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Wednesday, December 06, 2006

Another 2D Puzzle

Here is another one of those can't-see-the-flaw-but-don't-believe-the-conclusion-style arguments:

Initial definitions:
[] ----- it is metaphysically necessary that
<> ----- It is metaphysically possible that
Apriori ----- It is a priori that
EP ----- It is epistemically possible that (or 'it is compatible with what is a priori that')
'Julius' ----- name (in the semantic sense) but known a priori to refer to the actual inventor of the zip.

Principle A:
(As proposed in the comment section here -- DJC's last comment): "the outermost operator determines which intension(s) of the immediately embedded sentence to look at", e.g., '(Apriori, actually p) iff (Apriori p)'.

By A, we have:
(1) Apriori []p <--> Apriori Apriori(p) [Apriori tells box to look at 1-intension]
(2) []Apriori(p) <--> [][]p [box tells a priori to look at 2-intension]
(3) []EP(p) <--> []<>p [box tells EP to look at 2-intension]

Metaphysical modality is governed by S5 (which includes):

[]p --> [][]p
<>p --> []<>p

So, we get (by 1-3):
[]p --> []Apriori (p)
<>p --> []EP (p)

Substituting in:

[]Water is H20 --> [](Apriori (water is H20))
<>Julius is not the inventor of the zip --> [](EP(Julius is not the inventor ...))

But that seems wrong. It doesn't seem that it is necessarily a priori that water is H20. Nor does it seem that it is necessary that it is epistemically possible that Julius is not the inventor of the zip (given that it is not epistemically possible that he is not).

23 comments:

Mike said...

Brit, nice puzzle. I guess I'm wondering about the biconditional in (2), especially right to left.

(2) []Apriori(p) <--> [][]p

In light of the S4 axiom

S4. []p -> [][]p

we are going to get the consequence that EVERY necessary truth is necessary a priori. That can't be right, of course. Since there's no doubting the S4 axiom, the biconditional in (2) has to be false. I think the entailment goes just left to right. So maybe that is a restriction on DJC's principle.

djc said...

Hi Berit,

I think (2) and (3) are false. We may have crossed wires about principle (A), brought out in the way you've applied it to (1)-(3). (A) says that an operator tells you which intension(s) of the immediately embedded sentence to look at. So, for example, the truth-value of NAp depends on the 2-intension of Ap. But I wouldn't say "N tells A to look at the 2-intension" -- that suggests that NA operates on the 2-intension of p, which isn't right. The reason this isn't right is that (as in the comment on the last message), the 2-intension of Ap does not depend only on the 2-intension of p. A operates mainly on 1-intensions (with the minor "entertainability" role for 2-intensions), in such a way that the 2-intension of Ap depends strongly on the 1-intension of p.

Maybe what's going on here is an ambiguity in the phrase "operates on 1-intensions" (and so on). The sense in which Ap operates on 1-intensions is that the semantic value(s) of Ap depend at least in part on the 1-intension of p. But the dependent semantic values of Ap may include not just the 1-intension of Ap but also the 2-intension of Ap, and so on. It looks to me as if your argument invokes a stronger reading on which "A operates on 1-intension" entails that the 2-intension of Ap coincides with that of p (likewise, mutatis mutandis, for "N operates on 1-intensions"). On this reading, I think "A operates on 1-intensions" (and "N operates on 2-intensions") is certainly false.

Mike said...

Brit & Dave,

Doesn't (2') (the first half of (2)) have to be right?

(2') []A(p) -> [][]p

But (2') does make it look like 'A' takes the 2-intension of p when immediately embedded by '[]'. In the last post I thought it looked that way because '[]A(p)' is true only if the 1-intension of p = the 2-intension of p. I wonder if that's generally true.
(3') (the second half of (3)) also seems right. If in every world p is metaphysically possible then in no world is ~p known a priori. But the converse of (3') seems to fail for any p that is a posteriori necessary in every world.
(3') []<>p--> []EP(p)

Brit Brogaard said...

Hi Dave and Mike
o.k. good, then I take my objection back. But I then think you are committed to the following.

It is a priori that it is necessary that p.

What is relevant here is the 2-intension of p. In every scenario, the 2-intension of 'p' must be true at every world.

It is a priori that actually p.

