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The Logic of Time: A Model-Theoretic Investigation into the by Johan van Benthem

By Johan van Benthem

The topic of Time has a large highbrow charm throughout diverse dis­ ciplines. This has proven within the number of reactions bought from readers of the 1st variation of the current booklet. Many have reacted to matters raised in its philosophical discussions, whereas a few have even solved the various open technical questions raised within the logical elaboration of the latter. those effects could be recorded less than, at a easier position. within the seven years after the 1st book, there were a few visible more moderen advancements within the logical learn of Time and temporal expressions. so far as Temporal common sense right is anxious, it kind of feels reasonable to assert that those quantity to a rise in assurance and class, instead of additional break-through innovation. in truth, maybe the main major resources of recent job were the utilized parts of Linguistics and laptop technological know-how (including man made Intelligence), the place many fascinating new principles have seemed providing additional demanding situations to temporal common sense. Now, because this publication has a slightly tight composition, it will were tricky to interpolate this new fabric with out endangering intelligibility.

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Additional resources for The Logic of Time: A Model-Theoretic Investigation into the Varieties of Temporal Ontology and Temporal Discourse

Example text

Finally, a related, but deeper (and more famous) example is the theorem in Zeeman (1964) to the effect that the causality-preserving automorphisms in four-dimensional Minkowski Space-Time form precisely the so-called 'Lorentz Group'. The other road leads towards more logical questions concerning definability of invariants. For instance, in Weyl (1963) geometry is described as the study of spatial invariants for certain automorphisms "leaving the basic relations undisturbed" (p. 72). The author then shows that not only these basic relations are invariant, but also "any relation logically defined in terms of these" (p.

What about the standard structure 7L itself? It may be shown that all minimal invariants in 7L are definable. For, if t is any finite sequence of integers, then its orbit will consist of precisely those t' exhibiting the same relational pattern qua distances and relative positions. And this pattern may be described in a single first-order formula. Unfortunately, this method does not work in all cases, as the above counter-example showed: the pattern may be too complex for finite description. This completes the outline of logical invariant theory, as far as it is relevant here.

The universal theory of the SPO trees is axiomatized by TRANS, IRREF, L-LIN. Proof. Like above, any universal non-theorem ({J of this axiom set is falsified in some finite structure satisfying the three axioms. Moreover, POINTS 17 each such structure consists of a finite number of disjoint trees; with starting points t 1 , ... , tk, say. ) Combining all these into one single tree by adding a new starting point t with immediate successors t 1 , ... D. still yields a counter-example for cp. Later on, we will also consider relativistic temporal structures with branchings on both sides.

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