Material conditionals and quantifiers

From:

1. Every G is H

it seems we should be able to infer for any x:

2. If x is G, then x is H.

This pretty much forces one to read “If p, then q” as a material conditional, i.e., as q or not p. For the objection to reading the indicative conditional as a material conditional is that this leads to the paradoxes of material implication, such as that if it’s not snowing in Fairbanks, Alaska today, then it’s correct to say:

3. If it’s snowing in Fairbanks today, then it’s snowing in Mexico City today

even if it’s not snowing in Mexico City, which just sounds wrong.

But if we grant the inference from (1) to (2), we can pretty much recover the paradoxes of material implication. For instance, suppose it’s snowing neither in Fairbanks nor in Mexico City today. Then:

4. Every truth value of the proposition that it’s snowing in Fairbanks today is a truth value of the proposition that it’s snowing in Mexico City today.

So, by the (1)→(2) inference:

5. If a truth value of the proposition that it’s snowing today in Fairbanks is true, then a truth value of the proposition that it’s snowing today in Mexico City is true.

Or, a little more smoothly:

6. If it’s true that it’s snowing in Fairbanks today, then it’s true that it’s snowing in Mexico City today.

It would be very hard to accept (6) without accepting (3). With a bit of work, we can tell similar stories about the other standard paradoxes. The above truth-value-quantification technique works equally well for both the true⊃true and the false⊃false paradoxes. The remaining family of paradoxes are the false⊃true ones. For instance, it’s paradoxical to say:

7. If it’s warm in the Antarctic today, it’s a cool day in Waco today

even though the antecedent is false and the consequent is true, so the corresponding material conditional is true. But now:

8. Every day that’s other than today or on which it’s warm in the Antarctic is a day that’s other than today or on which it’s cool in Waco.

So by (1)→(2):

9. If today is other than today or it’s warm in the Antarctic today, then today is other than today or today it’s cool in Waco.

And it would be hard to accept (9) without accepting (7). (I made the example a bit more complicated than it might technically need to be in order not to have a case of (1) where there are no Fs. One might think for Aristotelian logic reasons that that case stands apart.)

This suggests that if we object to the “material conditional” reading of “If… then…”, we should object to the “material quantification” reading of “Every F is G”. But many object to the first who do not object to the second.

A Gricean theory of indicative conditionals

The theory consists of two theses and two definitions. I will use → for indicative conditionals. And all my disjunctions will be inclusive.

1. MatCond: "p→q" expresses the same proposition as "~p or q".
2. NonTriv: A use of "p→q" normally implicates that "~p or q" is an evidentially non-trivial disjunction for the speaker.
3. Definition: "a or b" is an evidentially non-trivial disjunction for an agent x if and only if x has non-negligible evidence for the disjunction that goes over and beyond evidence for ~p and evidence for q.

I don't here commit to any particular view of evidence, and if there are non-evidential justifications, one can probably easily modify the theory.

Here is an interesting consequence of the theory which I think is just right. When my evidence that at least one of ~p and q is true is simply the evidence for ~p (or for q), I don't get to say "If p, then q." But if I tell you that at least one of ~p and q is true, then normally you get to say "If p, then q". For when I tell you that at least one of ~p and q is true, then "~p or q" comes to be an evidentially non-trivial disjunction for you: my testimony is evidence for the disjunction and this evidence does not derive for you from evidence for the one or the other disjunct.

Notice that "has non-negligible evidence for the disjunction" has some vagueness to it. Moreover, negligibility is contextual, and that is how it should be. If I tell you that at least one of the following is true: snow is not purple and 2+2=4, then "If snow is purple, then 2+2=4" does not generally become assertible for you. For while you do gain additional testimonial evidence for the disjunction that snow is not purple or 2+2=4 from my speaking to you, the gain is normally negligible over and beyond your earlier evidence that 2+2=4. But if you respond to my assertion with "So, if snow is purple, then 2+2=4", you are speaking quite correctly, since the use of "So" and the conversational context makes the evidence I just gave you salient and hence non-negligible. (Perhaps "salient" or "relevant" could be used in place of "non-negligible" in (3).)

