By Albert G. Ingalls
Fresh e-book. quantity three HARDCOVER. send swifter FROM TX.
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To ensure this, R must also be symmetric, and Rww' hold whenever Rw'w. We can drop the redundant accessibility relation altogether and say that ◇φ is true in a model of S5 just in case φ is true in some possible world of that model. 44) and M4 that what is sometimes true is always sometimes true. NEC becomes a temporal generalization rule that claims that logical truths are always true. The axiom schema Z1 is a tense version of M1 and the temporal generalization rule TG performs the function of NEC, reflecting the fact that theorems of tense logic are always true.
6. Zeno concludes that motion is impossible. So if space is a continuum then motion is only possible if there are also continuum many times. Unless temporal substantivalists could produce an argument that shows that there are necessarily as many times as places, they would be committed to the view that there are possible worlds whose inhabitants are condemned to immobility for lack of times. This is the one respect in which it might be more natural to think of time as a continuous manifold than it is to think of space in this way, but this does little to lessen the weirdness of metaphysically basic time points.
In particular, this means that ◇φ is true in a model if φ is true in some possible world that is accessible from the actual world of the model, and □φ is true if φ is true in all such worlds. 42) As for propositional logic, the logical properties of ◇ can also be characterized axiomatically, but in this case there is a whole range of different systems one could consider. The necessitation rule NEC asserts that all theorems of our modal logic are themselves necessary. Its theorems are those sentences that are true in all models, without restrictions on R.