Wolfgang Schwarz

The deontic logic of Desire as Belief

Assume that for any proposition A there is a proposition \(\Box A\) saying that A ought to be the case. One can imagine an agent – call him Frederic – whose only basic desire is that whatever ought to be the case is the case. As a result, he desires any proposition A in proportion to his belief that it ought to be the case:

Let w be a maximally specific proposition. Such a "world" settles all descriptive and all normative matters. In particular, w entails either \(\Box w\) or \(\neg \Box w\). Suppose w entails \(\Box w\). Does Frederick desire to live in such a world? Yes. On the assumption that w is actual, the entire world is as it ought to be. That's what Frederick wants. So he desires w.

Updated notes on counterpart semantics for modal predicate logic

In around 2009, I got interested in counterpart-theoretic interpretations of modal predicate logic. Lewis's original semantics, from Lewis (1968), has some undesirable features, due to his choice of giving the box a "strong" reading (in the sense of Kripke (1971)), but it's not hard to define a better-behaved form of counterpart semantics that gives the box its more familiar "weak" reading.

Wondering if anyone had figured out the logic determined by this semantics, I found an answer in Kutz (2000) and Kracht and Kutz (2002). I also learned that counterpart semantics seems to overcome some formal limitation of the more standard "Kripke semantics". For example, while all logics between quantified S4.3 and S5 are incomplete in Kripke semantics (as shown in Ghilardi (1991)), many are apparently complete in the "functor semantics" of Ghilardi (1992), which I do not understand but which is said to have a counterpart-theoretic flavour. Skvortsov and Shehtman (1993) present a somewhat more accessible "metaframe semantics", inspired by Ghilardi's approach, and claim that the quantified version of all canonical extensions of S4 remain canonical (and hence complete) in metaframe semantics. Kracht and Kutz argue that their – much simpler – counterpart semantics inherits these properties of functor and metaframe semantics.

Nencha on counterpart semantics

Informal talk about de re necessity is sometimes "weak" and sometimes "strong", in Kripke's terminology. When I say, 'Elizabeth II could not have failed to be the daughter of George VI', I mean – roughly – that Elizabeth is George's daughter at every world at which she exists. By contrast, when I say, 'Elizabeth II could not have failed to exist', I don't just mean that Elizabeth exists at every world at which she exists. My claim is that she exists at every world whatsoever. The former usage is "weak", the latter "strong".

When people give a semantics for the language of Quantified Modal Logic (QML), they typically treat the box as strong. '\( \Box Fx \)' is assumed to say that x is F at every accessible world, not just at every accessible world at which x exists.

Lecture notes on modal logic

I've been teaching a course called Logic 2: Modal Logics for the past few years. It's an intermediate logic course for third-year Philosophy students, all of whom have taken intro logic. I'm not entirely convinced that a second logic course should focus on modal logic, but it works OK.

One nice aspect of modal propositional logic is that models, proofs, soundness, completeness, etc. are not as trivial as in classical propositional logic, but easier than in classical predicate logic. I also like the many philosophical applications. I spend a week on epistemic logic, another on deontic logic, one on temporal logic, and one on conditionals.

Anyway, I've just uploaded my lecture notes to github, in case anyone is interested. The LaTeX source is there as well.

Relativism and absolutism in deontic logic

Consider a world where eating doughnuts is illegal and where everyone thinks it is OK to torture animals for fun. Suppose Norman at w is eating doughnuts while torturing his pet kittens. Is he violating the laws? Is he doing something immoral?

In one sense, yes, in another, no. His doughnut eating violates the laws of w, but not the laws of our world. Conversely, his kitten torturing violates a moral code accepted at our world, but not a code accepted at w.

Non-existent mathematical objects

An amusing passage from a recent paper by Erik and Martin Demaine on the hypar, a pleated hyperbolic paraboloid origami structure:

Recently we discovered two surprising facts about the hypar origami model. First, the first appearance of the model is much older than we thought, appearing at the Bauhaus in the late 1920s. Second, together with Vi Hart, Greg Price, and Tomohiro Tachi, we proved that the hypar does not actually exist: it is impossible to fold a piece of paper using exactly the crease pattern of concentric squares plus diagonals (without stretching the paper). This discovery was particularly surprising given our extensive experience actually folding hypars. We had noticed that the paper tends to wrinkle slightly, but we assumed that was from imprecise folding, not a fundamental limitation of mathematical paper. It had also been unresolved mathematically whether a hypar really approximates a hyperbolic paraboloid (as its name suggests). Our result shows one reason why the shape was difficult to analyze for so long: it does not even exist!

So the hypar joins the ranks of phlogiston, the planet Vulcan, the largest prime, or the quintic formula: objects of inquiry that turned out not to exist.

Counterparts of sequences and multiple counterpart relations

Allen Hazen (1979, pp.328-330) pointed out a problem for Lewis's counterpart-theoretic interpretation of modal discourse: the fact that x is essentially R-related to y should be compatible with the fact that both x and y have multiple counterparts at some world, without all counterparts of x being R-related to all counterparts of y. But the latter is what Lewis's semantics requires for the truth of `necessarily xRy'.

Quick logic question

Suppose you add to the language of first-order logic a sentence operator L for which you stipulate that all instances of

(L(p -> q) & Lp) -> Lq

are valid and that validity is closed under prefixing L's:

if p is valid, then so is Lp.

For example, L could be the modal operator 'necessarily', or it could mean the same as ''. If it means the same as '', then

Intensions, extensions, and quantifiers

Suppose we want to follow Frege and distinguish an expression's denotation from its sense. Suppose also we take the denotation of a predicate to be its extension: the set of its instances. The following argument appears to show that this leads to trouble.

