantic behaviour of “according to the fiction”. What’s interesting about that observation is that indicative conditionals seem to exhibit exactly the same, monsterous, behaviour.

  2. Richard Woodward


    On (B) and setting up fictionalism. You say “Other versions would e.g. say that “blurgs exist” was false (after all, there are no blurgs!)”. My view of this (which was developed in the Arche paper) is that the best bet is for the fictionalist to go context-dependent. That is, the fictionalist’s heterophonic schema (“x’s exist” is true iff According to F, x’s exist) holds only in some contexts; in other contexts, we have a homophonic schema (“x’s exist” is true iff x’s exist). Obviously, What we never have is a context where both schemas hold.

    As for incompleteness. I’m not sure i quite followed everything, so let me see if i have got you right. If we have failures of CEM for the indicative conditional, then we’d get a contradiction out of the incompleteness problem (assuming the fictive prefix gets glossed as an indicative). But, incompleteness of some fictions is unavoidable. So, by the reductio, we don’t have any failures of CEM for the indicative conditional. The best explanation of this is that CEM holds in general.

    I’m not going to comment now on the status of CEM, but i do have some thoughts in relation to some of the things you say about incompleteness.

    I’m sympathetic to much of what you say, but only in particular cases. I think in the cases of ordinary fictions (Anna Karenina, etc.) its immensely plausible that incompleteness is unavoidable. The content of such fictions is, on the face of it, linked to the activities of the author in question. And we are quite happy that the truth of statements such as “Anna is happy” stand or fall depending on what the author decided.

    What i’m suspicous of is whether this is at all plausible in the case of, say, modal or mathematical fictionalism. I’m not particular disposed – to say the least – to the idea that modal truth is dependent on what Lewis said in Plurality.

    What to make of all this? In three words, I’m not sure. One thing – which i think i said (badly) yesterday – is that it puts pressure on the idea that the (modal, say) fictionalist can rely on the idea that the ‘according to GR’ operator is of the same family of operators as ‘according to Anna Karenina’ and the like. This then makes it harder for her to discharge her explanatory duty to tell a decent story about the content of ‘according to GR’.

    Good to see you yesterday btw. Blogs have benefits over late night post-pub chats. Namely, the decrease the chances of 5-pint-metaphysics.


  3. Robbie,

    I’m not sure that CEM really is the best explanation of why not to accept gaps. Why not just LEM? On the standard Adams view of indicative conditionals (‘if p then q’ expresses a commitment to believing q upon coming to believe p), CEM seems implausible, whereas LEM is untouched. From LEM we can derive B>(Sv¬S) and ¬B>(Sv¬S), but not (B>S)v(B>¬S).

    So where would I fault your argument? Well, it seems unfair to demand both that the ‘according to’ conditional is indicative, and that the biconditional is not indicative. I’d have thought that they should both be one or the other, i.e. you can’t have both (B) and (C). If the biconditional is indicative, then the move from (2) to (3) fails. And of course if the ‘according to’ conditional is material then we don’t get an argument for CEM. I suspect that fictionalists would take the latter line.

    But I’d really rather that your argument was valid, because I could take it as a reductio of (A) – I think there are better arguments for the Adams account of the indicative conditional than for any form of fictionalism.

  4. Daniel:

    I guess I’m not following. First, why will LEM help us? The argument is supposed to show that a local failure of CEM will lead directly to contradiction (given the side constraints).

    Here’s another way to put it. The equivalence schema gives a way to translate (local) LEM into (local) CEM. It’s familiar that denying LEM in classical logic gives contradiction. What the Rosen-style argument does is exploit the equivalence to transfer this into a reductio of a denial of CEM. No?

    Also, remember I’m being officially non-committal on what the indicative is: it could be the material conditional! (which, recall, satisfies CEM). So I don’t see why you say that we don’t get an argument for CEM if the indicative conditional is the material conditional…

    I take it (along with most everybody, except possibly Vann McGee and Bill Lycan) that a indicative conditional entails the corresponding material conditional. So by appealing to the material equivalence rather than an “indicative equivalence”, I’m just being less committal. No grounds for complaint there!

    Interesting to think about the Adams conditional in this context. What does that say about CEM? (It defines a logic, after all: it just doesn’t give a standard sort of semantics). It’s not clear to me off the bat that it’ll be inconsistent with LEM; if only because it’s always unclear what to make of logical compounds containing conditionals on the Adams approach…

  5. Hi Robbie,

    Interesting post! Here’s my gut reaction. Incompleteness claims for fictions can be made regardless of whether you cash out the ‘according to’ notion in terms of a conditional. So a similar reductio can be run for any account of ‘according to’; we just run the reductio directly on (2). This makes me a bit suspicious as to whether manouevering in the theory of conditionals is the appropriate response in your case. Isn’t it more plausible, in the light of the generality of the problem, that (B) itself is to blame?

  6. Hi Carrie! How’s Oz?

    Not sure what you mean by “incompleteness claims”. That there’s a p such that p neither holds nor fails to hold, according to the fiction? Then you’re not being neutral: a (Stalnakerian) indicative conditional reading of “according to” is inconsistent with that. But, if that’s not what’s under consideration, then I don’t see how to get the problem started.

    I do concede there’s a lot of plauisibility to (2). Indeed, here’s an argument against CEM.

    (a) incompleteness phenomenon arises for F
    (b) If incompleteness phenomenon arises for F, then ~Acc to F, p or ~Acc to F, ~p.
    (c) “According to F, p” should be analyzed as “if F, p”.
    (c) So ~Acc to F, p & ~Acc to F, ~p
    (d) So ~(if F, p) & ~(if F, ~p).

    (d) being a counterinstance to CEM.

    As I say, though, I think diagnosing indeteminacy is an adequate response to the intuitive case for (2). So I reject (b) above.

    So I can’t see a neutral way of getting the problem started. And I don’t see any strong reason to diagnose incompleteness in the way that makes it incompatible with CEM.

    To bed! it is much later here than in australia (Have just got back in from listening to Kit Fine’s talk at the R.I.P. Being conference).

  7. My point isn’t about how or whether (2) is motivated. I was just thinking that non-conditional ways of spelling out ‘according to’ will also lead to trouble *if* you go on to accept (2) in the presence of (B). This being so, it looks like the problem you’ve isolated is generated by holding (2) and (B) simultaneously, and doesn’t actually have much to do with conditionals.

    Maybe you’d want to reply along these lines: “Ah, but it *does* have something to do with conditionals: it shows you have to eschew counterexamples to CEM, otherwise you’re liable to end up accepting (2)”.

    If so, my reply is that, dialectically, you need to show why the thing for the fictionalist to do isn’t just to reject (B), rather than reject (2) (and also any principles which leave us liable to end up accepting (2)). To persuade the fictionalist to buy CEM you’d need to block this escape route.

    (Oz is great. Although I’m disappointed that things are not in fact upside down. And people here don’t say ‘strewth’ or ‘bonzer’ nearly enough.)

  8. Robbie,

    I think that for the reasons you give my comment above was pretty dumb. However, I can’t shake the thought that (2) and (1) are fine. But I see now that since an indicative biconditional (i.e. an indicative conditional each way) entails material equivalence, (2) entails (3) however we read the biconditional in (B). So, surprise, surprise, the upshot is that I don’t want to accept any form of fictionalism, at least (like Carrie – Hiya!) not one committed to (B). Is there any plausible fictionalist alternative to (B)? How about:

    ((B>S)>S) & ((B>¬S)>S)

    You can still stipulate LEM so that the target discourse obeys classical logic. I suppose the problem is what to say about cases where we have Sv¬S along with ¬(B>S) and ¬(B>¬S). It seems mysterious that it could somehow be determined whether or not S without it being determined by the fiction (if fictionalism is correct). Answer: LEM is part of the fiction, i.e. we have B>(Sv¬S) for all S, and that’s where Sv¬S comes from. Hmmm, I still wouldn’t exactly want to hold that position.

