Possible World Semantics and the Complex Mechanism of Reference Fixing

Abstract

Possible world semantics considers not only what an expression actually refers to but also what it might have referred to in counterfactual circumstances. This has proven exceptionally useful both inside and outside philosophy. The way this is achieved is by using intensions. An intension of an expression is a function that assigns to each possible world the reference of the expression in that world. However, the specific intension of terms has been subject to frequent disputes. How is one to determine the intension of a term? Carnap has shown how the intension of a term depends on the type of that term. Two-dimensional semantics has shown how intensions also depend on the actual state of affairs. I will show how, in addition, intensions are no less dependent on metaphysical criteria of identity. Furthermore, I will reveal how these three factors interact to fix the exact intension of a term. In other words, I propose an outline of the overall mechanism by which intensions are being fixed.

1 Introduction

The terms ‘Darwin’ and ‘the originator of evolutionary theory’ designate the same person, and yet, they might not have. For had evolutionary theory been originated by, say, Wallace, ‘the originator of evolutionary theory’ would have designated Wallace, while ‘Darwin’ would still have designated Darwin.¹ This difference in designation in counterfactual circumstances reveals a difference in meaning between the two terms. This is the type of intuition that guides possible world semantics (PWS). PWS has proven extremely fertile—both inside and outside philosophy. Here are some of its applications, listed by Perry.

[PWS] has been applied to a number of intensional phenomena in addition to necessity and possibility, including conditionals, tense and temporal adverbs, obligation and reports of informational and cognitive content. PWS spurred the development of philosophical logic and led to new applications of logic in computer science and artificial intelligence. It revolutionized the study of the semantics of natural languages. PWS has inspired analyses of many concepts of philosophical importance, and the concept of a possible world has been at the heart of important philosophical systems. (2005, 965)

The central tool used by PWS is intension. The intension of an expression is a function that assigns to each possible world, the extension (i.e., the reference) of the term in that world. Thus, considering the example above, the intension of ‘the originator of evolutionary theory’ is a function that assigns to each possible world, the person who originated the theory in that world. Namely, it assigns Darwin to the actual world, Wallace to the possible world in which Wallace was the originator, and no one to a world where the theory came to no one’s mind. The intension of ‘Darwin,’ by contrast, is a function that assigns Darwin to every possible world (i.e., it is a constant function). So, although the two terms have the same extension in the actual world, they nonetheless have different extensions in other possible worlds, viz., they have different intensions. And this, according to PWS, accounts for their difference in meaning. Intensions thus play a key role in PWS and as such have a significant, cross-disciplinary, explanatory force.

However, determining the specific intensions of terms is far from being trivial and has been frequently disputed. Take ‘water’ for example. Consider a possible world in which the watery stuff, i.e., the drinkable colorless stuff that falls from the sky and runs in rivers etc., is constituted not by H₂O molecules but rather by some XYZ molecules.² Would ‘water’ designate that stuff? Would such stuff be included in the intension of ‘water’? Similarly, consider a possible world in which the stuff that is constituted by H₂O molecules is not watery but rather, say, pink solid.³ Would ‘water’ designate such stuff? Would it be included in the intension of ‘water’? Replies vary. In order to understand what is at stake here, it would greatly help to understand how exactly intensions are being fixed, i.e., (a) what factors are at play in fixing the intension of a term and (b) how these factors combine to do that. The aim of this paper is to answer these questions.

In the first stage, I locate the factors upon which intensions depend. This is achieved by drawing lessons about reference in possible worlds from three different debates. Specifically, “Section 2” and “Section 3” briefly illustrate the dependence of the intension of a term on the type of that term (as Carnap has shown) and on the actual state of affairs (as two-dimensionalism has shown), respectively. In “Section 4”, I argue that in addition to these two factors, intension is no less dependent on metaphysical criteria of identity. In the second stage, I move on to reveal the general mechanism, or formula, by which all three factors interact to jointly fix intension. Specifically, “Section 5” reveals this formula with relation to the term ‘water’, whereas “Section 6” generalizes from it, and shows how to use it to calculate the intensions of other terms. “Section 7” summarizes the discussion.

