# Propositions as set of possible worlds in FOL

In possible world semantics for propositional calculus, possible worlds are usually taken to be models for propositional formulas (the set of valuations in which a certain formula is true)

In first order logic, is the proposition expressed by a sentence a set of valutations?

For instance: "Fa" expresses the proposition that is identifiable with the set of valuations that make "Fa" true, or the propositin is the set of structures <D,I,v> where D is a domain of possible things, I the interpretation function that assign the extensions of predicate letters and v the valuation function that to every formula of the language assign a truth value considering the extensions of the predicates given by I and the objects in the domain?

In short: in first order logic, are propositions expressed by sentences the set of structures in which the sentences are true?

IN this sense, in the context of FOL, are possible worlds just various structures <D,I,v> ?

• "possible world semantics for propositional calculus" Are you talking about some modal (albeit propositional) logic? Otherwise I'm not sure where the possible worlds enter the picture in propositional calculus.
– Fizz
Apr 18 at 12:01
• In general, it is related to Modal logic, but it has more to do with the notion of intension and what we can identify as a possible world. In a propositional setting, possible worlds are maximally ways a word could be so they are analogous to valuations for every letter of the language. I wanted to know if this could be said in a FOL setting Apr 18 at 12:09
• @Fizz I think the question relates to PW accounts of what propositions are, rather than to modal logic. A popular idea is that a proposition can be identified with the set of PWs in which it is true. So, the proposition "Plato is a philosopher" just is the set of PWs in which Plato (or his counterpart) is a philosopher. The idea has been criticised for not being compatible with rigid designators. Apr 18 at 14:04
• @Bumble: that seems to get to the philosophical/metaphysical talk of "possible worlds" that isn't too clearly related to the mathematical talk, i.e. what's a "possible world" without bringing an accessibility relation to the table. You may want to write your own answer about that, as the whole affair is unclear to me to say much about...
– Fizz
Apr 18 at 14:07
• The whole discussion about plato.stanford.edu/entries/rigid-designators seems fairly confusing to me. Those writing about it never seem to say what it means for another possible world to have "an Obama" or "a Phosphorus" as a designator that makes sense in that world. It's because when we give a simple math def to valuations in possible worlds (even in general frames) we mean the same thing by all those propositions that take values. Not clear to me what "inhabitants of those worlds" may mean something else by "an Obama"... which isn't mathematically well defined as far I can tell.
– Fizz
Apr 18 at 14:18

in first order logic, are propositions expressed by sentences the set of structures in which the sentences are true?

Yes in the sense of model theoretic semantics for FOL, which is the "standard" one, but this is not normally discussed as "possible worlds"; the latter notion usually comes with an accessibility relation between worlds.

There is actually a connection of sorts here between propositional modal logics and FOL in the following sense: some, but not all propositional modal logics can be described in terms of the frames they hold over in terms of the FOL formulas that these frames satisfy.

Somewhat more roundabout, some non-classical propositional logics have modal companions. This e.g. allows translation of intuitionistic propositional logic into S4 propositional modal logic.

More recently, the notion of forcing in set theory has been characterized in/as the S4.2 modal logic. So I guess you could say there are some connections, but they aren't as direct as you seem to see them.

• Ok thanks a lot for the answer. I was wondering if it was possible to define an accessibility relations between worlds were we to consider them as valuations or as <D,I,v> structure. Apr 18 at 12:29
• My 2nd para is discussed in more detail under en.wikipedia.org/wiki/Sahlqvist_formula
– Fizz
Apr 18 at 13:31
• So, even if in modal logic we consider a set of world as something "primitive", we still associate a domain to that world and identify the extensions of the predicates of the language with respect to the domain in that world with function I. For all possible world w, we fix a domain d(w) and the extensions of predicates in that world according to a Model <W,R,D,I>. The function I assign to every world a domain d(w) and the extensions of predicates on d(w). But then we could specify a structure <D,J> such that D=d(w) [the domain is just the one assigned to w]... Apr 19 at 9:08
• ...and J is the function that for every predicate P in the language, J(P)=I(P,w); the function that for every predicate assign to an extension of members in D that is just the extensions of member assigned by the function I that acts in the frame. In this sense, can possible worlds be identified with model theoretic structure even in modal logic contenxts? Apr 19 at 9:10