    On the Adams vs CEM front, my reason for thinking they’re incompatible is that Adams reads indicative conditionals as (roughly) conditional commitments. If they’re commitments, then there should be some kind of internal/external negation distinction, which will prevent ¬(B>S) from entailing (B>¬S), as CEM demands. This is roughly the distinction between not having a conditional commitment and having the opposite commitment – we don’t want to former to entail the latter. Caveat: my understanding of the Adams view is probably idiosyncratic.

  9. (Hiya back, Daniel!)

  10. Hi Carrie,

    Right, I see the dialectical concern now. And I guess the standard move is for fictionalisms to reject (B) in some manner. It all depends on the reason for finding (B) attractive in the first place: if you’re up for error-theoretic fictionalism, it’ll look crazy anyway. I do think there are ways of conceiving of the fictionalist project that’ll enforce (B). But that’s another story.

    A couple of thoughts about tweaking the dialectic. First, perhaps it should be reformulated schematically: taking one from fictionalism about x’s to CEM. Then the dialectical situation will depend on the particular fictionalism being engaged, which may be as it should be.

    Another thought: maybe it’d be better to give the turned-around Rosen-style argument directly, giving us each instance of the following:

    (Acc F, B) v (Acc F, ~B).

    Suggest that this isn’t incompatible with intuitive “incompleteness” phenomena, since the determinate truth of the disjunction doesn’t give us determinate truth of either disjunct.

    Next, argue that the best semantic account of the behaviour of “According to”, so conceived, is Stalnaker’s logic and semantics for conditionals. In a sense, then, “A according to B” will be a conditional, and one governed by (the analogue of) CEM.

    It’d be a final step (maybe a resistable one) to conclude that “if” and “according to” are one and the same conditional, so that CEM holds for the English indicative “if”.

    Does that sound any better to you?

  11. Richard:

    The way of avoiding (B) you suggest sounds interesting. Here’s a gloss (tell me if I’m getting it wrong). Standardly, fictionalists choose between either going error-theoretic (the homophonic reading, where the equivalence is denied) and seeing the fictionalist conditionals as giving the truth-conditions of the language (the hetrophonic reading, where the equivalence is sustained). You’re suggesting that we go context-dependent on this question.

    A question: does the heterophonic reading of modal language make “~p&~~p” true, though non-contradictory, because of the ellipses that are involved?

    The problem would then be that your account would appear to predict that there should be a reading where asserting that conjunction would be ok (as well as one which is awful). But there doesn’t appear to be.

    One reason I don’t like this kind of response in general, though, is that think that semantic considerations aren’t where the action is. For a start, I’m quite sympathetic to some methodological remarks that Jason Stanley makes in a paper on this stuff: what the truth-conditions for our sentences is in large part an empirical matter.

    (I can see ways of resisting this too: John Hawthorne at the RIP Being conference this weekend was talking about “rescue semantics”; in effect, rather than having an error-theory about ordinary talk about possibilia, you shift the error-theory up into the tacit beliefs that underlie semantic competence and generate empirical data about felicity conditions. So e.g. while the language-organ thinks that the truth-conditions for “there are chairs” might commit you to chairs, in reality there is extra (?syntax and) semantics to which the language-organ is blind. The sentence you produce can then be true, at the cost of making the language-organ represent the semantic facts wrongly. As you can see, making out this kind of case will involve some fancy footwork in the theory of meaning.)

    Another (personal) motivation for not liking the debate framed in a semantic way is that I wanted an answer to the incompleteness objection that applies to the fictionalism in which I’m most interested: fictionalism about semantic properties. Thinking of fictionalism as at root itself a semantic hypothesis seems in that case to be putting the cart before the horse.