The theory explains why it is that paradoxes of material implication can almost always be made to cease to be paradoxes of material implication as soon as one fills out the evidential backstory in a creative enough way. Take, for instance, the paradox of material implication:

4. If the president will invite me for dinner tonight, I will have dinner with the president in my pajamas.

The antecedent is false, so the material conditional is true, but (4) sure sounds bad (it sounds bad to assert and seems to be saying something bad about my manners). Yes, but now suppose that an epistemic authority has just handed me two numbered and folded pieces of paper, with a sentence written on each and folded in half, and told me that either at least the first paper contains a falsehood or they both contain truths. I puzzle out what she says, and I conclude, very reasonably:

5. If the sentence on the first piece of paper is true, the sentence on the second piece of paper is true.

I then unfold the pieces of paper, and notice that the first piece contains the sentence "The president will invite me for dinner tonight" and the second contains "I will have dinner with the president in my pajamas." And so I reasonably infer from (5):

6. So, if the president will invite me for dinner tonight, I will have dinner with the president in my pajamas.

(And, moreover, I now gain a new piece of evidence that the president won't invite me for dinner tonight—for it would be absurd to suppose I'd have dinner with him in my pajamas.) With this epistemic backstory, the paradoxical conditional is quite unparadoxical. That's because with this epistemic backstory, the corresponding disjunction

7. The president won't invite me for dinner tonight or I will have dinner with the president in my pajamas (or both)

is epistemically non-trivial. But in normal circumstances, (7) is epistemically trivial, since my only evidence for (7) is evidence for the first disjunct.

A similar kind of epistemic backstory can be given for any of the standard paradoxes of material implication, thereby turning paradoxical sentences into non-paradoxical ones (cf. this post). Our Gricean theory (1)-(3) explains this phenomenon neatly. So do theories on which indicatives are non-cognitive and ones on which they are subjective. But the Gricean theory is, I think, simpler.

Notice that in this Gricean theory we haven't brought in non-material conditionals through any back door, because we have explained the implicated content entirely in terms of disjunctions. Furthermore, (2) is basically a consequence of (1) plus the very plausible claim that disjunctive sentences normally implicate the epistemic non-triviality of the disjunction.

Non-triviality of conditionals

Here's a rough start of a theory of non-triviality of conditionals.

A material conditional "if p then q" is trivially true provided that (a) the only reason that it is true is that p is false or (b) the only reason that it is true is that q is true or (c) the only reasons that it is true are that p and q are true.

A subjunctive conditional "p □→ q" is trivially true provided that (a) the only reason that it is true is that p and q are both true or (b) the only reason that it is true is that p is impossible or (c) the only reason that it is true is that q is necessary or (d) the only reasons that it is true are that p is impossible and q is necessary.

For instance, "If it is now snowing in Anchorage, then it is now snowing in the Sahara" understood as a material conditional is trivially true, because the falsity of the antecedent (I just checked!) is the only reason for the conditional to be true. The contrapositive "If it not now snowing in the Sahara, then it is not now snowing in Anchorage" is trivially true, since it is true only because of the truth of the consequent. On the other hand, "If I am going to meet the Queen for dinner tonight, I will wear a suit" is non-trivially true. It is true not just because its antecedent is false--there is another explanation.

Likewise, "Were horses reptiles, then Fermat's Last Theorem would be false" and "Were Fermat's Last Theorem false, horses would be mammals" are "Were I writing this, it would not be snowing in Anchorage" are trivially true, in virtue of impossibility of antecedent, necessity of consequent and truth of antecedent and consequent, respectively. But "Were horses reptiles, either donkeys would be reptiles or there would no mules" is non-trivally true--there is another explanation of its truth besides the impossibility of antecedent, namely that reptiles can't breed with mammals and mules are the offspring of horses and donkeys.

