Wolfgang Schwarz

A Paper

I've written a little paper in German about the connections between metaphysical (modal) and analytical implication for the Olaf Müller-Kolloquium here at Humboldt University: "Fundamentale Wahrheiten" (PDF). It brings together some things I've already written about here. The main ideas are entirely due to Lewis, Jackson and Chalmers.

Since I haven't slept last night and feel unable to do anything productive, here is an abbreviated translation.

1. We can use language to convey information: When you say "it's raining" or "the speed of light is 299792 km/s" I might learn from your utterance that things are a certain way. Let's call the set of possible situations in which things are the way a sentence represents them as being the sentence's "A-intension". (A more traditional and perspicuous term would be "truth-conditions".) So the A-intension of "it's raining" in English is the set of possible situations in which it's raining.

A bit more precisely, the set of situations M is the A-intension of S in a certain community iff, roughly, 1) members of the community typically try to utter S only in situations that belong to M; and 2) they expect each other to utter S only in such situations. Hence they consider utterances of S as evidence that the current situation is of kind M, and they have reason to utter S if they want others to believe that the current situation is of kind M.

This is still very rough, but it shouldn't be too controversial that our sentences can be assigned A-intensions meeting conditions more or less like these. Moreover, it seems very plausible that there are general, systematic rules for how a sentence's A-intension depends on its constituents and its structure. These rules might then also deliver A-intensions for sentences that are too complex for anyone to understand; and they might tell us that the A-intension of "language doesn't exist" contains various situations in which language doesn't exist. This probably can't be determined by the criteria above.

2. Our languages contain intensional operators, and a common semantic strategy in the interpretation of these operators requires that we assign embedded sentences semantic values of various kinds. For instance, to interpret "necessarily, S" as being true iff S is 'true in every possible world', we have to assign to S a set of possible worlds: the worlds 'in' which S is true. Let's call this set the sentence's "C-intension". Similarly. to interpret (the temporal use of) "always S" as true iff S is true at all times, we have to assign S a "T-intension", i.e. the set of times 'at' which S is true.

It would be nice if we could define the C- and T-intensions in terms of A-intensions, but this is tricky if our language contains terms that escape the influence of intensional operators. E.g. "In 1984, the current prime minister was male" is true, even though presumably no actual 1984-situation belongs to the A-intension of "the current prime minister is male". The problem is that even when "In 1984" has shifted the time to 1984, "the current prime minister" still denotes the prime minister in 2004. Similarly, if Tony Blair is essentially male, "necessarily, the actual prime minister is male" is true (ignoring irrelevant complications), even though many possible situations are not in the A-intension of "the actual prime minister is male". So it isn't obvious how a sentence's C- and T-intension can be defined in terms of its A-intension.

Could we conversely use C-intensions to do the work of A-intensions? No. For one, C-intensions only contain entire worlds, but most of what we communicate isn't about our world as a whole, but about our specific location in the world. E.g. we can hardly replace the class of possible situations in which it is raining with the class of worlds in which it is raining (somewhere? everywhere?) as the A-intension of "it's raining". In fact, we can't assign any suitable C-intension to the sentence type "it's raining", but only to particular utterances of the sentence at particular locations. So at best we'd have to employ functions from contexts to C-intensions in our semantics, and one might wonder whether the diagonals of these functions aren't just the A-intensions we tried to get rid of.

Apart from issues to do with context-dependence, C-intensions are also ill-suited to do the work of A-intensions because whenever A- and C-intensions fall markedly apart -- due to terms that escape modal operators --, the C-intensions don't fit the job description of §1. E.g. we can convey substantial information by uttering "Hesperus is not Phosphorus", whose C-intension is empty; if astronomers credibly make this statement, we roughly know what kind of world they say we live in. On the other hand, we can't convey much by "the actual prime minister is prime minister", and certainly not the same as with "Tony Blair is prime minister", even though the C-intensions of these sentences coincide (on current orthodoxy).

3. Understanding a sentence can be explicated as knowing under what conditions it is true, i.e. knowing in which possible situations things are as the sentence represents them as being. So if I understand a sentence S whose A-intension contains such-and-such situations then all I need to find out in order to know that S is true is that my current situation is one of those situations. If the A-intension of S contains all situations, then that's trivial. So in this case understanding the sentence in principle suffices for knowing that it's true. Such sentences therefore deserve to be called "analytical".

If a sentence is analytical then understanding it puts you in a position to discover its truth without further empirical investigation. So all analytical sentences are a priori. The converse isn't obvious, however. Perhaps there are hidden parts of reality we know about only via extra-sensory 'intuition' or 'insight'. Mathematical, modal and moral truths come to mind. Such truths might be called "a priori" even though understanding doesn't suffice to find out their truth-value. In this case their A-intensions presumably won't be universal. (Personally, I don't believe in extra-sensory sources of knowledge. So on my view and the above definition, apriority and analyticity coincide.)

4. Let's define the A- and C-intension of a set of sentences as the intersection of the corresponding intensions of its elements. Then we can say that a set M of true sentences metaphysically implies a sentence S iff the C-intension of M is a subset of the C-intension of S. Analogously for analytically implies with A-intensions. Let's call M metaphysically or analytically fundamental iff M analytically or metaphysically (respectively) implies all true sentences.

If M is metaphysically fundamental, its C-intension contains our world, and only such other worlds w' for which there is no sentence in the relevant language whose C-intension distinguishes between our world and w'. In fact, if the language includes philosopher's English, the C-intension of M must be the singleton of our world, for the C-intension of "everything is just as it actually is", understood in an appropriate way, itself contains only our world. And if M is fundamental, it metaphysically implies this sentence, i.e. M's C-intension must be a subset of the singleton of our world, so it must itself be the singleton of our world.

Other sentences whose C-intension is the singleton of our world might include "all actually true sentences are true" and "the world as it is exists", as explicated in the postscript to David Lewis's "Things qua truthmakers".

If M is analytically fundamental, its A-intension contains the current situation c and only such other situations c' for which there is no sentence in the relevant language with which its users could communicate that they are in c but not in c'.

5. Are there metaphysically fundamental sets that aren't analytically fundamental? Yes, for example the set containing only "everything is just as it actually is", or any of the other examples mentioned in the last paragraph. The C-intension of these sentences is the singleton of our world, but their A-intension leaves lots of possibilities -- indeed all possibilities -- open. This is why we can't convey much by uttering those sentences.

Note how trivial it is to get a metaphysically fundamental set of truths. This raises a few questions for philosophers like Frank Jackson who characterize metaphysics as the search for a small set of truths that metaphysically imply all truths. Moreover, if sentences like "everything is just as it actually is" count as physical truths, then it's trivial that the physical truths metaphysically imply all truths. They do so even if our world is full of gods and cartesian spirits. The most obvious way to exclude the trivializers is to restrict the (alleged) fundamental truths to sentences not containing "actual" or other terms that break out of modal operators.

6. Consider type-B physicalism, the thesis that the physical truths (probably in some augmented English) are metaphysically, but not analytically fundamental. According to a common defence of this view, the metaphysical implication takes place because psychological terms like "pain" rigidly denote states that are also rigidly denoted by physical terms, perhaps "C-fiber firing". Hence necessarily, if Paul instantiates C-fiber firing, then Paul instantiates pain.

This reasoning breaks down if we disallow rigid terms in the fundamental truths, as I've just suggested we should do in order not to trivialize physicalism. Type-B physicalists will have to find another way to exclude the trivializers.

Incidentally, even if they manage to do this, they still face two serious problems:

First, how come "pain" denotes C-fiber firing? Is this a primitive truth we can only know by extra-sensory 'intuition' or could we find it out empirically? The latter seems intuitively correct (and is demanded by the project of naturized semantics). So what kind of information do we need to find out that "pain" refers to C-fiber firing? Perhaps information about causal chains leading to our usage of the term; perhaps information about the causal and structural properties of C-fiber firing; perhaps information about its distribution (e.g. that it typically occurs in just those people who express pain). But information of this kind is givable in physical terms. And if the physical truths tell us 1) that our term "pain" denotes C-fiber firing, and 2) that Paul instantiates C-fiber firing, then they also tell us that Paul instantiates what our term "pain" denotes, i.e. that Paul has pain. So type-B physicalists have to say that no amount of physical information can tell us what our term "pain" denotes. Which is not something a physicalist should want to say.

Second, consider panpsychism, the view that physical truths tell us only about structural-relational features of entities which, at bottom, have a psychological nature. Philosophers usually don't classify this as a kind of physicalism, even though it satisfies many definitions of "physicalism". But in what respect is type-B physicalism different from panpsychism? It holds that "C-fiber firing" and other physical-biological characterizations pick out an entity which is also, essentially, pain -- where this is presumably not due to its structural-relational properties (which would be expressible physically), but simply to its very nature. The main difference seems to be that panpsychism says of more and perhaps different things that they have a psychological essence. But this isn't the reason why philosophers don't count panspsychism as physicalism. Rather, they do so because of the fundamental psychological entities in its ontology whose psychological features are physically inexplicable. And that applies to type-B physicalism just as well.

