With the concept of a formal language in place, the corresponding notion of interpretation can be defined as follows:
(I) An interpretation of a formal language L is a partial function I whose domain com- prises all the existential quantifiers and constants of L and which satisfies the following conditions:
(i) To the constant ‘I’, I assigns the identity relation;
(ii) To the quantifier ‘∃’, I assigns a sense of ‘existence’ such that for every term t of L and every entity x: if t denotes x relative to I, then x exists in that sense. This definition is admittedly very liberal, since it does not require that any constant other than ‘I’ be assigned a denotation. However, if a given interpretation I assigns a denotation to some constants, then the second clause, in conjunction with the principles of the previous chapter, requires that not only the entities denoted by those constants, but also any entities that are (relative to I) denoted by terms that can be built up from those constants, exist in the sense that I assigns to the quantifier ‘∃’. Hence, while I may assign very specialized senses of ‘existence’ to other quantifiers, the sense that it assigns to ‘∃’ will nearly always – i.e., as long as I assigns a denotation to any constant at all – have to be a fairly broad one.
Next, we can define the concepts of validity, entailment, and equivalence:
(V) A constant or formula ϕ in a formal language L is valid if and only if, for every interpre- tation I and every variable-assignment g, ϕ denotes, relative to I and g, an obtaining state of affairs.
(En1) If Γ is a class of variables, constants, or formulas in a formal language L, and if ψ is a variable, constant, or formula of that same language L, then Γ entails ψ if and only if, for every interpretation I of L and for every variable-assignment g: if each member of Γ denotes, relative to I and g, a state of affairs, then ψ will likewise denote, relative
to I and g, a state of affairs; and if each member of Γ denotes, relative to I and g, an obtaining state of affairs, then the state of affairs denoted by ψ will also obtain.
(En2) If ϕ and ψ are variables, constants, or formulas in a formal language L, then ϕ entails ψ if and only if the singleton {ϕ} entails ψ.
(Eq) If ϕ and ψ are variables, constants, or formulas in a formal language L, then ϕ and ψ are equivalent if and only if they entail each other.
The general form of these last definitions should not be surprising, and, as far as entail- ment and equivalence are concerned, the usual results of reflexivity and transitivity are easily reproduced.1
The concept of a (logically) necessary state of affairs can now be very naturally defined as follows:
(N) A state of affairs is necessary if and only if it is denoted, relative to some interpretation and variable-assignment, by a valid constant or formula.
The adverb ‘logically’ seems appropriate, given the way in which this definition relies on the concepts of validity and interpretation, and given how these latter concepts have here been defined. For it can be seen from the relevant definitions that a formula is valid only if it contains neither free variables nor any constants other than ‘I’; and, at least from the assumptions we have made so far, it does not follow that any ‘mathematical fact’ – i.e., any fact that would naturally be expressed by a formula using mathematical vocabulary such as ‘∈’ – is denoted by any such formula. Consequently, it does not follow that any mathematical fact is denoted by a valid formula, and so it likewise does not follow that any such fact is (logically) necessary.
Given this definition of ‘necessary’, how should one define the concept of a logically possible state of affairs? It would be tempting to follow the tradition of taking possibility to be the
1The present concept of entailment is somewhat related to Parry’s (1933) notion of analytic implication,
dual of necessity, and to say that a state of affairs is possible just in case its negation fails to be necessary. But this would arguably be a mistake. For instance, it seems evident that the state of affairs (Socrates 6= Socrates), i.e., Socrates’ non-self-identity, should count as impossible. The proposal in question, however, does not bear this out. What it does bear out is rather only that Socrates’ non-self-identity is the negation of a state of affairs denoted by ‘Socrates = Socrates’. Given that this latter formula is not valid, it is therefore not clear – i.e., it does not follow from our assumptions – that Socrates’ non-self-identity is the negation of a necessary state of affairs, and so it does not follow that this state of affairs is impossible. To accommodate the intuition that any state of affairs that is denoted by a formula of the form ‘α 6= α’ is impossible, we will therefore have to adopt a different definition.
To see how to construct such a definition, it will be helpful first to consider, in the light of the above definition of ‘valid’, what it means for a state of affairs to be necessary. By combining (N) and (V), we find that a state of affairs is necessary just in case it is denoted, relative to some interpretation and variable-assignment, by a constant or formula ϕ such that, “for every interpretation I and every variable-assignment g, ϕ denotes, relative to I and g, an obtaining state of affairs”. It should then seem natural to define the concept of a (logically) impossible state of affairs as follows:
(Ip) A state of affairs is impossible if and only if it is denoted, relative to some interpretation and variable-assignment, by a constant or formula ϕ such that, for every interpretation I and every variable-assignment g, ϕ does not denote, relative to I and g, an obtaining state of affairs.
Given that the formula ‘Socrates 6= Socrates’ does not denote an obtaining state of affairs relative to any interpretation and variable-assignment, it is now clear, as it should be, that the state of affairs denoted by that formula is impossible. Further, I will say that a state of affairs is (logically) possible if and only if it is not impossible, and that a state of affairs is (logically) contingent if and only if it is not necessary.
The present way of defining modal concepts is admittedly non-standard in at least three respects. First, it takes the ‘primary bearers’ of necessity and possibility to be states of affairs rather than statements. Second, it does not rely on a notion of possible world (whether abstract or concrete). And third, it does not take possibility to be the dual of necessity, since for a state of affairs to be possible does under the present definition not mean that the negation of that state of affairs fails to be necessary. However, none of these features is unprecedented. Notably, the third feature is also present in Arthur Prior’s (1957) “system Q”. Prior saw it as a virtue of his system that, due to the fact that it does not treat possibility as the dual of necessity, it manages to avoid the implausible conclusion that everything exists necessarily: a conclusion that, in his words, would make “gods of us all” (p. 48).2 The present framework can claim the same virtue – once it is suitably extended. In particular, it will be necessary to formulate a distinctness condition for states of affairs, which will be done in the next section. For the purpose of formulating such a condition, it will be convenient to have at hand the notion of (logical) necessitation, which can be regarded as a ‘metaphysical analogue’ of the notion of entailment:
(Nc) If s and t are states of affairs, then s necessitates t if and only if there are variables, constants, or formulas ϕ and ψ such that ϕ entails ψ and, relative to some interpretation and variable-assignment, ϕ denotes s and ψ denotes t.
Note the trivial consequence that every state of affairs necessitates itself.3
2The thesis that everything exists necessarily has in recent years been defended by Williamson (2002), but
for most others, it seems to have retained its implausibility. See Efird (2010) for a recent proposal to avoid the conclusion in the manner of Prior; also see Fine (1985). Further, Correia (2007) employs an avowedly ‘Priorean’ treatment of modality in constructing an account of essence. (In contrast to Prior, however, Correia does treat necessity and possibility as duals.)
3In The Nature of Necessity, Alvin Plantinga has introduced a rather similar notion, namely, that of
inclusion: “Let us say that a state of affairs S includes a state of affairs S0 if it is not possible (in the broadly
logical sense) that S obtain and S0 fail to obtain” (p. 44f.). I would normally have been happy to adopt
Plantinga’s terminology, but the term ‘necessitation’ seems to suggest itself more naturally, given its usage in English. A notion that is related to the present concept of necessitation, and which has drawn a great deal of attention from contemporary metaphysicians, is the notion of grounding. It would be an interesting question to investigate whether, and if so to what extent, the former notion might help to elucidate the latter. (For an excellent introduction to the notion of grounding, see Correia and Schnieder (2012). A seminal paper is Fine (2001).)