Indicative and material conditionals

I will use "p→q" for the indicative conditional "if p, then q". I will use "p⊃q" for the material conditional "(not p) or q". I will say that "indicatives are material" providing that p→q and p⊃q are logically equivalent for all p and q, where a and b are logically equivalent if and only if it is necessary that (a if and only if b). I will say that p entails q provided that it is necessary that p⊃q.

Almost no philosopher thinks indicatives are material. There are very plausible counterexamples. For instance, suppose it is lightly raining in Seattle and Seattle is not having a drought. Let p be "Seattle is having heavy rain" and let q be "Seattle is having a drought". Then p⊃q, since p is false. But it seems quite wrong to say that if Seattle is having heavy rain, then Seattle is having a drought, so p→q doesn't seem to be true.

I am going to offer some arguments that indicatives are material. Say that → is non-hyperintensional provided p→q and p*→q* are logically equivalent whenever p and p* are logically equivalent and q and q* are logically equivalent. Consider the following two theses:

1. For any possible world w: (p at w) → (q at w) if and only if (p→q at w).
2. For any predicates F and G, from "Every F is a G" (where "x is an F" is more euphonious way of saying that x satisfies F) together with the assumption that c exists, it logically follows that if c is an F, then c is a G.

I will argue in S5, and using some fairly uncontroversial further premises:

3. If (1) is true and → is non-hyperintensional, then indicatives are material.
4. If (2) is true and → is non-hyperintensional, then indicatives are material.

Moreover, I will try to make plausible:

5. If (2) is true, then one has to assign the same truth value as the material conditional does to a number of paradoxical-sounding examples of indicative conditional sentences that are relevantly just like the standard alleged counterexamples to the thesis that all indicatives are material.

I think (1) and (2) are quite plausible. I don't know, however, how plausible it is that → is non-hyperintensional. My argument for (3) is very similar to more general arguments in Williamson. However, given (5), even if we don't assume that → is hyperintensional, there seems to be little advantage to denying the elegant and simple view that indicatives are material.

Argument for (5): Take my heavy rain and drought in Seattle case. Suppose that as it happens, there is no place where there presently is heavy rain. Let Fx say that x is having heavy rain. Let Gx say that x is having drought. Then all Fs are Gs. (If you think, with Aristotle, that "All Fs are Gs" requires there to be an F, then add the premise that on Venus somewhere right now there is a drought but a very, very brief heavy rain is currently occurring. I will leave out such modifications in the future.) Then by (2), we have to say that if F(Seattle), then G(Seattle):

6. If Seattle is having heavy rain, then Seattle is having drought.

And that is pretty much a standard alleged counterexample to the view that indicatives are material, of the false-antecedent sort. The case divides into two: we might suppose that Seattle is having neither drought nor heavy rain, in which case (6) is false-antecedent, false-consequent, or we might suppose that Seattle is having drought and (unsurprisingly) no heavy rain, in which case we have false-antecedent, true-consequent.

We can also use (2) to manufacture a true-antecedent, true-consequent case. Suppose that it is raining in both Seattle and the Sahara. Then the following is a standard alleged counterexample of the true-antecedent, true-consequent sort:

7. If it's raining in Seattle, then it's raining in the Sahara.

Let Fx say that x is a planet on which it is raining in Seattle, and let Gx say that x is a planet on which it is raining in the Sahara. Then every F is a G, since the only F is earth. By (2):

8. If earth is a planet on which it is raining in Seattle, then earth is a planet on which it is raining in the Sahara.

And that sounds about as paradoxical as (7). That completes my argument for (5).