    I guess, in general, I think of (B) as motivated metaphysically: it’s true because *what it is* for blurgs to exist is for the blurg-fiction to say that they exist. Such reductive analyses are then naturally thought to sustain the biconditional.

    I can see that there’s room for denying incompleteness for some mathematical cases. E.g. you might go for second-order PA (with a standard semantics for the second order quantifiers). I’m a little sceptical about the significance of the categoricity results on which this would have to rest, actually, but I take the general point. I’m not sure why we’d think that the worlds-fiction is complete (about all props involving worlds and possibilia, not just the sentences of QML. Moving to weak modal fictionalism isn’t necessarily going to get us out of all troubles here: Sider talks about this kind of issue at the end of the pluriverse paper as I recall). Can you expand on that?

    A final point: the objectivity of a discourse we’re fictionalists about will depend on how we select the fiction. In the case of semantic properties, I take the whole interpretationist stick as a way of identifying, more or less objectively, which fiction is going to be meaning-fixing (that is, which semantic theory is selected for by the patterns of assent and dissent).

    In principle we could have a similar account for the worlds case. In fact, I take it that’s what Ted Sider does in the pluriverse paper, under the reading on which that’s a fictionalist story.


  12. I meant “interpretationist schtik”. Though the idea of an interpretationist stick seems an interesting one. For enforcing charity…

  13. Richard Woodward


    I’ll get back to you about the incompeteness/sider stuff. I do think that Ted’s “conditionalist” view is very similar to the fictionalists (it’s pretty obvious if you think that ‘according to’ is come kind of conditional!).

    As for the characterisation of fictionalism i go for. I’m not particular sure about the issues that crop up regarding “which semantics is right”. Stanley thinks its an empirical matter. Others don’t. Personally, I’m inclined to think that there is a massive (and underdeveoped) issue here regarding the relationship between a semantic theory and natural languages. One place in the literature where this kind of thing crops up is the “revolutionary or hermeneutic” question. FWIW, i’m inclined to think that the fictionalist should try and dissolve that distinction. What is going on is that the fictionalist is interpreting natural language sentences in line with her scheme and arguing that her scheme gives us everything we need. Whether we do in fact mean what the fictionalist means is besides the point. If we do, great. If we don’t, we can start to do so. Obviously, this kind of strategy means that whether fictionalism can deliver the BENEFITS of possible-worlds talk (semantic, ontological and conceptual) then becomes a massive issue (this is on of the main questions I hope to address in my thesis).

    “does the heterophonic reading of modal language make “~p&~~p” true, though non-contradictory, because of the ellipses that are involved?”

    Suppose ‘p’ is “there is a plurality of worlds”. Okay, if we interpret the first conjunct homophonically and the second heterophonically, we get the (apparently contradictory) sentence coming out as non-contradictory, since the proposition expressed is then ‘there is not a plurality of worlds but according to GR, there is’.

    The fictionalist might think that we dont get this result though, since there is no context where we can interpret the first conjunct differently from the second. That is, either both get the heterophonic treatment or both get the homophonic treatment. That way, we don’t ever get “p and not-p” coming out as true. (Incidentally, this mirrors something that Ross and I discussed in my talk in Leeds, since you get the same kind of issue in relation to John Divers’ advanced modalising scheme).

    However, the fictionalist doesn’t need to say this. It seems to me as though fictional talk is a prime example where people say apparently contradictory things but the ellipses get us off the hook.. e.g. “some hobbits live underground, but there are no such things as hobbits”. But again, maybe the context shifts half way through the sentence. (I discuss this more in my paper on the Brock-Rosen objection, which i’ll send to you if you’re interested?).

    Its always interesting to think what fictionalists in other areas will make of this (e.g. your fictionalist about semantic properties). But there is a big danger of equivocation here since “fictionalism” is a term of art, which means many things to many people.