Since 'actually' operates on 2-intension (in "my" sense of 'operates'), what is relevant here is 2-intension of 'p'. So, at every scenario it is the case that at the actual world the secondary intension of 'p' is true. Otherwise, you are not treating 'actually' as an operator on a par with 'necessarily'. If you think it should NOT be treated as an operator on a par with 'necessarily', then I have no problem (except I would disagree about that treatment of 'actually')
(maybe that pertains to your point, Mike?)
If none of this is correct, why does 'necessarily' and 'actually' behave differently in deep epistemic contexts? (that is, why does one operate on 2-intension and the other not?)

djc said...

Hi Berit, I'm not sure what you take me to be committed to -- the truth of the displayed sentences or your analyses of them? I don't think either displayed sentence need be true.

Re ANp: The truth of this requires that Np is true at all scenarios v. Np is true at a scenario v iff p is true at (v, w) for all worlds w (that are possible relative to v). So ANp is true iff p is true at all (scenario, world) pairs (v, w). Note that what ends up being relevant here is not just the 2-intension of p (simpliciter) but its 2-intension relative to all scenarios -- so in effect, its entire 2D-intension. Of course ANp will end up being true only for those relatively rare p with a 2d-intension that is true everywhere.

A@p (where @ is 'actually'): This is true iff @p is true at all scenarios. @p is true at a scenario v iff p is true at v. (More generally @p is true at (v, w) iff p is true at (v, v).) So A@p is true iff p is true at all scenarios.

Re your analysis of the latter: 'actually' operates on 2-intension in that the *truth-value* of @p is a function of the 2-intension of p. However, the semantics above tells us that the *1-intension* of @p is not a function of the 2-intension of p. And it is the latter that is relevant to the truth of A@p. So here again one needs to be careful about distinguishing senses of "operates on".

Note that in this sense, @ and N are on a par. That is, N operates on 2-intension in that the truth-value of Np (simpliciter) depends on the 2-intension of p. But the 1-intension of Np depends on the whole 2D-intension of p, as discussed above. That falls out of the 2D semantics for N given in the second paragraph above.

Brit Brogaard said...

Thanks, Dave. The only thing I want to add is that it still doesn't seem to me that 'actually' is on a par with other operators (assuming your account). Take 'it is possible that'.

'It is possible that p' is true iff the 2-INTENSION of p is true at some world.

'It is a priori that it is possible that p' is true iff at all scenarios there is a world at which the 2-INTENSION of p is true.

'Actually p' is true at c iff the 2-INTENSION of p is true at the world of c

'It is a priori that actually p' is true at c iff the 1-INTENSION of p is true at all scenarios.

So, in the case of 'it is possible that' we do NOT shift from one kind of intension to another (when we have a deeply epistemic context). But we DO shift in the case of 'actually'. Why the difference?

djc said...

Hi, I think one can make them on a par by writing out the full versions (where A/P/@ are apriori/possibly/actually operators):

APp is true iff at all scenarios, there is a world at which the 2-intension of p *relative to that scenario* is true.

A@p is true iff at all scenarios, the 2-intension of c *relative to that scenario* is true at the world of that scenario.

Here the 2-intension of p relative to a scenario v is the function mapping a world w to the truth of p (i.e. of the 2d-intension of p) at (v,w). It's not hard to see that one needs to invoke 2-intensions relative to a scenario in the first analysis above. If one doesn't relativize in this way, one will get bad results whenever Pp but not APp (your analysis will suggest that Pp entails APp, which is false). But once one relativizes in this way, it's easy to see that the same sort of treatment works for 'actually'.

Of course talk of "relativization" her is just shorthand for invoking the full 2D intensions of the relevant expressions. So all this is more straightforward if one gives a fully 2D semantics for the operators. E.g. Pp is true at (v,w) iff there exists w* such that p is true at (v,w*); @p is true at (v,w) iff p is true at (v,v); Ap is true at (v,w) iff p is true at (v*,v*) for all v* [plus an entertainability condition]; and so on. Then one can just compose these definitions to yield 2D truth-conditions for APp, A@p, and so on.

Brit Brogaard said...

Hi David. Right. So, just to sum up, on your account,

@p is true at (v,w) iff p is true at (v,v)

Pp is true at (v,w) iff p is true at (v,w*), for some v-possible w*.

So, I note that when p is embedded in @, then p is true at a scenario. When p is embedded in P, then p is true at a (world, scenario) pair.