Argument for (3): First we need a special case:

9. If p and q are non-contingent and → is non-hyperintensional, then p→q is logically equivalent to p⊃q.

To argue for (9), consider the following four sentences:

10. 2+2=4→2+3=5. (necessary, necessary)
11. 2+2=5→2+3=6. (impossible, impossible)
12. 2+2=5→ (2+2=5 or 1+1=2 or both). (impossible, necessary)
13. 2+2=4→2+2=5. (necessary, impossible)

If p and q are non-contingent, then they are respectively logically equivalent to the antecedent and consequent of exactly one of (10)-(13). By non-hyperintensionality of →, it follows p→q must have the same truth value as the conditional in that line. But the truth values of (10)-(13) are just as the material conditional says they are: thus, clearly, (10)-(12) are (necessarily) true and (13) is (necessarily) false. So, p→q must have the same truth value as p⊃q, assuming p and q are non-contingent.

The argument for (3) is now easy. Observe that (p at w) and (q at w) are non-contingent, even if p and q are contingent. So,

14. (p at w) → (q at w) is logically equivalent to (p at w) ⊃ (q at w).

But, plainly:

15. (p⊃q at w) is logically equivalent to (p at w) ⊃ (q at w).

From (1), (14) and (15) we conclude that:

16. (p→q at w) is logically equivalent to (p⊃q at w)

and hence indicatives are material.

Argument for (4): The most intuitive form of the argument is to assume theism, and let Fx say that x is an omniscient being that knows that p, and let Gx say that x knows that q. Then as long as p⊃q, it will be the case that every F is a G (just think about the four possible truth-value combinations). Hence:

17. If God is an omniscient being that knows that p, then God knows that q.

Since God's existence and omniscience are necessary, the antecedent and consequent are logically equivalent to p and q, respectively, and so we get p→q by non-hyperintensionality.

If we don't want to suppose there is a God, let's suppose that numbers and sets exist necessarily. Let P be the singleton set whose only member is p. Let Q be the singleton set whose only members is q. Then, let Fx say that x is greater than zero and x equals the number of truths in P. Let Gx say that x is greater than zero and x equals the number of truths in Q. Then, if p⊃q, it is easy to see that all Fs are Gs, so:

18. If one is greater than zero and one equals the number of truths in P, then one is greater than zero and one equals the number of truths in Q.

But the antecedent and consequent are logically equivalent to p and q respectively, so by non-hyperintensionality we get p→q, once again.

Indicative conditionals and material implication

I am partial to the view that "if p, then q" has the same truth conditions as "not-p or q", but there is pragmatic stuff going on. But after thinking hard about Field's remarks on the implications of the Montague Paradox for certain solutions to the liar paradox (in a Beall anthology), I've been drawn to think more about an odd but significant difference between "If p, then q" and "not-p or q".

Here is the less striking way to show the difference. Consider:

1. (x)(Human(x) → Mortal(x))
2. (x)(~Human(x) or Mortal(x))

Now, in (2) there is a symmetry between the disjuncts, where (1) has an asymmetry. Consider a particular case:

3. Human(Seabiscuit) → Mortal(Seabiscuit)
4. ~Human(Seabiscuit) or Mortal(Seabiscuit)

I am happy to grant that (3) and (4) are both true. But they are true for different reasons. I want to say that (3) is trivially true because of the falsity of its antecedent. Period. The fact that Seabiscuit is mortal is not explanatorily relevant to the truth of (3). But (4) is equally true because of the non-humanity of Seabiscuit as because of his mortality. There is a symmetry there.

Here is another way to see the difference. Some people think that sentences like "x is F" are nonsense (gloss: don't express a proposition) when x fails to be of a certain kind for which attribution of Fness makes sense. These people will, for instance, deny that "The chair is true-or-false" makes any sense. (I am inclined to think it makes perfect sense, but is false; chairs aren't propositions and don't express propositions, so they are neither true nor false.) Now, plausibly, if "q" doesn't make sense, neither does: "r or q". That's just nonsense. So, on such a view:

5. This chair is true-or-false or this chair is not a proposition

is nonsense. And so is:

6. This chair is not a proposition or this chair is true-or-false.

But:

7. If this chair is a proposition, this chair is true-or-false

makes perfect sense as an ordinary indicative conditional. It is trivially true because the antecedent is false. If this is right, then we can have meaningful conditionals whose consequents aren't meaningful, but not so for disjunctions. This may force a restriction on the use of modus ponens in subproofs.

