14 Quantified Modal Logic
Quantified modal logic supplements the vocabulary of quantificational logic with a modal operator \(\Box\). The simplest quantified modal logic extends the axioms of quantificational logic with substitution instances of \(\textsf{K}\) and the rule of necessitation. We will explain how to construct possible worlds models for the language of quantificational modal logic.
By way of motivation, we briefly look at the expressive resources we gain when we combine quantification and modality.
There are at least two interpretations of the English sentence:
- Someone rich could have been poor.
On the one hand, there is a de dicto interpretation on which we qualify the proposition that somone rich is poor:
1a. Possibly, someone rich is poor.
On the other hand, there is a de re interpretation, which qualifies the way in which some individual person is rich:
1b. Someone who is rich is possibly poor.
Quantificational modal logic allows one to capture that distinction:
\[ \begin{array}{lll} 1a & \Diamond \exists x (Rx \wedge Px) & \text{(de dicto)} \\ 1b & \exists x (Cx \wedge \Diamond Px) & \text{(de re)} \\ \end{array} \]
This distinctions help dispel what might otherwise be thought to be puzzles on the interaction between predication and modality. Furthemore, the framework allows one to make a common distinction between essential and accidental attributes.
One common distinction between a statue made of clay and the portion of clay of which it is made is that while they exemplify a certain shape, the shape is an essential attribute of the statue and an accidental attribute of the clay of which it is made. One way to articulate the distinction makes use of de re modality:
\(F\) is an essential attribute of \(x\) if, and only if, \(\Box Fx\).
\(F\) is an accidental attribute of \(x\) if, and only if, \(Fx \wedge \neg \Box Fx\).
This is, however, just a provisional characterization of the distinction, since it is not well-suited to accommodate the contingent existence of the statue and the clay.
W. V. O. Quine had a more serious objection to the very intelligibility of quantified modal logic. His objection is premised on a certain interpretation of the modality involved in the language of quantified modal logic. Like Carnap and others before him, he took the modality to be linguistic at heart. To claim that necessarily, all bachelors are unmarried is to assert, on their view, that the sentence ‘all bachelors are unmarried’ is analytic or true in virtue of the meaning of the terms involved. What else, they may add, could the necessity involved be?
That interpretation is ill-suited to make sense of quantified modal logic. For compare the truth conditions of the two sentences below:
\(\Box \exists x Fx\)
For Quine, this sentence is true if, and only if, the sentence \(\exists x Fx\) is analytic.
\(\exists x \Box Fx\)
This sentence is true if, and only if, the open formula \(\Box Fx\) is true relative to an assignment of an object to the individual variable \(x\). If \(\Box Fx\) is true relative to an assignment if, and only if, \(Fx\) is analytic relative to that assignment. But what could that mean? Closed sentences may be true or not in virtue of the meaning of the terms involved, but it is difficult to make sense of the truth of an open formula in virtue of the meaning of the terms involved, much less the truth of an open formula relative to an assignment.
14.1 Basic Language
Given an initial list of \(n\)-place predicates \(R^n\) and an infinite stock of variables \(v_1, \dots, v_n, \dots\), we define atomic formula as follows:
\[ \varphi ::= R^n v_1, \cdots, v_n \ | \ v_i = v_j \ \]
Given a set \(AT\) of atomic formulas, we define a formula as follows:
\[ \varphi ::= AT \ | \ \neg \varphi \ | \ (\varphi \to \psi) \ | \ \forall v_i \varphi \ | \ \Box \varphi \]
We now define familiar connectives in terms of \(\neg\), \(\to\), \(\forall\), and \(\Box\):
\[ \begin{array}{lll} \top & := & (p \to p)\\ \bot & := & \neg \top \\ (\varphi \vee \psi) & := & (\neg \varphi \to \psi)\\ (\varphi \wedge \psi) & := & \neg (\varphi \to \neg \psi)\\ (\varphi \leftrightarrow \psi) & := & (\varphi \to \psi) \wedge (\psi \to \varphi)\\ \exists v_i \varphi & := & \neg \forall v_i \neg \varphi \\ \Diamond \varphi & := & \neg \Box \neg \varphi\\ \end{array} \]
The simplest quantified modal logic combines the axioms of quantificational logic with identity with the axioms of the minimal normal modal logic K. That will motivate a simple possible worlds model theory for the language, one that has, however, been controversial. In order to be in a position to discuss the framework, we should review the axioms of quantificational logic with identity.