Using any formula for specific calculations requires plugging-in the relevant values of the variables in the formula. So using the intensions formula for the calculation of the specific intension of a term requires plugging in the values pertaining to that given term. The proposed formula can be thus viewed as a meta-semantic theory, which requires an input of specific values for each of its variables, in order to serve as a semantic theory that provides specific intensions.⁴ Sometimes, these values are easily known, and the calculation of intensions is accordingly straightforward. At other times, the values of the variables are much harder to find, thus impeding calculation. But in all cases, the meta-semantic theory provides the general principle according to which intensions are being fixed.

2 The Dependence of Intension on Semantic Rules

Here is one lesson about intensions that we learned from disputes between theories of reference.

How are referring terms related to their objects? How do they designate them? A common way of addressing such questions is by invoking thought experiments that involve counterfactual circumstances. Would a counterfactual watery XYZ stuff deserve to be called ‘water?’ And what about a counterfactual pink solid H₂O stuff?⁵

According to the descriptive theory of reference, ‘water’ has a descriptive content—presumably something like ‘the watery stuff’ (i.e., ‘the drinkable, colorless, …’)—and it designates, with respect to each possible world W, that which fits this descriptive content in W (at least according to the so-called ‘Frege-Russell’ version of this descriptive theory).⁶ So on this view, ‘water’ designates the actual watery H₂O stuff as well as the counterfactual watery XYZ stuff, but not the counterfactual pink solid H₂O stuff (since the latter does not fit the description). Let us call terms that are governed by such a semantic rule (i.e., terms that have a descriptive content and designate, with respect to every possible world, that which fits their descriptive content) ‘descriptive.’

According to an alternative, referential view,⁷ ‘water’ picks out some stuff in the actual world,⁸ and it is then stipulated to designate, with respect to every counterfactual world W, just that same stuff (regardless of any description that this referent may or may not fit in W). This also makes ‘water’ a rigid designator (i.e., a term that designates the same referent in every possible world).^{9 , 10} Thus, says the referentialist, since in the actual world ‘water’ designates H₂O stuff, ‘water’ designates H₂O stuff in every possible world.¹¹ So ‘water’ designates the counterfactual pink solid H₂O stuff but not the counterfactual watery XYZ stuff.¹² Following Kripke, let us call terms that are governed by such a semantic rule (i.e., terms that are stipulated to designated, with respect to every possible world, that which they designate in the actual world) ‘de jure rigid.’¹³

It follows that the descriptive view and the referential view disagree on the semantic rule that governs the term ‘water’: the former takes ‘water’ to be descriptive, whereas the latter takes it to be de jure rigid. This disagreement can teach us something about intensions in general. We have seen that each view entails a different designation of ‘water’ with respect to the counterfactual stuffs in question, as is summarized in Table 1.

Table 1 The possible intensions of ‘water’ relative to different semantic rules that may govern the term

Now, recall that the intension of a term is a function that assigns to each possible world, the referent of the term in that world. So, the top row in Table 1 in fact represents the intension of ‘water’ in case the term ‘water’ is descriptive, whereas the bottom row represents the intension of ‘water’ in case ‘water’ is de jure rigid.

The first lesson we can draw then is that the intension of at least some terms depends on the semantic rule that governs these term (e.g., on whether these terms are descriptive of referential).

3 The Dependence of Intension on the Actual Referent of a Term

We can also learn something about intensions from disputes about what the actual state of affairs is like.

Take the famous Newton-Leibniz dispute over the origination of the calculus. Let the term, ‘the actual originator of the calculus’ designate, with respect to each possible world W, the person who originated the calculus in the actual world (i.e., regardless of whether or not that person originated the calculus in W).¹⁴ What particular person would this term designate in different possible worlds?

Surely, that would depend solely on who originated the calculus in the actual world. Specifically, if the actual originator of the calculus was Newton, then the term ‘the actual originator of the calculus’ designates Newton in the actual world and then goes on to designate Newton in all possible worlds, including in those worlds where Leibniz was the originator, as well as in those where neither of them was. If, on the other hand, the actual originator was in fact Leibniz, then ‘the actual originator of the calculus’ designates Leibniz in the actual world and then goes on to designate Leibniz in every possible world, including in those worlds where Newton was the originator, as well as in those where neither of them was.

Again, we wish to see what we can learn about intensions from this dispute. We can see that ‘the actual originator of the calculus’ will have different designations with respect to different possible worlds, depending on who originated the calculus in the actual world. Table 2 summarizes these differences.