That's a disanalogy.

Let's introduce an operator '@@'. @@p is true at (v, w) iff p is true at the v of utterance and the w of utterance.

Is @@ less acceptable than @?

djc said...

Hi, well, strictly speaking @p is true at (v,w) iff p is true at (v, v*) where v* is the world of v. So either way, the embedded p is evaluated at (world, scenario) pairs. Of course different pairs are relevant for @ and for P, but that's just to say that these are different operators on 2D intensions.

Re your '@@': I take it that by "the v of utterance" you mean "the scenario of utterance" and so on. So '@@P' is supposed to be true at all (scenario, world) pairs if P is true in the scenario/world of utterance, and false at all pairs if it is false in the scenario/world of utterance.

This differs from all the other 2D operators we've discussed in that the 2d-intension of @@p isn't just a function of the 2d-intension of p, but of other things (the world/scenario of utterance) instead. Normally the world/scenario of utterance just comes in when we evaluate the 2d-intension of an expression for truth or extension, not when we're composing it. So this is already structurally odd, and is in some tension with the fact that 2d-intensions (as defined on the epistemic 2d framework) are a priori accessible by definition.

My own view is that no linguistic expression (in any language) could work like this. When combined with standard epistemic 2D principles, mayhem results. E.g. if P is true, @@P has a necessary 1-intension, so @@P is a priori, so one can know a priori that @@P. But obviously it follows from the semantics that if an utterance of @@P is true, an utterance of P is true. So can move trivially from @@P to P. So for all P, one can thereby know apriori/trivially that P! That's very odd.

The closest there might be to a linguistic expression that expresses this is the material conditional 'if D, p', where D gives a full neutral description of one's actual scenario. Actual utterances of 'if D, p' work the way you suggest (if they are true, they have a 2d intension that is true everywhere, and so on). But obviously counterfactual utterances don't.

This brings out the point that there are many 2D intensions that one can define that aren't the 2D intensions of any expressions (as least if intensions are defined in the epistemic 2D way). For example, there is no expression whose 1-intension picks out the planet Venus at all scenarios. The epistemic definition of the intensions in effect imposes epistemological constraints that rule these expressions out.

Brit Brogaard said...

Hi David,
Yes, the world/scenario of utterance does show up in the truth-conditions for @@. But prior to 2D, the truth-conditions offered for @ were similar. Pre-2-D, '@P' is true at w iff P is true at the world of utterance. @@ is just a 2D version of that.

You say: "@@P has a necessary 1-intension". This is not odd in all cases. An utterance of '@@ Julius (if he exists) is the inventor of the zip' seems a priori. But, as you sugget, what is odd is that you ought to be able to attach @@ to all actual truths. 'John is hungry' should entail '@@(John is hungry)'. And that then has a necessary 1-intension.

I take that to be conclusive evidence against the truth-conditions offered for @@.

I would like to propose an alternative set of truth-conditions. @@p is true at (v, w) iff p is true at (v, w^), where w^ is the world of utterance. I think that is closer to what I had in mind anyway.

Some examples: 'It is epistemically possible that (@@, Julius (if he exists) is the inventor)', then is true at (v^, w^) iff 'Julius (if he exists) is the inventor' is true at (v, w^), for some v possible relative to v^. This, of course, is true, as v^ is possible relative to v^.

'It is a priori that @@, Julius (if he exists) is the inventor' then is true at (v^, w^) iff 'Julius (if he exists) is the inventor' is true at (v, w^), for every v possible relative to v^. This, of course, is false, as v may not have any inventors. This I take to be a welcome consequence.

'Necessarily, @@, Julius (if he exists) is the inventor' is true at (v^, w^) iff 'Julius (if he exists) is the inventor' is true at (v^, w^). So, this is true.

'Necessarily, apriori, @@, Julius (if he exists) is the inventor' is true at (v^, w^) iff 'Julius (if he exists) is true at (v, w^), for every v possible relative to v^. This is false (for the reasons indicated earlier).

Now note that operators then fall into two kinds. Those which "shift" only the scenario parameter of the index of evaluation, and those which "shift" only the world parameter of the index of evaluation. The first kind includes apriori, and 'deeply epistemically possible', and the second includes box, diamond, @, and @@.