For another illustration of this last point, go back to (1) and (2). The quantification seems to be unrestricted, including such entities as numbers and properties. But it is not clear that "Mortal(7)" makes sense. If not, and if disjunction requires both disjuncts to make sense, then (2) is in trouble. But (1) is just fine.

In a lot of programming languages, logical disjunction operators are actually asymmetrical. This means that if you do something like "f(x) || g(x)" ("||" being disjunction) in perl or C, the function g(x) is not evaluated when f(x) turns out to return truth. The disjunction operator shortcuts in this way. As a result, you can do things like "x == 0 || y/x == z" without worrying about the fact that the second disjunct is non-sense if x is zero, because the second disjunct is unevaluated when x is zero. But in human language, I suspect that there is no similar shortcutting in disjunction. Apart from implicatures, there is a symmetry in disjunction. But, I suggest, there is such a shortcutting in conditionals.

But I don't know exactly how far the shortcutting in conditionals goes. I am not sure I want to say:

8. If the sky is green, then **##^^).

But at least to my ear it sounds better to say this than:

9. The sky isn't green or **##^^).

"If... then..." and material conditionals

I will argue that if the indicative "If p, then q" in English has mind-independent truth value (a somewhat vague phrase, admittedly), then this truth value is the same as that of (not-p or q) (i.e., the material conditional). The way I shall argue this is as follows. Assume that "If p, then q" has mind independent truth value. Now, I will show that (i) if (not-p or q) is false, then "If p, then q" is false, and (ii) if (not-p or q) is true, then "If p, then q" is true. Claim (i) is easy. For if (not-p or q) is false, then p is true and q is false, and it clearly cannot be the case that "If p, then q" (modus ponens would be violated).

I now argue for (ii). The easiest way to do this is to specialize to the case where p and q and their denials do not tell us anything about what beliefs and credences people have (the proposition that there are dogs satisfies this constraint; the proposition that nobody believes anything does not satisfy this constraint). If (ii) holds for propositions satisfying this constraint, it will hold in general, surely (assuming "If... then..." has mind-independent truth value). Suppose that you rationally assign a probability very close to 1/2 to p as well as to q, and neither believe nor disbelieve either of these, and rationally assign a probability very close to 1 to the claim that (not-p or q), and, moreover, you know this disjunction to be true. Given the constraint on p and q, it should be quite possible to have a set of evidence that makes one have these probability assignments, and having this set of evidence should not affect the truth values of p, q or the indicative "If p, then q".

You then reflect on the following valid argument:

1. p (premise)
2. not-p or q (premise)
3. Therefore, q.

You want to summarize what you've learned from this argument. You know (2) to be true, and you assign very high probability to it. You don't know (1) or (3) to be true, and you in fact do not have a belief either way about either of these. It seems quite right to summarize your current position as: "If (1) holds, then (3) holds." After all, you've got a valid argument from (1) to (3) given an auxiliary premise, namely (2), which you know to be true. But once we agree that "If (1) holds, then (3) holds", surely we likewise have to agree that "If p, then q." Hence, if the disjunction (2) is known with probability close to 1, and neither p nor q is known or has high or low probability, then "If p, then q" is true. But if "If... then..." has mind-independent truth value, then the assumptions about knowledge and probability are irrelevant to its truth value, and hence we can simply conclude that if (2) holds, then "If p, then q."

The conclusion might be taken as a reductio of the claim that "If... then..." has mind independent truth value.

Quiz on "If... then..."

