1.2.6. Logical space and the algebra of propositions
Logic is concerned with propositions in the way mathematics is concerned with numbers, but propositions are not numbers. While numbers can be ordered in a linear way, the collection of propositions has a more complex structure. The series of examples of increasing strength we looked at in 1.2.3 did form a single chain, but it should be clear that we could have gone in many different directions to find stronger or weaker claims propositions. For example, The package will arrive next Wednesday is implied by The package will arrive next Wednesday morning but also by The package will arrive next Wednesday afternoon, and neither of the latter sentences implies the other. And The package will arrive next Wednesday implies the sentences The package will arrive next week and The package will arrive on a Wednesday, and the latter two sentences are not ordered one way or the other by implication.
This metaphor of many directions suggests a space of more than one dimension; and, although the structure of a collection of propositions differs not only from the 1-dimensional number line but also from the structure of ordinary 2- or 3-dimensional space, spatial metaphors and diagrams can help in thinking about its structure. These metaphors and can be associated with the term logical space that was introduced by the philosopher Ludwig Wittgenstein (1889-1951).
We will actually use two different sorts of spatial metaphor. One metaphor is the one used in 1.2.2 to depict propositions. In it, possible worlds are the points of logical space, and propositions determine regions in the space by drawing a boundary between the possibilities they rule out and the ones they leave open. But we use a different metaphor when we speak of increasing strength in many different directions. According to this second metaphor, propositions are points in space rather than regions, and possible worlds function in it behind the scenes as something like the dimensions of the space. If we were to apply this idea in any very realistic way, the space would have too many dimensions to be visualized, but in artificially simple cases this sort of space can be depicted by a figure in ordinary 2- or 3-dimensional space.
Fig. 1.2.6-1. The possibilities (the hatched bottom and right halves) that are ruled out by two propositions.
Let’s begin to look further at these ideas by considering an very simple example of the first sort of logical space. Suppose there were only 4 possible worlds. A proposition will either rule out or leave open each of these possibilities. Figure 1.2.6-1 is intended to illustrate two such propositions. Each of these propositions rules out two of the four possibilities (in the hatched areas) and leaves open two others. The proposition expressed by the sentence φ rules out the two possibilities at the bottom of the diagram and the one expressed by ψ rules out the ones at the right. As a result both rule out the possible world in the lower right of the diagram and neither rules out the one in the upper left.
Of course, these are not the only propositions that can be expressed given this range of possibilities. A proposition has two options for each possible world: it may rule it out or leave it open. With 4 possible worlds this means that there are 2 × 2 × 2 × 2 = 16 propositions in all, and 6 of these rule out exactly two possible worlds.
We can illustrate all 16 of these propositions by using a logical space of the second sort. Figure 1.2.6-2 depicts (in two dimensions) a 3-dimensional figure that is one possible representation of a 4-dimensional cube. It is labeled to suggest what sorts of sentences might express these propositions.
You can imagine that the propositions φ (which appears at the left) and ψ (near the center) are the two propositions depicted in Figure 1.2.6-1.
The levels in the structure correspond to grades of strength, with Absurdity at the bottom ruling out all possible worlds and Tautology at the top ruling out none. A line connects propositions that differ only with respect to one possible world. This world separates the proposition higher in the diagram from the one below it, but the lower proposition implies the one above it. Each of the four propositions immediately above Absurdity then leaves open just one possible world. Lines connecting propositions that differ with respect to this world are parallel (in the 3-dimensional figure, though not in the 2-dimensional perspective image on the screen or page) to the line connecting the proposition to Absurdity. In this sense, the worlds can be thought of as the dimensions on which the content of propositions can vary.
The relation between the two sorts of diagram can be seen by replacing each proposition in Figure 1.2.6-2 by its representation using a diagram of the sort illustrated in Figure 1.2.6-1. Putting the two sorts of illustration together in this way gives us the following picture of the same 16 propositions.
Fig. 1.2.6-3. The propositions generated by 4 possible worlds depicted as regions in one logical space (the repeated rectangle) and as points in another (the overall diagram).
The spacing of the nodes differs between Figures 1.2.6-2 and 1.2.6-3 but the left-to-right order at each level is the same, and the regions associated with φ and ψ are the same as those depicted in Figure 1.2.6-1. Since a sentence that rules out more possibilities makes a stronger claim, the size of the region occupied by the possibilities it rules out can be thought to correspond to the strength of the claim it makes. Notice that the regions ruled out here increase towards the bottom of the diagram and that they are the same in size for all nodes on the same level.
The whole structure of Figure 1.2.6-2 can be seen as a complex diamond formed of four diamonds whose corresponding vertices are linked. A simple diamond is the structure of the 2 × 2 = 4 propositions we would have with only 2 possible worlds. The structure in Figure 1.2.6-2 doubles the number of possible worlds and squares the number of propositions. If we were to double the number of possible worlds again to 8, we would square the number of propositions to get 256. The structure they would form could be obtained by replacing each node in the structure of Figure 1.2.6-2 by a small structure of the same form and replacing each line by a bundle of 16 lines connecting the corresponding nodes.
To get a sense of the structure of the set of propositions for a realistically large set of possible worlds, imagine carrying out this process over and over again. The result will always have an upper and lower limit (⊤ and ⊥) and many different nodes on each of its intermediate levels. As the number of possible worlds increases, the distribution of possible worlds among the various degrees of strength (which is 1, 4, 6, 4, 1 in Figure 1.2.6-2) will more and more closely approximate a bell curve. But the bell shape of the curve will also narrow significantly, and bulk of the propositions will be found in intermediate degrees of strength. In short, as the space of propositions gets closer to a realistic degree of complexity, it departs further and further from a single line with ⊤ at the top and ⊥ at the bottom.