### Possible Worlds and "Cases"

Some philosophers would like to (reductively*) analyze possibility and necessity in terms of possible worlds. As I understand their position, it is something like this:

Analysis of Possibility and Necessity (APN): There is a property P ("the property of being a possible world") and a relation R ("the relation of being true at") such that (i) being possible = being an x such that there is a y such that Py and Rxy and (ii) being necessary = being an x such that for all y, if Py, then Rxy.**

One argument some have offered in favor of (APN) is that treating possibility and necessity as existential and universal quantification over worlds, respectively, allows us to explain the validity of various inferences containing 'necessarily' and 'possibly'.*** The idea, I take it, is that sentences of the form:

(N) Necessarily, S.

are in fact ascriptions of the property of being necessary to the proposition that S:****

(N') That S is necessary.

which, given (APN), is equivalent to:

(N'') That S is such that for all possible worlds w, it is true at w.

Similarly for sentences of the form:

(P) Possibly, S.

Now it is interesting to note that to explain certain inferences containing 'necessarily' and 'possibly', proponents of (APN) are going to have to attribute hidden structure to sentences of English. For instance, consider the following inference:

(1) Necessarily, S.

(2) Therefore, S.

Given what has been said before, proponents of (APN) will claim that (1) is equivalent to:

(1') That S is such that for all possible worlds w, it is true at w.

But then whence the inference from (1) to (2)? Presumably, the proponent of (APN) will have to say that (2) is equivalent to something like this:

(2) Therefore, that S is true at a ("the actual world").

Now I want to draw an analogy between the argument for (APN) that I have been discussing and an argument for a different thesis. Call the (relevant) relation that obtains between the propositions that P and that Q iff if P, then Q being conditioned on. Now consider the following thesis:

Analysis of Being Conditioned On (ABC): There is a property P ("the property of being a case") and a relation R ("the relation of being true in") such that being conditioned on = being an x and a y such that for all z such that Pz and Rxz, Ryz.

The idea behind (ABC) is that a (material) conditional is true if and only if (the proposition expressed by) its consequent is true in all cases in which (the proposition expressed by) its consequent is true. Just as 'is a possible world' and 'is true at' are technical terms used to express whichever property and relation appear in the correct analysis of being necessary and being possible, so too 'is a case' and 'is true in' are technical terms used to express whichever property and relation appear in the correct analysis of being conditioned on.

Notice that just as (APN) allowed us to "explain" the validity of inferences containing 'necessarily' and 'possibly', so too (ABC) allows us to "explain" the validity of inferences containing the material conditional. For instance, hypothetical syllogism:

(1) If P, then Q.

(2) If Q, then R.

(3) Therefore, if P, then R.

HS can be explained by the proponent of (ABC) by treating (1), (2), and (3) as, respectively:

(1) For all cases x such that that P is true in x, that Q is true in x.

(2) For all cases x such that that Q is true in x, that R is true in x.

(3) Therefore, for all cases x such that that P is true in x, that R is true in x.

Also notice that a proponent of (ABC) will also need to attribute hidden structure in sentences of English to explain the validity of certain inferences, such as modus ponens:

(1) If P, then Q.

(2) P

(3) Therefore, Q.

In particular, they will need to postulate that (2) and (3) contain a case constant referring to the "real case".

It seems, then that the argument from the explanation of the validity of inferences for (APN) has an analogue: an argument from the explanation of the validity of inferences for (ABC). However, I take it that many of those who find the first argument plausible will not find the second argument plausible. But why? The cases seem relevantly similar. Via translation, we can explain the validity of inferences involving 'possibly' and 'necessarily' using possible worlds, and we can do the same for the validity of inferences involving the material conditional using cases. In addition, both views seem to involve positing hidden structure to English sentences. So what gives?

[One potential answer is that restricted quantification of the sort used in the statement of (ABC) is usually explained in terms of unresticted quantification and the material conditional. Hence, some may take (ABC) to be circular. However, restricted quantification needn't be construed in this way. Inference rules for restricted quantification can be introduced. So can a semantics. Thus, we are not forced to explain restricted quantification partly in terms of unrestricted quantification.]

* I include 'reductively' here because some philosophers make a distinction between reductive and non-reductive analyses. However, I must admit that I don't know what a non-reductive analysis is supposed to be.

** We could introduce an accessibility relation into the analysis as well without affecting the point I am trying to make.

*** I take it that only a "reductive" analysis would explain the validity of these inferences.

I got the idea for this post from teaching "baby logic" to undergraduates. They seem to grasp why certain inferences using the material conditional are valid/invalid best when those conditionals are translated into talk of cases and Venn diagrams are used.