14.2 The Simplest Modal Quantificational Logic
The axioms for quantificational logic with identity include:
Substitution instances of axioms of propositional logic for the expanded language.
Substitution instances of the axiom of universal instantiation:
\[ \begin{array}{lll} \forall x \ \varphi \to \varphi[y/x] & & (\textsf{UI})\\ \end{array} \]
where \(\varphi[y/x]\) is the formula that results from \(\varphi\) from the uniform substitution of occurrences of the variable \(y\) for every free occurrence of the variable \(x\) in the formula relettering if necessary to make sure that the resulting occurrences of \(y\) remain free after the substitution.
- Substitution instances of Reflexivity and the Indiscernibility of Identicals for the expanded language:
\[ \begin{array}{lll} x = x & & (\textsf{Ref})\\ x = y \to (\varphi \to \varphi[y/x]) & & (\textsf{II})\\ \end{array} \]
Two rules of inference, Modus Ponens and Universal Generalization:
\[ \begin{array}{lll} \varphi, \varphi \to \psi / \psi & & (\textsf{MP})\\ \varphi \to \psi / \varphi \to \forall x \ \psi, & \text{$x$ is not free in $\psi$} & (\textsf{UG})\\ \\ \end{array} \]
We are in a position prove some derived rules of inference from them: \[ \begin{array}{lll} \varphi / \forall x \ \varphi & \text{provided $x$ is not free in $\varphi$}\\ \varphi \to \psi / \forall x \ \varphi \to \forall x \ \psi & & \\ \forall x (\varphi \to \psi) \to (\forall x \varphi \to \forall x \psi) & & \\ \end{array} \]
We now adopt all substitution instances of the axioms of quanficational logic with identity for the expanded language, which we supplement with the axioms and rules of inference for the minimal normal modal logic:
\[ \begin{array}{lll} \Box (\varphi \to \psi) \to (\Box \varphi \to \Box \psi) & & (\textsf{K}) \\ \varphi / \Box \varphi & & (\textsf{RN}) \\ \end{array} \]
We now derive three remarkable consequences of the axioms of quantified modal logic with identity.
The Converse Barcan Formula
One consequence of the axioms is the Converse Barcan Formula (\(\textsf{CBF}\)):
\[\tag{$\textsf{CBF}$} \Box \ \forall x \ \varphi \to \forall x \ \Box \ \varphi \] Here is a simple derivation schema:
\[ \begin{array}{lllll} 1 & & \forall x \ \varphi \to \varphi & & \text{UI} \\ 2 & & \Box \ \forall x \ \varphi \to \Box \ \varphi & & \text{RK} \ 1\\ 3 & & ( \Box \ \forall x \ \varphi \to \Box \ \varphi) \to (\Box \ \forall x \ \varphi \to \forall x \ \Box \ \varphi) & & \text{UG} \ 2 \\ 4 & & \Box \ \forall x \ \varphi \to \forall x \ \Box \ \varphi & & \text{MP} \ 2, 4\\ \end{array} \]
CBF seems problematic, since it automatically gives us the necessity of existence.
The Necessity of Existence
Existence is generally understood in terms of quantification and identity: to exist is to be identical with something. But given \(\textsf{CBF}\), we now are in a position to prove that everything necessarily exists.
\[ \begin{array}{lllll} 1 & & \forall x \ \exists y \ x = y & & \text{QL} \\ 2 & & \Box \ \forall x \ \exists y \ x = y & & \text{N} \ 1\\ 3 & & \Box \ \forall x \ \exists y \ x = y \to \forall x \ \Box \ \exists y \ x =y & & \text{CBF} \\ 4 & &\forall x \ \Box \ \exists y \ x =y & & \text{MP} \ 2 3\\ \end{array} \]
Here is another derivation of the theorem: \[ \begin{array}{lllll} 1 & & x= x & & \textsf{Ref} \\ 2 & & x = x \to \exists y \ x = y & & \textsf{EG} \\ 3 & & \Box \ x = x & & \textsf{RN} \ 1\\ 4 & & \Box \ x = x \to \Box \ \exists y \ x = y & & \textsf{RK} \ 2\\ 5 & & \Box \ \exists y \ x = y & & \textsf{MP} \ 3, 4\\ 6 & & \forall x \ \Box \ \exists y \ x = y & & \textsf{UG} \ 5\\ \end{array} \]
Since we can necessitate the conclusion, we find that the simplest quantified modal logic proves that it is necessary that everything necessarily exists. This appears to fly in the face of common sense: do we not exist contingently?
