|Possible World: entity that can be quantified over. There ist a dispute over the question whether possible worlds exist or are only assumed for purposes of proofs of completeness. See also actual world, modal logic, modal realism, realism, actualism, possibility, possibilia, quantification._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Lexicon of Arguments. |
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Definition Possible World/Bigelow/Pargetter: a possible world e, in which a sentence a is true, is then defined as the maximum consistent set containing a as an element.
Possible world/(s): are then completely described in all details, nothing is unmentioned and no detail is described contradictory.
N.B./Bigelow/Pargetter: the joke is that then every non-theorem in any possible world is wrong. It will be wrong in a maximum consistent extension. There will be a maximum consistent extension in which the sentence is wrong, i.e. a world in which it is wrong.
Theorem: is then a sentence that is true in all possible worlds. And that is what a completeness theorem is supposed to show.
Maximum Consistent Extension/Bigelow/Pargetter: a consistent set of sentences is extended by adding either a or ~a if it does not become inconsistent as a result. The extension is maximally consistent if a (or ~a) was the last sentence that could be added.
((s) there may be many extensions depending on whether an individual is described differently in the sentence added. This provides equivalence classes.
Definition Possible World/Bigelow/Pargetter: is a maximum consistent property that includes all the things and properties of a world.
World/Properties/Theory/Bigelow/Pargetter: how do the previous theories in philosophy history look like, in which one tried to describe the world as an aggregate of properties?
E.g. Wittgenstein, Tractatus
E.g. Carnap, The logical structure of the world. ....
Bigelow/Pargetter: we could consider the space-time points as the last individuals of the world. Each of them can either have a certain property or not.
Then we can construct sets of ordered pairs
> assigned, x >
Democritical World/Terminology/Cresswell/Bigelow/Pargetter: that is how Cresswell called worlds constructed from such points. (Cresswell 1972,1973)
That is roughly the same as Russell's logical atomism. An n-digit predicate, followed by n individuals.
Atomism: we should assume that such atomic sentences are logically independent of each other.
If it is only a question of whether a point is occupied or not, the corresponding sentence set will surely be consistent.
Book: a complete "book of the world" would not be a world, but only a representation.
Properties: then arise from books as follows: instead of the atomic sentences, we form longer sentences from combinations of descriptions of points by ordered pairs. This simply leads to a longer book.
Points: instead of them we could also take waves, or elementary particles.
Properties: instead of the property of a space-time point to be occupied we could also choose properties such as charge, mass, and so on. From these we can make sequences:
_____________Explanation of symbols: Roman numerals
indicate the source, arabic numerals indicate
the page number. The corresponding books
are indicated on the right hand side.
((s)…): Comment by the sender of the contribution.
J. Bigelow, R. Pargetter
Science and Necessity Cambridge 1990