7. One might suggest that there is another, very different, way in which an implication relation could be metaphysical but not analytical, a way that has nothing to do with rigid designators: For M to metaphysically imply S, all that's required is that there exists no world in which M holds but not S. This is a metaphysical fact, perhaps ultimately due to the nature of the things mentioned in M and S. At any rate, it need not be a simple analytical fact, accessible a priori to every competent speaker.

In response, I have to say more about what I mean by "possible world" (and "possible situation", etc.). I use the possible worlds jargon because it proves fruitful in and outside of philosophy, not because I think its interpretation doesn't raise serious philosophical questions. It's the same with my use of set theoretical terms. The interpretation of set theory is challenging: Are there really any sets? Where and what are they? Or should set theoretical talk be understood in some fictionalist or structuralist way that doesn't presuppose a special domain of entities? I can't answer these questions here, but I want to offer a constraint on any adequate answer: it must respect our usage. Well, admittedly, it might turn out that there is no satisfactory way to interpret set theory in a way that respects our core assumptions about sets, In this case the thing to say is that there are no sets and we should stop talking set theory. But it cannot turn out that even though sets exist, they don't satisfy our core assumptions and hence can't be employed for the work we employ them for. E.g. it can't turn out that there is no singleton of the empty set, or that there are only 17 ordinals. Any interpretation which delivers results like these should be disregarded as the correct interpretation of set theory. Briefly, if sets exist at all, they must satsfy something like ZF (with urlements), and thereby be capable to do their work.

The same holds for possible worlds. Perhaps possible world talk doesn't make any sense and should be given up. But if not, if it has a coherent interpretation, then that interpretation must respect our usage of possible worlds in philosophy and elsewhere. One major use is to capture informational content in more or less the way I suggested. Hence on the assumption that possible worlds talk makes sense at all, if a sentence conveys substantial information, there must be a possible situation in it's A-intension. We need no more extra-sensory insight here than we need it to know that (assuming set theory is coherent) the empty set has a singleton.

So one can't coherently deny that if "P and not Q" is not analytically false -- if it conveys substantial information --, then there is a situation in which things are the way the sentence says. Not unless one denies that there are any possible situations at all. To open a gap between metaphysical and analytical implication that doesn't rely on semantic rigidity, one should therefore say that while there are indeed all these possible worlds, including worlds that would falsify the alleged metaphysical implication, not all of them are metaphysically possible. Metaphysical possibility is then a subspecies of absolute possibility, like physical possibility.

But this is not how "metaphysical possibility" is commonly used. It is commonly defined as possibility in the widest sense. At any rate, if I were to defend the claim that Geroge Bush being president is metaphysically necessary by stipulating that "metaphysically possible" applies only to worlds in which Bush is president, my claim wouldn't be very interesting.

Appendix on Two-Dimensionalism

I have introduced A-intensions in roughly the way David Lewis characterizes truth-conditions, and the way Frank Jackson sometimes introduces A-intensions. These 'semantic' A-intensions contrast in several ways with entities defined in other uses of the two-dimensional framework.

There is probably also an important disagreement about C-intensions. I have defined them as what's needed for the standard interpretation of intensional operators. Many authors however seem to regard this use of C-intensions as an additional (and accidental?) feature of C-intensions. These authors often talk of C-intensions as being the primary 'content' or 'meaning' of a sentence. This kind of meaning can't be defined in the way I characterized content in §1, as that would instead lead to A-intensions. Often, it seems to be defined compositionally in terms of the semantic values of constituents, without meeting any independent characterization. Sometimes it's motivated by alleged semantic intuitions about "what is said" and the like. Some philosophers even seem to think that it's just intuitively obvious that "water is H2O" is 'true at' twin world. Anyway, the disagreement about A-intensions is at least somewhat clearer.

One alternative to semantic A-intensions as I employed them are 'metasemantic' A-intensions, defined as the set of situations in which the relevant sentence is uttered as a truth, where the relevant sentence is individuated orthographically or phonetically. So the metasemantic A-intension of "it's raining" contains situations where someone truely utters "it's raining" not because it's raining but because in his language the sentence means that 2+2=4.

