Wyman wrote:'Possible worlds' makes sense in the context of modal logic, which involves necessity and possibility.
Technically, modal logic isn't
inherently a logic of necessity and possibility, it's just a formal apparatus that can be used to analyse the concepts of necessity and possibility (among other things). When it
is used in this way, it is officially referred to as
alethic logic.
Wyman wrote:The highlighted parts are restatements of my position.
I take it that the highlighted parts to which you are referring are the parts of the text you have marked in bold. Let's look at them:
The deontic analog of the modal axiom (M): OA→A is clearly not appropriate for deontic logic. (Unfortunately, what ought to be is not always the case.)
Well actually, what you have been saying until now is that neither "possibility" or "permissibility" are the same as "actually true":
Wyman wrote:nothing about a proposition's being conceivable implies that the proposition is true.
Wyman wrote:'Possible' and 'Permissible' are not the same as true in fact
So whereas the above extract, that you highlighted in bold, is saying that axioms of the form □A→A are not appropriate for deontic logic, you seem to have been arguing against axioms of the form ◊A→A. Firstly, no one in this thread is trying to claim that ◊A→A is ever appropriate. Secondly, although the above text in bold is not actually a restatement of anything you have so far said, it
is basically a restatement of something that I have already said:
egg3000 wrote:Look, suppose it is impermissible to murder someone from our point of view in the actual world. Does this mean that no one does in fact murder anyone in our actual world? Of course it doesn't, people murder other people all of the time in the actual world, in spite of that fact that murder is not morally permissible.
Given the above quote, I think it is clear that I would
not consider the axiom □A→A to be appropriate for deontic logic, so none of this is particularly striking. On the other hand, what
is interesting is the sentence immediately succeeding the one that you put in bold:
The deontic analog of the modal axiom (M): OA→A is clearly not appropriate for deontic logic. (Unfortunately, what ought to be is not always the case.) However, a basic system D of deontic logic can be constructed by adding the weaker axiom (D) to K.
Why is this interesting? Because system K is a very weak system, whose only axioms are:
1. A→(B→A)
2. (A→(B→C))→((A→B)→(A→C))
3. (~A→~B)→((~A→B)→A)
4. □(A→B)→(□A→□B)
Because it is such a weak system, □A→A is
not a theorem in K. So, what happens when we add the axiom □A→◊A to the axioms of K? The result is a different system often referred to as D. System D is slightly stronger than K, and therefore, for various reasons, arguably more appropriate for deontic logic. However, while system D is stronger than K, it is still too weak to make □A→A a theorem. In other words, if we give □ a deontic interpretation, then "it ought to be the case that P" does
not imply "P" within system D, which is exactly what we want.
I find that this all makes a lot more sense when we consider these systems in terms of model theory rather than proof theory. You may recall that I mentioned the accessibility relation:
egg3000 wrote:A model for Modal Propositional Logic (MPL) consists of: a set W of possible worlds; a two place function f which, to each simple well-formed formula in each possible world, assigns either a value of 1 or 0; and a binary relation R over the set W of possible worlds. We say that some world x is accessible from some world w if and only if Rwx (call R the accessibility relation).
egg3000 wrote:j.) V(□β, w)=1 if and only if: if Rwx then V(β, x)=1
k.) V(◊β, w)=1 if and only if: there is some x in W such that Rwx and V(β, x)=1
Well, as you may know, there are different kinds of binary relations (e.g. some are serial, others are reflexive, symmetric, transitive, and so on), and the accessibility relation is no exception. In system K, there is no requirement on the accessibility relation. In system D, on the other hand, the accessibility relation is serial. In neither case, however, is the accessibility relation reflexive, which means that it is impossible for some world
w to access
itself in these systems. But for a proposition of the form □A to be true at some world
w, it is only necessary that A be true in every world
accessible from
w. Since
w cannot access itself, it can still be the case that '□A' is true in world
w even if 'A' is not itself true in world
w (and the same goes for moral reasoning: it can still be true that something
ought to be the case even if it is not
in fact the case).
Consider the following examples (one is new, one I have used already):
1. Suppose it is impermissible to murder someone from our point of view in the actual world. Does this mean that no one does in fact murder anyone in our actual world? Of course it doesn't, people murder other people all of the time in the actual world, in spite of that fact that murder is not morally permissible. However, suppose that we conceive, in our minds, of an imaginary, morally perfect world, where no one does anything that we actual-worlders consider to be morally impermissible. In this imaginary world, would anyone commit murder? No, because we have just said that murder is impermissible, and no one does anything impermissible in this imaginary world of ours.
2. Suppose that it is permissible for someone to eat vegetables. Does this mean that someone does in fact eat vegetables? No, not necessarily. However, if it is permissible to eat vegetables, then we can still conceive of (or access) some hypothetical, morally perfect world where someone does in fact eat vegetables. Because it is permissible to eat vegetables, no one would be violating any moral obligations by doing so, and hence we can conceive of a morally perfect world where this does in fact happen.
It isn't appropriate to analyse either of these paragraphs using a modal system whose accessibility relation is reflexive (like in S5, for example), but it certainly is appropriate to analyse such reasoning with a modal system like D (whose accessibility relation is merely serial and not reflexive or anything else).