Table 2 The possible intensions of ‘the actual originator of the calculus’ relative to different actual referents that this term may have

So the intension of the term ‘the actual originator of the calculus’ depends on whether the actual originator was Newton or Leibniz. Our second lesson is thus that the intension of at least some terms depends on the actual state of affairs.¹⁵

Note that this does not apply to simple (non-actualized) definite descriptions, e.g., ‘the originator of the calculus.’ This term simply designates, with respect to every possible world W, the person who originated the calculus in W, i.e., regardless of the state of the actual world, and in particular, of who originated the calculus in the actual world.

4 The Dependence of Intension on Criteria of Identity

Finally, we can also learn something about intensions from metaphysical disputes about identity.

Which of an object’s properties are essential to it and which are not? A common way of addressing this issue is by invoking counterfactual circumstances that involve changes to the object’s properties. Consider a clay statue of David, or, put more neutrally, an object that is made of clay and has the shape of David. Call it, ‘David.’¹⁶ What if we were to smash David into a ball? Would that object be David still? Would it deserve to continue to be called ‘David?’ Alternatively, what if we were to gradually replace all of David’s parts, bit by bit, until it was entirely made of, say, bronze? Would that object be David? Would it deserve to keep its name?¹⁷

According to a materialist view, an object remains the same iff it keeps its material constitution.¹⁸ So according to this view, David would survive being smashed into a ball (as such a manipulation would involve no change to its material constitution; both the clay statue and the clay ball are made of the very same lump of clay). However, David would not survive a complete part replacement (as this would involve a complete change of its material constitution, namely, from clay to bronze). Hence, on this view, the term ‘David’ would continue to designate the counterfactual clay ball but not the counterfactual bronze statue. Let us call such criteria of identity over time, ‘material.’

According to an alternative view, an object remains the same iff it keeps some of its manifest properties.¹⁹ So according to this view, David would survive a complete part replacement (as such a manipulation would involve no change to its manifest form; the clay statue continues to have the exact same form, namely of a statue, throughout the process of turning into a bronze statue). However, David would not survive being smashed into a ball (as this would involve a complete change of its manifest form, namely, from a statue to a ball). Hence, according to this view, the term ‘David’ would continue to designate the counterfactual bronze statue but not the counterfactual clay ball. Let us call such criteria of identity over time, ‘manifest.’²⁰

Note that both views assume that criteria of identity are an objective feature of reality and cannot be stipulated at will. In particular, it is assumed that the question of whether a certain object is identical to David admits of an objective answer, regardless of what the speaker that uses the term ‘David’ may believe.

The two views disagree on the criteria of identity over time that apply to David: The former takes these criteria to be material, whereas the latter takes them to be manifest. Of course, there are other views on offer regarding criteria of identity over time.²¹ Yet, it is enough to consider these two views to realize that this metaphysical debate can also teach us something about intensions. For we can see that each view entails a different designation of ‘David’ with respect to the counterfactual objects in question, as is described in Table 3.

Table 3 The possible intensions of ‘David’ relative to different criteria of identity that may in fact obtain in the world

So, the intension of the term ‘David’ depends on whether the criteria of identity that in fact obtain in the world are material or manifest. Thus, the lesson we can draw from this debate is that the intension of at least some terms depends on criteria of identity.

Overall then, we have reached three conclusions: (a) the intension of at least some terms depends on the semantic rule that governs these terms (e.g., on whether a term is descriptive or referential), (b) the intension of at least some terms depends on the actual state of affairs (e.g., on whether the actual inventor of the calculus is Newton or Leibniz), and (c) the intension of at least some terms depends on the criteria of identity that in fact obtain in the world (e.g., on whether the relevant criteria are material or manifest).

Of the various versions of possible world semantics, standard possible world semantics (also known as ‘one-dimensional semantics’) only incorporates the first conclusion; whereas, the more elaborate, two-dimensional semantics also incorporates the second conclusion (i.e., in the sense that different actual states of affairs will yield different intensions).²² However, all versions of possible world semantics have thus far failed to acknowledge the third conclusion, namely, the dependence of intension on criteria of identity. So, this conclusion would be our first contribution to possible world semantics.²³

In what comes next, we shall proceed to see how the intension of a term, in this case ‘water’, in fact depends on all three factors. Moreover, in the course of discussion, we shall also reveal the mechanism by which these three factors interact to jointly determine the intension of ‘water.’ This mechanism will be then generalized and will make the final contribution of the present discussion.