It's clear that these A-intensions look very different from semantic A-intensions. Also, from a semantic point of view, the definition in terms of truth values of sentence tokens in possible situations gets things backwards: whether an utterance is true in a situation depends on whether the things in that situation are as the uttered sentence says they are. We need (semantic) A-intensions to get to truth values in the first place.

A variantion of the metasemantic approach is Stalnaker's use of 'diagonal propositions'. These don't comprise all possible true utterances of the sentence, but only those compatible with the relevant subject's believes and presuppositions. (Compare Kripke's explanation that "Hesperus is Phosphorus" appears contingent because there are possible subjects in exactly the same epistemic situation like us who say something true with this sentence.) So these A-intensions are relative to subjects in particular situations. Stalnaker claims that his metasemantic A-intensions together with C-intensions make semantic A-intensions redundant. I disagree. He also argues that semantic A-intensions presuppose a radical internalistic account of mental content. I don't see this connection.

Another common way to introduce A- and C-intensions is in terms of two intuitive ways to describe or consider a possible situation: If we describe the XYZ-world as just another possible world, we say that "water is XYZ" is false there. If however we assume that the actual world turns out to be an XYZ-world -- if we consider the XYZ-world as actual -- we judge that "water is XYZ" is true. So we can say that situation c is in the A-intension of S iff we'd use S to describe c considered as actual.

For competent members of a linguistic community, this approach delivers similar results as the semantic approach. For part of the semantic characterization is that competent speakers are disposed to utter and assert to S in the situations that belong to S's A-intension. And within limits, we are generally able to say what we would do under hypothetical assumptions. Considering a situation as actual means assuming hypothetically that one is in that situation. So, within limits, we can expect a competent speaker to judge that S is true in a situation considered as actual iff that situation belongs to S's semantic A-intension.

However, experience shows that many philosophers don't understand what 'considering a world as actual' is supposed to mean. E.g. George Bealer complains that "water is H2O" is true in all worlds, no matter how they are considered, and Stalnaker claims that "tigers are pieces of furniture" is true in worlds considered as actual in which "tiger" means "sofa". Moreover, some philosophers think we can only consider worlds as actual under some description, which complicates things, because now the resulting A-intensions probably depend on the chosen description. (I think these philosophers are wrong: the content of an assumption, like the content of a sentence, is best understood directly as a set of possible situations, not as some kind of linguistic entity.)

This leads to another approach, on which A-intensions, like my C-intensions, are characterized in terms of sentential embeddings. E.g. one might say that situation c is in the A-intension of S iff something like "if B(c) then S", or "if it turned out that B(c) then S", or "it is a priori that if B(c) then S" holds, where B(c) is some description of c. (The latter is one of Chalmers's characterizations of his 'epistemic' A-intensions.)

To this end, B(c) shouldn't describe c as counterfactual. E.g. we don't get interesting A-intensions if we allow "there is no water" in the description of the XYZ-world. It's a little tricky to specify the allowed vocabulary in B(c) without presupposing A-intensions. But it probably can be done.

Like the metasemantic approach, this approach assumes that the truth values of the larger constructions are already given. Again, I would think the truth of "if B(c) then S" crucially depends on how B(c) and S describe things as being. Moreover, since B(c) should be a complete description of c, the relevant constructions are enormous, often infinite sentences, and I wouldn't like to appeal to the truth-value of such monsters in defining A-intensions (or analyticity, etc.).

That said, it seems plausible that some constructions in our language operate on (semantic) A-intensions. In fact, I've argued to analyse "S is analytic" in terms of S's A-intensions. Such uses can be considered as additional (and, yes, accidental) features of semantic A-intensions. If A-intensions are already independently specified, there's no longer any need for complete or semantically neutral descriptions. E.g. we can simply propose that "if R then S" is true iff the A-intension of R is a subset of the A-intension of S.

Finally, A-intensions are sometimes introduced to capture 'reference-fixing descriptions'. E.g. if "Julius" is introduced as "the inventor of the zip", we can take the class of individuals in all possible worlds satisfying this description to be the A-intension of "Julius". These A-intensions can be used to define A-intensions of simple, extensional predications in the obvious way.

I have no idea what A-intensions defined in this way might be good for. Firstly, most terms aren't introduced by reference-fixing descriptions. And even for the few terms that are, competent speakers needn't know the description. In general I think it's not a good idea to begin with subsentential expressions when defining A-intensions.