Now, which of @ and @@ corresponds to the natural language word 'actually'? Well, I think neither does. For reasons I can't state here [but see the last sections of my "Sea Battle Semantics"], I think the truth-conditions for the NATURAL language word 'actually' are as follows.

'Actually, p' is true at (v, w) iff p is true at (v, w).

So, neither @ nor @@ corresponds to 'actually'. Is either operator then a technical choice within 2D-ism. With djc's help, I have come to believe that the answer is 'yes'.

djc said...

Hi Berit,

You're right that non-2D semantic values have the feature that the world of utterance can show up in them. But these semantic values aren't supposed to be accessible a priori, whereas 2D intensions are supposed to be accessible a priori. So there is a particular worry with having the world of utterance (presumably determined a posteriori) show up in a 2D intension.

I think I agree about natural-language 'actually'. But I'm inclined to think that there also can't be a technical term 'actually' that works the way you've spelled out in this message. The reasons are related to those above and in my previous comment. But here is a way to make the case specifically.

Let P be a neutral a posteriori truth: e.g. 'there are more than 100 philosophers'. Then for all v, P is true at (v, @), where @ is the actual world. It follows from your semantics that for all v and w, @@P (uttered in this world) is true at (v, w). It follows from 2D principles that an utterance of @@P expresses a priori knowledge. But as in the previous comment, it follows from your semantics that one can move trivially from the truth of an utterance of @@P to the truth of an utterance of P. So for all a posteriori and neutral P, one can come to know P by a combination of a priori and trivial reasoning. That seems wrong.

Brit Brogaard said...

Hi David. I accept all of your reasoning up until "it follows from 2D principles that an utterance of @@P expresses a priori knowledge". Which principles? I prefer to say that it follows from 2D principles that an utterance of @P expresses a priori knowledge, not that @@P does.

We cannot infer from '@@P' that 'apriori, @@p'. Countermodel (your example would also do but for variety's sake we will consider a different one). utterance world: Julius is the inventor and rich. v: Julius is the inventor and rich. v2: Julius is the inventor but poor. '@@, Julius is rich' is true at (v, utterance world) iff 'Julius is rich' is true at (v, utterance world). So, '@@ Julius is rich' is true at (v, utterance world). But Julius is not rich at v2. So, '@@, Julius is rich' is not a priori, which is as it should be.

So, I still think @@ is a plausible technical operator in 2D. It's fine with me if we need @ for different purposes. Neither is the natural language word 'Actually'.

djc said...

Hi Berit, the 2D principle in question is "If Q is true at (v,w) for all v and w, then Q is a priori." [Actually the usual principle is "Q is a priori iff its primary intension is true at all scenarios. But this entails the above.] In my previous message I made the case that when P is a neutral truth at the actual world, @@P is true at (v,w) for all v and w. So it follows from the principle that @@P is a priori. This reasoning doesn't apply to your 'Julius' sentence, as 'Julius' is not neutral.

Brit Brogaard said...

Thanks, Dave. But do I really have to accept this?

"Let P be a neutral a posteriori truth: e.g. 'there are more than 100 philosophers'. Then for ALL v, P is true at (v, @), where @ is the actual world. It follows from your semantics that for all v and w, @@P (uttered in this world) is true at (v, w). It follows from 2D principles that an utterance of @@P expresses a priori knowledge. [CAPITAL added]"

Even if P is neutral, why does it follow that for all v, P is true at (v, @)? At some v*, the inventor of the zip and the tallest spy are the only philosophers. So, P is false at (v*, @), So, @@P is false at (v*, @). So, the primary intension of '@@P' is not necessary. So, to know P one needs a posteriori sources of evidence.

djc said...

Here I'm using '@' as a name for the actual world (one with more than 100 philosophers). P is true at @, and at (@*, @), where @* is my scenario of utterance. Since P is neutral (which means that its 2-intension relative to v is the same for all scenarios v), it follows that P is true at (v, @) for all v. [N.B. P is true at (v, @) iff the 2-intension of P relative to v is true at @.] The rest follows as before.

Brit Brogaard said...

Right, so it seems that the truth-conditions I proposed for @@ would require us to reject:

P --> @@P, for neutral Ps

Perhaps this is o.k. For, don't your truth-conditions for @ require us to reject:

P --> @P, for non-neutral Ps

Suppose I acquired 'XYZ' by the description: "the chemical substance philosophers call 'XYZ' ".