1. All humans are featherless bipeds, and all featherless bipeds are human, but there could have been featherless bipeds that are not human. In short, (Ax)(Hx <-> FBx) & <> (Ex)(~Hx & FBx)).
2. By existential generalisation over the predicate positions, it follows that (EX)(EY)((Ax)(Xx <-> Yx) & <> (Ex)(~Xx & Yx)).
3. If things in predicate position denote sets of individuals, this can be read as: there is a set X and a set Y such that X and Y have the same members and it is possible for something to be a member of Y and not of X.
4. But if X and Y have the same members, then they are identical; and then nothing could belong to "one of them" without also belonging to "the other".
5. Hence things in predicate position do not denote sets of individuals.

The argument is modeled on a brief passage (p.13) in Tim Williamson's latest paper on the Barcan Formula. Williamson there argues against the plural interpretation of second-order quantifiers. On this interpretation, the sentence in (2) can be read as "there are things xx and things yy such that all xx's are yy's and all yy's are xx's and it is possible for something to be one of the yy's but not of the xx's". Williamson objects that if the xx's just are the yy's, then it is not possible for something to belong to "the ones" without also belonging to "the others".

Diodorus and actuality

Let [] and <> express alethic necessity and alethic possibility, let @ stand for 'actually', and L for 'it is unalterable that'. We are going to prove that if something happens, then it is unalterable that it happens.

We need the following principles:

1. A <-> <>@A.
   Something is the case iff it is possibly actually the case.
2. <>A -> L<>A.
   If something is alethically possible, one cannot make it alethically impossible.
3. L(A -> B) -> (LA -> LB).
   If A -> B and A are both unalterable, then so is B.
4. If A is provable then LA.
   Logical truths are unalterable.

Here is the proof, with a sea battle for illustration.

Names and descriptions in modal logic

In the old days, it was common to exclude individual constants from quantified modal logic in favour of Russellian descriptions. I can see how this works if we have either fixed domains (the same individuals populating all worlds) or possibilist quantifiers. But in such systems individual constants don't cause much trouble anyway. Can one also make the description move in more liberal systems? I don't see how, but I guess I'm just missing something obvious.

Consider a formula "possibly, a is F". We want to replace the name "a" by a description "the A". Does the description get narrow scope ("possibly, the A is F") or wide scope ("the A is possibly F")? Either way, we seem to get the wrong result.

Kripke's (Alleged) Argument for the Necessity of Identity Statements

I have often encountered in articles, talks and classes the following argument for the necessity of true identity statements, always attributed to Kripke:

1) a = b (assumption)
2) a = a
3) a = b (from 1, 2 by Leibniz' Law)

The argument is no good, and I think it is very doubtful that Kripke ever endorsed it.

Slingshooting against the necessity of identity

The set is the empty set if p is false, otherwise it is the set of all numbers. Hence iff either p and q are both false or p and q are both true. So

Metaphysical Modality and S5

Note to self: I sometimes say that metaphysical modality is of S5 type, when I should rather say only that it satisfies the characteristic axiom of S5, Mp -> LMp.

It isn't clear to me that metaphysical modality obeys all the S5 principles because it isn't even clear that it obeys T. One of the problems is what to say about Lp if p contains names of objects which may exist only contingently. The two most obvious proposals are: a) Lp is true iff p holds at all worlds where the named objects exist (in this sense, Hesperus is necessarily Hesperus, even though Hesperus exists contingently); b) Lp is true if p holds at all worlds (in this sense, Hesperus is not necessarily Hesperus, but it is necessary that if Hesperus exists then Hesperus is Hesperus). Either way violates T. On (a), let "F" express the property of coexisting with Hubert Humphrey; then "L(F(Hesperus) & F(Humphrey) -> F(Hesperus)) -> L(F(Hesperus) & F(Humphrey)) -> LF(Hesperus)" is false, even though it's an axiom of T. On (b), "L(Hesperus = Hesperus -> Hesperus = Hesperus)" is false, even though it's a theorem of T.

Fara and Williamson against the Counterpart Theory

A few comments on Counterparts and Actuality by Michael Fara and Timothy Williamson (via Brian, of course).

Fara and Williamson argue that if Quantified Modal Logic is enriched by an "actually" operator, then given some further assumptions there is no correct translation scheme from QML to Counterpart Theory. Here, a correct translation scheme is one that translates theorems of QML into theorems of CT and non-theorems of QML into non-theorems of CT. (theorems of which QML? -- good question; read on.).

Gödel's Ontological Proof

Let K be a class of sets such that whenever x is in K and x is a subset of y, then y is also in K. It follows that if the empty set is in K, then every set is in K. Let's rule this out by stipulating that some set is not in K. Thus every set that is in K is not empty. So instead of saying outright that some set is not empty we can instead say that it is in K, which sounds less controversial but really comes down to the same thing.

I think this is the trick in Gödel's ontological proof of god. His class K is the class of 'positive' properties, where properties are individuated intensionally. Gödel claims 1) that whenever some property Q is necessarily implied by a positive property P, then property Q is also positive (which is just the closure principle above), and 2) that not all properties are positive. On these assumptions saying that a property is positive means saying that it is not empty, that is, not necessarily uninstantiated. Hence when Gödel says that 3) necessary existence is a positive property he in effect says that a necessary being possibly exists, which in turn means that a necessary being actually exists.

The fallacy is to assume that there is any class of ('positive') properties satisfying (1)-(3).