About

On this page are all the possible worlds diagrams for a single proposition and a pair of propositions. These diagrams depict all the possible ways a given proposition can stand in relation to a set of possible worlds, and all the possible ways two given propositions can stand in relation to each other and the set of possible worlds.

These are made after Swartz and Bradley (1979), pp. 49-51, using OpenOffice Draw. The text below is copied from that work although edited to fit this page.

Diagrams

1 proposition

A single proposition (or proposition-set) P, may be true in all possible worlds, just some, or none. There are no other possibilities. If, then, we depict the set of all possible worlds by a single box, it follows that we have need of three and only three basic worlds-diagrams for the modal properties of a proposition (or proposition-set) P. They are:

Diagram 1 in figure above depicts the contingency of a single proposition (or proposition-set) P. The proposition P is contingent because it is true in some possible worlds but false in all the others. Our diagram gives the modal status of P but says nothing about its actual truth-status (i.e., truth-value in the actual world).

Diagram 2 depicts the necessary truth of a single proposition P. The proposition P is necessarily true, since it is true in all possible worlds.

Diagram 3 depicts the necessary falsity of a single proposition P. Here P is necessarily false since it is false in all possible worlds.

2 propositions

In order to depict modal relations between two propositions (or two proposition-sets) P and Q, we need exactly fifteen worlds-diagrams. In these worlds-diagrams no significance is to be attached to the relative sizes of the various segments. For our present purposes all we need attend to is the relative placement of the segments, or as mathematicians might say, to their topology. For the purposes of the present discussion our diagrams need only be qualitative, not quantitative.

Diagrams 1 to 4 depict cases where both propositions are noncontingent. Diagrams 5 to 8 depict cases where one proposition is noncontingent and the other is contingent. The final seven diagrams (9 to 15) depict cases where both propositions are contingent.

Now each of these fifteen diagrams locates two propositions, P and Q, with respect to the set of all possible worlds, and thence with respect to one another, in such a way that we can determine what modal relations one proposition has to the other. How can we do this?

The modal relations we have singled out for consideration so far are those of inconsistency, consistency, implication, and equivalence. Recall, then, how each of these four relations was defined: P is inconsistent with Q if and only if there is no possible world in which both are true; P is consistent with Q if and only if there is a possible world in which both are true; P implies Q if and only if there is no possible world in which P is true and Q is false; and P is equivalent to Q if and only if in each of all possible worlds P has the same truth-value as Q. Recall, further, that our device for depicting a proposition as true in a possible world is to span that world by means of a bracket labeled with a symbol signifying that proposition.

The requisite rules for the interpretation of our worlds-diagrams follow immediately:

Rule 1: P is inconsistent with Q if and only if there does not exist any set of possible worlds which is spanned both by a bracket for P and by a bracket for Q;

Rule 2: P is consistent with Q if and only if there does exist a set of possible worlds which is spanned both by a bracket for P and by a bracket for Q;

Rule 3: P implies Q if and only if there does not exist any set of possible worlds which is spanned by a bracket for P and which is not spanned by a bracket for Q (i.e., if and only if any set of possible worlds spanned by a bracket for P is also spanned by a bracket for Q);¹

Rule 4: P is equivalent to Q if and only if there does not exist any set of possible worlds which is spanned by the bracket for one and which is not spanned by the bracket for the other (i.e., the brackets for P and for Q span precisely the same set of worlds).

It is the addition of these rules of interpretation that gives our worlds-diagram the heuristic value that we earlier claimed for them. By applying them we can prove a large number of logical truths in a simple and straightforward way. Consider some examples:

(i) Diagrams 2, 3, 4, 7, and 8 comprise all the cases in which one or both of the propositions P and Q is necessarily false. In none of these cases is there any set of possible worlds spanned both by a bracket for P and by a bracket for Q. Hence, by Rule 1, we may validly infer that in all of these cases P is inconsistent with Q. In short, if one or both of a pair of propositions is necessarily false then those propositions are inconsistent with one another.

(ii) Diagrams 1, 2, 3, 5, and 6 comprise all the cases in which one or both of the propositions P and Q is necessarily true. In each of these cases, except 2 and 3, there is a set of possible worlds spanned both by a bracket for P and by a bracket for Q. Hence, by Rule 2, we may validly infer that in each of these cases, except 2 and 3, P is consistent with Q. But diagrams 2 and 3 are cases in which one or other of the two propositions is necessarily false. We may conclude, therefore, that a necessarily true proposition is consistent with any proposition whatever except a necessarily false one.

(iii) Diagrams 3, 4, and 8 comprise all the cases in which a proposition P is necessarily false. In none of these cases is there a set of possible worlds spanned by a bracket for P. Hence in none of these cases is there a set of possible worlds which is spanned by a bracket for P and not spanned by a bracket for Q. Hence, by Rule 3, we may validly infer that in each case in which P is necessarily false, P implies Q no matter whether Q is necessarily true (as in 3), necessarily false (as in 4), or contingent (as in 8). By analogous reasoning concerning diagrams 2, 4, and 7 — all the cases in which a proposition Q is necessarily false — we can show that in each case in which Q is necessarily false, Q implies P no matter whether P is necessarily true (as in 2), necessarily false (as in 4), or contingent (as in 7). In short, we may conclude that a necessarily false proposition implies any and every proposition no matter what the modal status of that proposition.

(iv) Diagrams 1, 3, and 6 comprise all the cases in which a proposition Q is necessarily true. In none of these cases is there a set of possible worlds which is not spanned by a bracket for Q. Hence, by Rule 3, we may validly infer that in each case in which Q is necessarily true, Q is implied by a proposition P no matter whether P is necessarily true (as in 1), necessarily false (as in 3), or contingent (as in 6). By analogous reasoning concerning diagrams 1, 2, and 5 — all the cases in which a proposition P is necessarily true — we can show that in each case in which P is necessarily true, P is implied by a proposition Q no matter whether Q is necessarily true (as in 1), necessarily false (as in 2), or contingent (as in 5). In short, we may conclude that a necessarily true proposition is implied by any and every proposition no matter what the modal status of that proposition.

The special heuristic appeal of these worlds-diagrams lies in the fact that, taken together with certain rules for their interpretation, we can literally see immediately the truth of these and of many other propositions about the modal relations which propositions have to one another. The addition of still further definitions and rules of interpretation later in this book will enable us to provide more perspicuous proofs of important logical truths — including some which are not as well-known as those so far mentioned.

Notes