PREFACE: WHAT ONE NEEDS TO KNOW xxii
that you do not believe in). Yet we can think of them, fear them, admire them, just as we can existent objects. Indeed, we may not know whether an object to which we have an intentional relation of this kind exists or not. We may even be mistaken about its existential status. The domain of objects comprises, then, both existent and non-existent objects. There is a monadic existence predicate, E, whose extension is exactly the set of existent objects; and the extension of an intentional predicate, such as ‘admire’, is a set of ordered pairs, the first of which exists, and the second of which may or may not. How to understand the notion of existence, is, of course, a thorny issue. For the record, I take it to be to have the potential to enter into causal relations.
We can also quantify over the objects in the domain, whether or not they exist. Thus, if I admire Sherlock Holmes I admire something; and I might want to buy something, only to discover that it does not exist. I will write the particular and universal quantifiers as 𝔊 and 𝔘, respectively. Normally one would write them as ∃ and ∀, but given modern logical pedagogy the temptation to read ∃ as ‘there exists’ is just too strong. Better to change the symbol for the particular quantifier (and let the universal quantifier go along for the ride).Thus, one should read 𝔊xPx as ‘some x is such that Px’ (and 𝔘xPx as ‘all x are such that Px’). It is not to be read as ‘there exists an x such that Px’—nor even as ‘there is an x such that Px’, being (in this sense) and existence coming to the same thing. (To put it in Meinongian terms, some objects have Nichtsein—non-being.) If one wants to say that there exists something that is P, one needs to use the existence predicate explicitly, thus: 𝔊x(Ex ∧ Px). Quantifiers, note, work in the absolutely standard fashion:𝔊xPx is true iff something in the domain of quantification satisfies Px; and 𝔘xPx is true iff everything in the domain of quantification satisfies Px.
So far so good. But more needs to be said about the properties of non-existent objects. Consider the first woman to land on the Moon in the twentieth century. Was this a woman; did they land on theMoon? A natural answer is yes: an object, characterized in a certain way, has those properties it is characterized as having (the Characterization Principle).That way, however, lies triviality, since one can characterize an object in any way one likes. In particular, we can characterize an object, x, by the condition that x = x ∧ A, where A is arbitrary. Given the Characterization Principle, A follows. We must take a different tack.
Worlds are many. Some of them are possible; some of them are impossible. The actual world, @, is one of the possible ones: