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Possible Worlds

Overview:

We discuss some ideas about what "possible worlds" might be, partly to elucidate the character of logical necessity, and partly as an approach to metaphysics and the philosophy of science.
Introduction
These thoughts on possible worlds are intended to help explicate the notion of logical necessity and the boundary between logic and science.
Sets of Sentences
Some philosophers have identified "possible worlds" with "maximal consistent sets of sentences". We briefly consider this before passing on to the more flexible idea of a possible world as an interpretation.
Interpretations
If we identify "possible worlds" with "interpretation of a first order language", then we get a notion of logical necessity which corresponds to that of first order validity.
Standard Interpretations
Restricting the notion of possible world to interpretations which are consistent with the intended meaning of abstract constants yields a more general notion of necessity which fits better with both the analytic and the a priori.
Metaphysics
To talk about the world we need abstract (often mathematical) models. We get these by adopting some convenient but otherwise arbitrary abstract ontology, which we use for building abstract models of the real world. Models of the real world require no new concrete ontology but simply the introduction of new constants which are denote abstract surrogates for the real world.

Introduction:

These thoughts on possible worlds are intended to help explicate the notion of logical necessity and the boundary between logic and science.
A note on Modal Logics
Philosophers commonly discuss possible worlds in the context of those modal logics intended for reasoning about necessity.

This discussion does not specifically address modal logics, since the purpose of the discussion of possible worlds is metaphysical, and is regarded as a preliminary to science. Scientific theories about the real world often depend upon a great deal of mathematics. These theories cannot be articulated in the narrow confines of modal propositional logics and in general do not involve modal concepts. For our present purposes the metatheoretic techniques found in meta-mathematics are more relevant than a discussion of modal logics.

For a completely different discussion of this topic see: Brock Sides on Possible Worlds.

Semantic Connections
There is a close connection between the notion of a possible world and that of an interpretation as used in the semantics (i.e. model theory) of logic. This parallels the connection between necessity (which is defined in terms of the former) and analyticity (or validity), which can be defined in terms of the latter.

One way of thinking of Logical Atomism is as an attempt to answer the question "what must the world be like for it to be possible to talk about it using formal logic?". i.e. an attempt to derive metaphysics from formal semantics.

This discussion of possible worlds is substantially influenced by these connections.

Sets of Sentences:

Some philosophers have identified "possible worlds" with "maximal consistent sets of sentences". We briefly consider this before passing on to the more flexible idea of a possible world as an interpretation.
propositional logics
If you start by thinking about propositional logic then sets of propositions are a natural first thought about possible worlds, since to establish the truth value of a propositional formula you need to know the truth value of the propositional variables. Thinking of a possible world as an interpretation, then a set of propositional variable, those true in the interpretation, is just what you need. This is a set of atomic sentences. Including non atomic sentences is redundant since the truth value of molecular propositions can be calculated once the truth value of the atomic ones is known. Furthermore, allowing molecular propositions introduces the possibilities of inconsistency, and of incompleteness. Hence we arrive at the more complex and not obviously any better idea that a possible world is a maximally consistent set of sentences.
logical atomism
An interesting feature of this kind of syntactic definition of a possible world is its connection with logical atomism. We may caricature logical atomism as having identified a possible world with a set of atomic propositions.

From this point of view we again see that talk of maximal consistent sets of sentences is redundant. Set of atomic sentences suffice not only for propositional but also for predicate logics. (though possibly not for other languages)

However, modern logic uses instead the notion of an intepretation, which is more or less equivalent, and our further discussion adopts this now more conventional approach to semantics.

Interpretations:

If we identify "possible worlds" with "interpretation of a first order language", then we get a notion of logical necessity which corresponds to that of first order validity.
Good Fit with Logical Truth
The primary advantage of this definition is that the concept of necessity which flows from it fits well with the most common idea of what constitutes logical truth.
Ontology
There are really two variants of this, depending on whether you take standard first order logic or a "free" variant. In the latter case the logic is ontologically non-committal to the extent that nothing need exist. In the former case something necessarily exists, but it could be on its own, and there are no other necessary ontological truths.
Logical Constants Special
This notion of possible world inherits the position of first order logic in discriminating semantically between logical constructions (e.g. conjunction, negation, quantification) and constants, predicates or relations. The semantics of the logical constants is fixed by the semantics of first order logic, but not that of other symbols, which are regarded as completely uninterpreted. Contrast the notion of analyticity, in which the semantics of any (or all) constituents of a sentence may contribute to an attribution of analyticity.
mathematics contingent
It follows from the restricted notion of interpretation here that the truths of mathematics are contingent, since there will be possible worlds in which the meaning of the mathematical symbols differs from the one we normally intend, and in some of these worlds "2+2=4" will be false.
Definitional Limitations
More generally the notion of necessity flowing from this definition of possible world will fail to encompass sentences whose truth can be seen to follow from the definitions of the terms which they contain.

Standard Interpretations:

Restricting the notion of possible world to interpretations which are consistent with the intended meaning of abstract constants yields a more general notion of necessity which fits better with both the analytic and the a priori.
Linguistic Aspects of Possible Worlds
Since a possible world must contain all that we need to determine the truth value of sentences in our languages, it is necessary to admit that a possible world must identify for every name in the language the entity in the world which it names. We are now going to permit some constraints on this assignment.
Abstract Ontology
In particular we note that many names are intended to denote abstract entities which are not expected to be constituents of the "real world". While we may not consider the existence of such abstract entities logically necessary, it seems also inappropriate to consider the existence contingent, since not empirical observation can provide evidence for or against their existence.
Standard Interpretations
This is a generalisation of various usages of "standard" in mathematical logic, e.g. the standard interpretation of higher order logic (in which higher types are "complete"), the standard model of arithmetic. The idea is that the semantics of the names in the language is determined by defining a subset of the interpretations of the language which are consistent with the intended meaning of the names. This feature is only intended for abstract entities, and is subject to the constraint of consistency, which requires that there must be at least one standard interpretation. The interpretation of concrete references may not be constrained, but there is some difficulty in clarifying this distinction.
Necessity and Analyticity
By allowing the semantics of "non-logical" constants to be built into the notion of a possible world the resulting concept of necessity is broadened to something closer to the broad notions of analyticity which allow all information about the semantics of the language to come into play. Mathematics now becomes necessary.
Concrete Ontology
There remains some awkwardness in relation to concrete language and ontology which may be fatal for this account of possible world. In order to deal with this problem it is necessary to get into metaphysics, so that our notion of "possible world" can recognise the distinction between abstract and concrete ontology, which need separate treatment if the truths of arithmetic are to be admitted as necessary without also admitting the necessary existence of physical objects with the properties which our language supposes them to have.

Metaphysics:

To talk about the world we need abstract (often mathematical) models. We get these by adopting some convenient but otherwise arbitrary abstract ontology, which we use for building abstract models of the real world. Models of the real world require no new concrete ontology but simply the introduction of new constants which are denote abstract surrogates for the real world.
My own preference is for a notion of possible world which I would guess is pretty much the kind of thing a mathematical physicist would go for.

First let me observe that I don't see any way we can get a handle on the real world other than through various kinds of models. So I'm just going to identify "possible world" with "model of a world", and then a bit of discussion of how to build mathematical models.

I'm going to split the building of a model into two phases, which I will call metaphysics and physics for reasons which I hope will be fairly clear.

In the first phase we decide what kind of thing is a possible world, and then in the second we try and pin down the actual world. There is no single correct answer in either phase, but the ways in which answers can be incorrect differ in the two phases. In the first phase we are essentially doing mathematics, leading to a definition of the concept possible world. If we first adopt and later reject a definition, we do not consider ourselves to have formulated a conjecture which we have later discovered to be false. Instead we proposed a usage, which we later concluded was not such a good idea, just as mathematicians might debate which among alternative definitions of a concept is likely to be most fruitful without admitting that a definition can be true or false.

The metaphysics splits again into two parts, which I will call abstract ontology and concrete ontology. The former is mathematics, the latter metaphysics proper. For our abstract ontology we might take the cumulative hierarchy of sets, or some other adequate ontological basis for mathematics. None of these entities are in themselves considered constituents of a material world, but our metaphysic will identify some of them which are to be interpreted as models of possible worlds.

To make this a little more definite lets consider a classical abstract ontology (just that of standard well-founded set theory), and a metaphysic which I will call Newtonian. The standard set theory permits us to construct the real numbers, and vector spaces. A point mass newtonian model state is a set of point material objects each of which has three spatial coordinate which are real numbers, and three velocity components, also real numbers. The natural laws of a possible point mass newtonian world are modelled by a function which maps a model state to a set of triples of differential coordinates, one for each point mass in the state. These indicate the rate of change of velocity of the point mass.


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