Analysis of Possibility and Necessity (APN): There is a property P ("the property of being a possible world") and a relation R ("the relation of being true at") such that (i) being possible = being an x such that there is a y such that Py and Rxy and (ii) being necessary = being an x such that for all y, if Py, then Rxy.**

One argument some have offered in favor of (APN) is that treating possibility and necessity as existential and universal quantification over worlds, respectively, allows us to explain the validity of various inferences containing 'necessarily' and 'possibly'.*** The idea, I take it, is that sentences of the form:

(N) Necessarily, S.

are in fact ascriptions of the property of being necessary to the proposition that S:****

(N') That S is necessary.

which, given (APN), is equivalent to:

(N'') That S is such that for all possible worlds w, it is true at w.

Similarly for sentences of the form:

(P) Possibly, S.

Now it is interesting to note that to explain certain inferences containing 'necessarily' and 'possibly', proponents of (APN) are going to have to attribute hidden structure to sentences of English. For instance, consider the following inference:

(1) Necessarily, S.

(2) Therefore, S.

Given what has been said before, proponents of (APN) will claim that (1) is equivalent to:

(1') That S is such that for all possible worlds w, it is true at w.

But then whence the inference from (1) to (2)? Presumably, the proponent of (APN) will have to say that (2) is equivalent to something like this:

(2) Therefore, that S is true at a ("the actual world").

Now I want to draw an analogy between the argument for (APN) that I have been discussing and an argument for a different thesis. Call the (relevant) relation that obtains between the propositions that P and that Q iff if P, then Q being conditioned on. Now consider the following thesis:

Analysis of Being Conditioned On (ABC): There is a property P ("the property of being a case") and a relation R ("the relation of being true in") such that being conditioned on = being an x and a y such that for all z such that Pz and Rxz, Ryz.

The idea behind (ABC) is that a (material) conditional is true if and only if (the proposition expressed by) its consequent is true in all cases in which (the proposition expressed by) its consequent is true. Just as 'is a possible world' and 'is true at' are technical terms used to express whichever property and relation appear in the correct analysis of being necessary and being possible, so too 'is a case' and 'is true in' are technical terms used to express whichever property and relation appear in the correct analysis of being conditioned on.

Notice that just as (APN) allowed us to "explain" the validity of inferences containing 'necessarily' and 'possibly', so too (ABC) allows us to "explain" the validity of inferences containing the material conditional. For instance, hypothetical syllogism:

(1) If P, then Q.

(2) If Q, then R.

(3) Therefore, if P, then R.

HS can be explained by the proponent of (ABC) by treating (1), (2), and (3) as, respectively:

(1) For all cases x such that that P is true in x, that Q is true in x.

(2) For all cases x such that that Q is true in x, that R is true in x.

(3) Therefore, for all cases x such that that P is true in x, that R is true in x.

Also notice that a proponent of (ABC) will also need to attribute hidden structure in sentences of English to explain the validity of certain inferences, such as modus ponens:

(1) If P, then Q.

(2) P

(3) Therefore, Q.

In particular, they will need to postulate that (2) and (3) contain a case constant referring to the "real case".

It seems, then that the argument from the explanation of the validity of inferences for (APN) has an analogue: an argument from the explanation of the validity of inferences for (ABC). However, I take it that many of those who find the first argument plausible will not find the second argument plausible. But why? The cases seem relevantly similar. Via translation, we can explain the validity of inferences involving 'possibly' and 'necessarily' using possible worlds, and we can do the same for the validity of inferences involving the material conditional using cases. In addition, both views seem to involve positing hidden structure to English sentences. So what gives?

[One potential answer is that restricted quantification of the sort used in the statement of (ABC) is usually explained in terms of unresticted quantification and the material conditional. Hence, some may take (ABC) to be circular. However, restricted quantification needn't be construed in this way. Inference rules for restricted quantification can be introduced. So can a semantics. Thus, we are not forced to explain restricted quantification partly in terms of unrestricted quantification.]

* I include 'reductively' here because some philosophers make a distinction between reductive and non-reductive analyses. However, I must admit that I don't know what a non-reductive analysis is supposed to be.

** We could introduce an accessibility relation into the analysis as well without affecting the point I am trying to make.

*** I take it that only a "reductive" analysis would explain the validity of these inferences.

I got the idea for this post from teaching "baby logic" to undergraduates. They seem to grasp why certain inferences using the material conditional are valid/invalid best when those conditionals are translated into talk of cases and Venn diagrams are used.

## 1 Comments:

I think there is something more to the argument for a possible worlds interpretation of modality than that it explains why certain seemingly valid inferences are valid. I think it also includes the claim that such an explanation is needed. Roughly, the idea is that (1) it is very mysterious why certain modal inferences seem valid and others do not. Moreover, (2) a possible worlds interpretation is the best way to account for this fact. So, a possible worlds interpretation of modality is true.

Now consider the analogues of (1) and (2) for the (ABC) hypothesis. It seems plausible to reject either of these analogue premises. One might say that it is not a mystery why certain conditional inferences are valid (thus rejecting the analogue of (1)). Alternatively, one could might say that although it the validity of these inferences is mysterious, the cases interpretation of conditionality does not provide the best way to account for our views concerning which inferences are valid and which are not. An alternative explanation involves introducing truth tables and noting that truth of the premises guarantees truth of the conclusion.

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