The Barcan Formula
The Barcan Formula (\(\textsf{BF}\)) is not a theorem of the simplest quantified modal logic, as we have described it, but its instances become derivable in the presence of axiom schema \(\textsf{B}\). \[\tag{$\textsf{BF}$} \forall x \ \Box \ \varphi \to \Box \ \forall x \ \varphi \] Given axiom \(\textsf{B}\), we have as a derived rule of inference: \[\tag{$\textsf{DR}$} \Diamond \ \varphi \to \psi/\varphi \to \Box \ \psi \] Here is a justification for the derived rule in KB:
\[ \begin{array}{lllll} 1 & & \Diamond \ \varphi \to \psi & & \\ 2 & & \Box \ \Diamond \ \varphi \to \Box \ \psi & & \textsf{RK} \ 1\\ 3 & & \varphi \to \Box \ \Diamond \ \varphi & & \textsf{B} \\ 4 & & \varphi \to \Box \ \psi & & \textsf{PL} \ 2, 3\\ \end{array} \] We use the derived rule of inference for a derivation of BF in the presence of axiom B: \[ \begin{array}{lllll} 1 & & \forall x \ \Box \varphi \to \Box \varphi & & \textsf{UI} \\ 2 & & \Diamond \ \forall x \ \Box \varphi \to \Diamond \ \Box \ \varphi & & \textsf{RK} \ 1\\ 3 & & \Diamond \ \Box \ \varphi \to \varphi & & \textsf{B}_\Diamond \\ 4 & & \Diamond \ \forall x \ \Box \varphi \to \varphi & & \textsf{PL} \ 2, 3\\ 5 & & (\Diamond \ \forall x \ \Box \varphi \to \varphi) \to (\Diamond \ \forall x \ \Box \varphi \to \forall x \ \varphi) & & \textsf{UG} \\ 6 & & \Diamond \ \forall x \ \Box \varphi \to \forall x \ \varphi & & \textsf{MP} \ 4, 5 \\ 7 & & \forall x \ \Box \varphi \to \Box \ \forall x \ \varphi & & \textsf{DR} \ 6 \\ \end{array} \]
The derivability of BF is philosophically problematic: a physicalist may well accept that everything is necessarily a physical object but nevertheless make room for the possibility that non-physical objects exist. More dramatically, consider the converse of \(\textsf{BF}\):
\[ \Diamond \ \exists x \ \varphi \to \exists x \ \Diamond \ \varphi. \]So, for example, if it is possible for there to be unicorns, then something is possibly a unicorn.
The Necessity of Identity
We are in a position to establish the necessity of identity.
\[ \tag{NI} x= y \to \Box \ x = y \]
Here is a derivation: \[ \begin{array}{lllll} 1 & & x = y \to (\Box \ x=x \to \Box \ x =y) & & \textsf{II} \\ 2 & & x = x & & \textsf{RI} \\ 3 & & \Box \ x=x & & \textsf{RN} \ 2 \\ 4 & & x=y \to \Box \ x =y & & \textsf{PL} \ 1, 3 \\ \end{array} \]
The Necessity of Distinctness
In the presence of axiom B, we can derive the necessity of distinctness: \[ x \neq y \to \Box \ x \neq y \tag{$\textsf{ND}$} \\ \] Here is a simple argument: \[ \begin{array}{lllll} 1 & & \Diamond \ x \neq y \to x \neq y & & \textsf{NI} \\ 2 & & \Box \ \Diamond \ x \neq y \to \Box \ x \neq y & & \textsf{RK} \ 1\\ 3 & & x \neq y \to \Box \ \Diamond x \neq y & & \textsf{B} \\ 4 & & x \neq y \to \Box \ x \neq y & & \textsf{PL} \ 2, 3\\ \end{array} \]
The role of axiom B appears to be crucial for the proof. Otherwise, we can provide a model of contingent distinctness by construing identity as indiscernibility and providing a model in which to discernible objects can become indiscernible at a world to which no other world in which the objects exemplify distinct properties is accessible.