5 How the Three Factors Interact to Jointly Determine Intension

Consider ‘water’ again. In “Section 2”, we have seen that the intension of ‘water’ depended on the semantic rule that governs this term. Specifically, if ‘water’ is descriptive, then ‘water’ is associated with some description (‘watery stuff’) and designates, with respect to every possible world W, that which fits the description in W. Consequently, in such a case, ‘water’ designates the watery XYZ stuff but not the pink solid H₂O stuff (row 1 in Table 4 below). If, by contrast, the term ‘water’ is de jure rigid, then ‘water’ rigidly designates, with respect to every possible world, the same stuff that it designates in the actual world, namely, the same stuff as the watery H₂O stuff. But what would that stuff be? Is the counterfactual watery XYZ stuff the same stuff as the actual watery H₂O stuff? And what about the counterfactual pink solid H₂O stuff?

Table 4 The possible intensions of ‘water’ relative to: (a) different semantic rules that may govern the term ‘water’, (b) different stuffs that may be the actual referent of ‘water’, and (c) different criteria of identity that may in fact obtain in the world

In our initial presentation of the water case, we simply assumed that being the same as the watery H₂O stuff amounted to being H₂O. However, following the discussion of the case of David above, we now know better. We know that being the same depends on criteria of identity. Specifically, if the criteria of identity that obtain in the world are material, then being the same as the watery H₂O stuff indeed amounts to being composed of H₂O molecules. Consequently, a de jure rigid ‘water’ would designate the counterfactual pink solid H₂O stuff but not the watery XYZ stuff (row 2 in Table 4 below). However, should the criteria of identity that in fact obtain not be material, but rather manifest, then being the same as the watery H₂O stuff simply amounts to being watery. After all, nothing in the way the world looks and behaves can rule out, in principle, the mere possibility that the true criterion of identity for stuffs is indeed manifest, rather than material. Just as, for all we know, being David may not consist in having the same material parts but rather in a continuity of form, so, for all we know, being the same stuff may not consist in sharing the same chemical composition but rather in sharing some non-material qualities. And relative to such manifest criteria of identity, a de jure rigid ‘water’ would designate the counterfactual watery XYZ stuff but not the counterfactual pink solid H₂O stuff (row 3 in Table 4).

It thus turns out that the intension of ‘water’ depends not only on the semantic rule that governs the term but also on criteria of identity.

But now, (in light of our discussion of the originator of the calculus,) suppose that the actual watery stuff—i.e., the stuff that we have been calling ‘water’ all along—turns out not to be composed of H₂O molecules but rather of XYZ molecules. Suppose, that is, that our current chemistry got it wrong, and that the true chemical composition of the actual watery stuff is different from what the theory says. (After all, most past scientific theories have been disproved and replaced, so why assume our current chemistry will fare any better?) In such a case, a de jure rigid ‘water’ would designate, with respect to every counterfactual world, that which is the same as that stuff, namely, as the actual watery XYZ stuff (rather than the watery H₂O stuff).

It follows that, in case the term ‘water’ is de jure rigid, the intension of ‘water’ depends not only on the semantic rule that governs the term and on criteria of identity but also on the actual state of affairs.

And what would the exact intension of ‘water’ be in such a case (i.e., when ‘water’ is de jure rigid and its actual referent is watery XYZ stuff)? Would ‘water’ designate the counterfactual pink solid H₂O stuff? And what about the (now) counterfactual watery H₂O stuff? Again, (just like in the case when the actual referent was watery H₂O,) this would depend on criteria of identity. Specifically, if the criteria that in fact obtain in the world are material, then being the same stuff as the (now) actual watery XYZ stuff would amount to being composed of XYZ molecules, and hence, a de jure rigid ‘water’ would designate neither the pink solid H₂O stuff nor the watery H₂O stuff—for neither is composed of XYZ molecules (row 4 in Table 4). If, by contrast, the criteria of identity that obtain in the world are in fact manifest, then being the same as the watery XYZ stuff amounts to being watery, in which case a de jure rigid ‘water’ would designate the watery H₂O stuff, but not the pink solid H₂O stuff—for the former, but not the latter, is watery (row 5 in Table 4). Table 4 below summarizes all of these options

Table 4 clearly indicates that the intension of ‘water’ depends on all of the following three factors: (a) the semantic rule that governs ‘water’, (b) the actual referent of ‘water’, and (c) the criteria for being identical to this actual referent. Moreover, Table 4 specifies how exactly variations in each of these three factors change the intension of ‘water.’