I am holding out to you my two closed fists. Let us suppose that I know that you know that I know which, if any, of my fists are empty and which are full (for simplicity, I take "full" to be the denial of "empty"). You don't know which, if any, of my fists are empty and which are full. In which of the following cases would I be telling you a lie if I said: "If my left hand is full, then my right hand is full" while competently using English? (Choose "depends" if you think the answer depends on factors that I didn't include in the description of the case.)

1. In fact my left hand is full and my right hand is full: lie not a lie depends don't know
2. In fact my left hand is empty and my right hand is full: lie not a lie depends don't know
3. In fact my left hand is full and my right hand is empty: lie not a lie depends don't know
4. In fact my left hand is empty and my right hand is empty: lie not a lie depends don't know

Adequacy of language

What does it mean to say that language M is at least as "adequate" as language L? One option would be to say that any proposition that L can express is a proposition that M can express. This, I think, is too strong a requirement. For sometimes one language cannot express exactly the same proposition as another language does, but in some sense loses nothing thereby in adequacy, or at least the language is at least as good for theology, ethics, science and ordinary life (that's what I mean by "practically"!) For instance, suppose that L is English and M is a restriction of English to those sentences that end with the conjunct "and each thing is identical to itself". Then M cannot express the proposition that there are horses. But the speakers of M do just as well with respect of theology, ethics, science and ordinary life any way: M is just as good as L for theology, ethics, science and ordinary life. Where a speaker of L would say that there are horses, the speaker of M will say that there are horses and each thing is identical to itself.

I do not have a clear notion of "adequacy" here—suggestions are welcome. Here, for what it is worth, are two interesting examples of how the notion might be useful.

1. Detensing: It is well known that tensed sentences like "I am now in pain" cannot be translated into token-reflexive sentences like "My pain is simultaneous with this utterance" (for instance, the latter sentence entails the occurrence of an utterance). But perhaps one can say, more weakly, that a tenseless language that makes use of token-reflexive forms like this is just as adequate as the tensed language. Certainly, it is just as adequate for ethics, science, ordinary life and probably theology. Instead of saying a sentence of ethics, science, ordinary life and theology like "It is now time for me to partially fulfill my duty of thanking God for the nomic orderliness of the universe", we just say: "This utterance is simultaneous with the time for the partial fulfillment of my duty of thanking God for the nomic orderliness of the universe." In saying this, we are saying something different. Different, yes, but in practice just as useful for ethhics, science, ordinary life and theology.

2. Indicatives: Maybe

1. "If the Queen visits me today, I will be prepared"

does not just mean

2. "I will be prepared for the Queen's visit or the Queen won't visit me today or both."

Nor does it just mean

3. "P(I will be prepared | the Queen visits me today) is high"

(claim (3) does not give modus ponens). In fact, plausibly, there is no paraphrase of the indicative conditional except in terms of indicative conditionals (including ones involving "unless" and other variants). Fine. But one can still say that all indicative conditionals could be dropped from English, and the resulting language would be just as adequate. I would not be saying the same thing as (1) if I affirmed the conjunction of (2) and (3), but I would lose nothing by doing so.

Conditionals, Adams' Thesis and Molinism

The Theorem below is surely known. But the consequence about Molinism is interesting. It is related to arguments by Mike Almeida.

Definition. The claim A→B is a conditional providing A→B entails the material conditional "if A, then B".

Remark: This is of course a very lax definition of a conditional (B counts as a conditional, as does not-A), so the results below will be fairly general.

Definition. A→B is localized provided A&B entails A→B.

Remark: Lewisian and Molinist subjunctives are always localized.

Definition. Adams' Thesis holds for a conditional claim A→B providing P(A→B)=P(B|A).

Definition. The claim B is (probabilistically) independent of A provided P(B|A)=P(B). (If P(A)>0, this is equivalent to P(A&B)=P(A)P(B).)

Theorem 1. Suppose A→B is a localized conditional. Then Adams' Thesis holds for A→B if and only if A→B is independent of A.