World v*: the drinkable liquid ... is H20 but philosophers call it XYZ.

Scenario v: the scenario in which v* is the world.

World w: the drinkable liquid ... is XYZ.

'the drinkable liquid ... is XYZ' is true at (v, w).

But '@, the drinkable liquid ... is XYZ' is false at (v, w).

For at the world of v (= v*), the drinkable liquid is H20, not XYZ.

djc said...

I'm not sure how you can reject the inference from the truth of an utterance of P to the truth of @@P, given your semantics. There's still my argument from six comments back. In brief: if P is true at the world of utterance w^, then neutrality entails that it is true at (v,w^) for all v. But by your semantics this is just to say that @@P is true.

Re your case: I'd say that as you've set things up, 'the drinkable liquid is XYZ' is false at (v,w), as you've set things up so that the primary intension of 'XYZ' picks out H2O at v, so that the 2d-intension of 'XYZ' picks out H2O at (v,w) for all w.

For the same reason '@, the drinkable liquid is XYZ' is true at (v,w). Note that on my account 'P <-> @P' is a priori but not necessary (and hence not a priori necessary), so one would expect it to fail at some (v,w).

Brit Brogaard said...

Apropos the first part, you say "if P is true at the world of utterance w^, then neutrality entails that it is true at (v,w^) for all v". Why? It is epistemically possible that there are 2 philosophers, even though P (there are more than 100) is true at the world of utterance w^. But then the scenario v which verifies "there are 2 philosophers", and which is not v^, does not verify P. So, how can P be true at (v, w^)? If it isn't, then (given the truth of P at @) neutrality doesn't entail that P is true at (v, w^), for all v.

Sure, if @@P is true, then P is true at (v, w^), which, for neutral Ps, can only be the case if w^ has more than 100 phils and v has more than 100 phils. But if I deny P --> @@P, for neutral Ps, then how do we get to @@P in the first place?

Actually, upon further reflection, I do not think I need to deny P --> @@P, for neutral Ps. I just need to deny that P --> @@P holds for all (v, w). For P --> @@P will clearly be true for some (v, w) pairs.

As for the second part, H= 'Hesperus is the brightest object visible in the evening sky' --> @H fails in the following case:

v: Mars is the brightest object in the evening sky
For H is true at (v, the actual world). But @H is false at (v, the actual world), as H is false at (v, v*).

which is, of course, just what you said below.

But then, if what I said is right, @@ is no less plausible than @.

P --> @@P, fails for some (v, w) pairs when P is neutral

P --> @P fails for some (v, w) pairs when P is non-neutral.

djc said...

Hi, to say that P is neutral is just to say that for all v, the 2-intension of P relative to P is the same. So it follows automatically that if P is neutral and true at (@, w), then it's true at (v, w) for all w.

Re P = 'There are more than 100 philosophers': even if v is a scenario with 2 philosophers, than as long as w has more than 100 philosophers, P is true at (v,w). The fact that v verifies ~P is irrelevant here. That fact tells us something about the diagonal of the matrix (v,v*), but (v,w) is off-diagonal. One might think of the value of P at (v, w) as: the value of P at w, when v is considered as actual. So here that value is "true". Or to use an indicative/subjunctive heuristic: Given that there are exactly 2 philosophers, then if there had been 101 philosophers, would there have been more than 100 philosophers? The answer is obviously yes.

Re the second part, my argument (eight comments ago now) didn't invoke the claim that P->@@P is true at all (v,w) -- that claim is obviously false, as you say. It just invoked the claim that if an utterance of P is true, an utterance of @@P is true (and it gave an argument for that claim). I'm still not sure what your response is to the argument given there!

djc said...

Sorry, the first sentence should say "relative to v", not "relative to P".

Brit Brogaard said...

Right, I do owe you a response. It turns out to be more straightforward than I made it out to be. @@P (as uttered in @) is not entertainable at worlds other than @. Following you, P is a priori at a world iff its primary intension is necessary and P is entertainable at that world. So, @@P (as uttered in @) fails to be a priori.

djc said...

Well, @@P is certainly entertainable at @ (any proposition is entertainable at a world where it is expressed), and it has a necessary primary intension, so it follows that it is a priori at @. That's all my argument needed.

Brit Brogaard said...

Thanks, Dave. o.k. I will make do with @ and dispose of @@ (if only reluctantly).