In fact, in the course of discussion, we have also uncovered the mechanism by which the intension of ‘water’ is fixed. This mechanism can be viewed as a three-stage process. We begin with the semantic rule that governs the term ‘water.’ If ‘water’ is descriptive, then ‘water’ designates, with respect to every possible world W, that which fits the description (‘watery stuff’) in W. If, by contrast, ‘water’ is de jure rigid, then, in the next stage, the actual referent has to be fixed.²⁴ Once the actual referent is fixed, then, in the last stage, criteria for being identical to that actual referent determine the referents of ‘water’ in other counterfactual worlds, thereby determining the intension of ‘water.’

Finally, we can now proceed to our main goal: outlining the general mechanism by which the three factors jointly determine intensions. This can be achieved by generalizing from the case of ‘water.’

6 Generalization

Let P₁ and M₁ be properties of some stuff in the actual world and let T be the term that designates that P₁ + M₁ stuff in the actual world. Let P₂ + M₁ be a counterfactual stuff (i.e., stuff that has the same M-property as the actual stuff but a different P-property) and let P₁ + M₂ be another counterfactual stuff (i.e., stuff that has the same P-property as the actual stuff but a different M-property). Using the case of ‘water’ above as a model, we can apply the same principles to this general case. The result will be the possible intensions of T relative to: (a) the different semantic rules that may govern T, (b) the different referents that may be the actual referent of T, and (c) the different criteria of identity that may in fact pertain to the actual referent of T. Table 5 below specifies these intensions.

Table 5 The possible intensions of some term T relative to: (a) different semantic rules that may govern T, (b) different referents that may be the actual referent of T, and (c) different criteria of identity that may in fact pertain to the actual referent of T

Furthermore, P₁ + M₁ may, alternatively, stand not for some stuff, but rather for some object. For example, P₁ + M₁ may stand for the clay statue David, where P₁ is David’s material constitution and M₁ is its shape.²⁵ Likewise, P₁ + M₁ may stand for Newton, where P₁ is Newton’s body and M₁ is his mind.²⁶ Accordingly, P₂ + M₁ would stand for a counterfactual object that has the properties P₂ and M₁, whereas P₁ + M₂ would stand for another counterfactual object that has the properties P₁ and M₂. Now, if T is the singular term that designates the actual object P₁ + M₁, then Table 5 above simply specifies the possible intensions of that singular term, T.²⁷

Table 5 can thus serve as an outline for a general formula of intension. The intension of a specific term—be it ‘water,’ ‘David,’ or ‘the actual originator of the calculus’—is calculated by interpreting the relevant ‘P’s and ‘M’s as appropriate and by plugging in the specific values of the three variables that pertain to the term in question (viz., choosing the specific row in the table that represents these values).²⁸

True, the challenge of knowing the correct values of the three variables may vary considerably depending on the terms in question. For example, the value of the variable the semantic rule that governs the term is very easy to tell when the term is ‘the actual originator of the calculus’; it is clearly de jure rigid, i.e., this term designates, with respect to every counterfactual world, the same person that it designates in the actual world. However, the correct value of the same variable is much less clear in the case of ‘water.’ By contrast, the value of the variable ‘the actual referent of the term’ is fairly easy to tell with respect to ‘water’; it is extremely likely to be watery H₂O stuff, yet it is much harder to be certain about in the case of ‘the actual originator of the calculus.’

7 Summary

Possible world semantics offers a powerful tool that has been extensively used in many realms of inquiry. The key player in possible world semantics is intension, i.e., a function from worlds to referents. As is well known, intensions are dependent on the semantic rule that governs the term and also on the actual referent of the term. It has been shown that in addition to these two variables, intension is also dependent on metaphysical criteria of identity—a fact that has thus far managed to escape possible world semantic theories. Furthermore, I have revealed the general complex mechanism by which all three factors interact to jointly determine intensions, which provided a type of formula for the calculation of intensions. In other words, the view put forward here provides the general principles by which intensions are being fixed.²⁹