Proof. First note that if A→B is a localized conditional, then, necessarily, A&(A→B) holds if and only if A&B holds. Therefore P(A→B|A)=P(A&(A→B)|A)=P(A&B|A)=P(B|A). Now P(A→B|A)=P(A→B) if and only if A→B is independent of A. ■

Remark: It follows that Molinist conditionals do not satisfy Adams' Thesis. For in Molinist cases, God providentially decides what antecedents of conditionals to strongly actualize on the basis of what Molinist conditionals are true, and hence A is in general dependent on A→B (and thus A→B is in general dependent on A).[note 1]

Indicative conditionals

On the material conditional interpretation, the propositional content of the indicative conditional "If p, then q" is p→q, i.e., (not-p or q).

I claim that this is basically the right interpretation if "If p, then q" expresses a proposition whose truth-value is mind-independent (except for any mind-dependence in p and q themselves). You can take this as evidence that the material conditional interpretation is right—that is how I take it—or that English indicative conditionals do not express a mind-independent proposition.

The argument is simple. Suppose that p and q concern non-mental matters, and suppose that w is a world p→q holds, i.e., p is false or q is true or both. Then there is a world w* which is very much like w, except that it contains two persons, A and B, conversing about p and q, neither of whom has any false or misleading or unjustified beliefs, and neither of whom has any beliefs giving significant evidence for any of the propositions p, q, not-p and not-q. We could then imagine A learning that either p is false or q is true or both, and that then the conversation turns to the subject of p and q. I claim that it would then be appropriate for A to say: "Well, I don't have any idea which if any of p and q is true, but I now know that if p holds, so does q." This seems quite right. Moreover, in saying this, A would not be saying anything false. Therefore, if "If p, then q" expresses a proposition, it expresses a true proposition in w*. But if the proposition it expresses is mind independent, it is also true in w, since the two worlds differ only in respect of mind-dependent stuff.

Hence, p→q entails that if p, then q. The converse is easy. If p→q is false, then p is true and q is false, and it is clear that then if p, then q isn't true. Therefore, necessarily, p→q holds iff if p, then q does. Hence, the material conditional gets the truth conditions for the indicative "if... then..." right.

Could it be that there is still a difference in meaning? The only way I could see that would be if "If p, then q" said something additional, something entailed by p→q, but nonetheless added on to it. But I just cannot see what that could be, unless it be something mind dependent.

But perhaps there is a difference here like that between "p or q" and "q or p"? Maybe there really is a difference in the proposition expressed by these claims, even though neither adds anything to the other. If there really is a difference in the propositions expressed by "p or q" and "q or p", then I guess there might be a difference between those expressed by "p→q" and if p, then q. But if so, that difference is not very significant, it seems. Basically, the two say the same thing. Of course, even if there is no difference in proposition, there may be pragmatic differences.

What about standard counterexamples to the material conditional interpretation? For instance, could I say about a batch of cookies that I know to be poisoned
(*) "If George eats these cookies, he won't feel sick"
simply because I know that George won't eat them? Well, I think such counterexamples at most challenge the claim that the indicative conditional expresses a proposition, not the claim that if it expresses a proposition, the proposition it expresses either is or is basically the same as a material condition. Suppose that I don't know that the cookies were poisoned, but Patricia tells me: "An omniscient being either told me that George won't eat these cookies, or that he won't feel sick, but I can't remember which." It seems perfectly appropriate for me to utter (*), then. Suppose I later learn that the cookies are poisoned and that George won't eat them. Do I have any reason to say that I was mistaken when I uttered (*)? Surely not. I can say that what I said was misleading, but not that it was false. Whether (*) is appropriate to say depends on mind-dependent stuff. But if (*) expresses a proposition, then that proposition is mind-independent. Consequently, the intuitions about the appropriateness of saying (*) should not be taken as evidence about what propositional content (*) has if it has any.