    Best, Rich

  14. Hi Rich,

    I’ll have to think some more about the way you handle the apparent contradiction.

    I’m very sympathetic to the methodological view you take. Personally, I’m inclined to agree with Stanley on the empiricality of what the semantic facts are (though the idea of rescue semantics struck me as really interesting and worth thinking about). But the interest of, say, stage theory isn’t that it’s an analysis of everyday talk, or some kind of normative recommendation, but that it’s something that subserves all the functions and explanatory role of talking about persistence. I just don’t think it’s helpful to talk about things being “true” or “false” later. I’d prefer to use a proper term of art, like Yablo’s “real content”. Of course that’s terminological, but it’ll serve to mark an important point.

    (FWIW I’m sure that I’ve been heavily influenced by John Divers on this way of making explanatory role the centerpiece of the methodology, though I’m not sure whether he formulates things in exactly these terms.)

    I take the warning about “fictionalism” being a term of art. Here’s what I was thinking. If you’re going to be fictionalist about just one area, then fair enough to take whatever view works. But once you’re a fictionalist about one area, it’s tempting to be a fictionalist about a whole lot of other ontology you don’t want to be committed to. And it just seems inelegant to me to then try to endorse multitudes of different forms of metaphysical theory. Much nicer to do e.g. what Cian Dorr tries to do in a recent paper, and provide some single uniform recipe that your fictionalisms can fit into.

  15. Richard Woodward

    “(FWIW I’m sure that I’ve been heavily influenced by John Divers on this way of making explanatory role the centerpiece of the methodology”

    I think you’ve got the central point pretty much right when you say… “But the interest of, say, stage theory isn’t that it’s an analysis of everyday talk, or some kind of normative recommendation, but that it’s something that subserves all the functions and explanatory role of talking about persistence”.

    But, we might want to go a bit further and say that an interpretation of F-talk is perfectly adequate qua analysis of F-talk just so long as it subserves all the functions and explanatory roles that F-talk serves. Personally, I’m inclined to this kind of stronger view (and the point obviously relates to my earlier comment about revolutionary/hermeneutic fictionalism). Whether John would want to do this far is another matter. (As an aside, John’s taught me metaphysics since I was a lowly 2nd year UG, so his influence has been pretty strong on my own thinking).

    “it just seems inelegant to me to then try to endorse multitudes of different forms of metaphysical theory.”

    Really? Surely you don’t mean that its inelegant to be a realist about numbers but an anti-realist about value? Care to expand?


  16. All depends what you mean by “analysis”. It sounds pretty semantic to my ear, but here (unlike with e.g. “truth”) I’m happy to go along with any convention so long as we know what we mean.

    In the past, I’ve used the phrase “reductive paraphrase” for the functional-constrained linkages that makes no claims about semantic features of the language (either revolutionary or herm-y).

  17. Oh, on the metaphysical theory bit. I’ve no problem with someone to be a realist here, and a fictionalist there. It’s more like: it’s a bit ugly being a fictionalist of type 1 here, a fictinoalist of type 2 there, a fictionalist of type 3 somewhere else, and so on.

    I’d have the same feelng about someone being an expressivist of 7 different types in different areas. It’d be ok for it to vary e.g. which attitudes are being expressed by utterances of sentences (which is the analogue of the fiction varying from case to case). But I wouldn’t want the answer to the Frege-Geach problem to vary from case to case.


  18. Ah, right. Im not sure what is mean by ‘7 different types of fictionalism’ in this context. I’m guessing it means ‘according to GR’ is analysed as an indicative conditional, but ‘according to standard mereology’ is analysed as a counterfactual, etc? If thats right, I’m sympathetic.

    When i warned against equivocation, I didn’t have that in mind though. The variety of theories that get called “fictionalist” don’t even all invoke fictive operators. What I’d like to call error theories, agnosticisms, non-cognitivism, paraphrase strategies, figularisms, etc., have all been called ‘fictionalist’ in the literature.

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