De Re vs. De Dicto Modality and Referential Opacity

1. Modality and the Failure of Extensionality

Classical logic is celebrated for its extensionality. A key feature of an extensional system is that its connectives are truth-functional. The truth value of a complex sentence like $P\wedge Q$ is determined entirely by the truth values of $P$ and $Q$.

However, consider the modal operators 'it is necessary that' ($\square$) and 'it is possible that' ($◊$). Are they truth-functional?

Let's look at two true statements:

1. "2+2=4"
2. "Grass is green"

Now let's apply the necessity operator:

1. "Necessarily, 2+2=4" (True)
2. "Necessarily, grass is green" (False, as it could have been another colour)

Even though the original statements were both true, applying the necessity operator yielded different truth values. This demonstrates that $\square$ is not truth-functional. Its application depends on something more than the truth value of the proposition it modifies—it depends on the way in which the proposition is true. This is our first sign that we are moving beyond the familiar landscape of extensional logic.

This failure of truth-functionality is just the tip of the iceberg. A more disruptive problem arises when we consider another cornerstone of extensional logic: the substitutivity of identicals.

2. The Problem of Referential Opacity

A fundamental principle of identity, often called Leibniz's Law, states that if two terms refer to the same object, one can be substituted for the other in any sentence without changing that sentence's truth value (salva veritate).

For example, since "Bob Dylan" and "Robert Zimmerman" name the same person, and the sentence "Bob Dylan is a musician" is true, then "Robert Zimmerman is a musician" must also be true. The context "... is a musician" is referentially transparent.

However, some contexts are not so accommodating. These are called referentially opaque contexts.

As the video explains, the problem with modal contexts can be illustrated with a simple, yet devastating, argument structure made famous by the philosopher W.V.O. Quine:

1. Necessarily, 9 is greater than 7. (True)
2. The number of planets = 9. (A contingent astronomical fact, true at the time Quine was writing).
3. Therefore, necessarily, the number of planets is greater than 7. (False)

The inference is valid by the principle of substitutivity, but it leads from true premises to a false conclusion. The context "Necessarily, ..." has made the position occupied by the term '9' referentially opaque. It seems that whether the statement is true depends not just on the object being referred to (the number 9), but on how it is referred to ('9' vs. 'the number of planets').

For a more formal definition of these concepts, the following reading is very helpful.

This referential opacity is not merely a technical puzzle; for Quine, it signals a deep philosophical confusion at the heart of modal logic. Before we examine his critique, we must first look at the standard way philosophers diagnose the ambiguity revealed by opacity.

3. The Distinction: De Dicto vs. De Re

The failure of substitutivity suggests that sentences like "Necessarily, the number of planets is greater than 7" are ambiguous. Philosophers distinguish between two possible readings. The distinction hinges on what the necessity is being ascribed to.

• De Dicto (of the saying): The necessity is predicated of the entire proposition or statement (dictum). The statement as a whole is claimed to be a necessary truth.

  • Reading of the premise: □(9 > 7). This is true. The proposition that 9 is greater than 7 is a necessary truth.
  • Reading of the conclusion: □(The number of planets > 7). This is false. The proposition that the number of planets is greater than 7 is a contingent truth, not a necessary one.

• De Re (of the thing): The necessity is predicated of the object (res) itself. It ascribes a necessary property to the thing, regardless of how it is described.

  • Reading of the premise: The object, 9, has the property of being necessarily greater than 7. This is true.
  • Reading of the conclusion: The object, which happens to be the number of planets, has the property of being necessarily greater than 7. This is also true, because that object is the number 9.

The fallacy in Quine's puzzle occurs if we interpret the first premise de dicto (or de re) and then substitute a co-referring term to draw a conclusion that we read as de dicto. The de re reading, however, seems to license the substitution. On the de re reading, both premise 1 and the conclusion are about the number 9, and they both attribute the same necessary property to it.

The following resources explain this crucial distinction using different helpful examples.

The image below provides a concise summary, mapping the natural language distinction to the formal notation of quantified modal logic, which should be familiar from your studies.

Finally, the Stanford Encyclopedia of Philosophy article on Possible Worlds provides a rich discussion of this topic, connecting it to the concepts of essential properties and transworld identity, which will become central in our upcoming lessons.

4. Quine's Skepticism: Essentialism and the "Metaphysical Jungle"

For you as a mathematician, distinguishing the scope of operators might seem like a standard way to resolve ambiguity. However, for Quine, the de re reading was not a solution but the source of the problem.

Ascribing a property to an object de re—saying that the number 9 is necessarily greater than 7, regardless of how you describe it—is to claim that this is an essential property of that object. Quine was deeply suspicious of this idea, which he associated with outdated Aristotelian metaphysics.

He famously used the following puzzle to show the apparent absurdity of de re modality:

• Consider a person who is both a mathematician and a cyclist.
• As a mathematician, he is necessarily rational.
• As a cyclist, he is necessarily two-legged (bipedal).
• So, is this single individual necessarily rational but only contingently two-legged, or necessarily two-legged but only contingently rational?

It seems that which properties are "essential" depends on how we describe the person. Quine's point is that it makes no sense to talk about an object's essential properties outside of a particular description. To do so, in his view, is to wander into a "metaphysical jungle."

Quine's attack suggests that if modal logic requires a commitment to de re modality and essentialism, then it is philosophically bankrupt. This critique set the stage for a defense of modal logic, which we will explore in the next lessons.

Conclusion

In this lesson, we have taken our first steps into the complex philosophical territory of modal logic. We have seen that the introduction of modal operators for necessity and possibility breaks the principle of extensionality that underpins standard logic. This leads to the problem of referential opacity, where the normally reliable principle of substituting co-referential terms fails.

The key to diagnosing this failure is the distinction between two ways of reading a modal claim:

• De Dicto: Ascribing necessity to a whole proposition ("It is a necessary truth that...").
• De Re: Ascribing a necessary property to an object itself ("This thing has the property of being necessarily...").

While this distinction helps clarify the ambiguity, the de re reading itself became the target of a powerful philosophical critique from Quine, who argued it illegitimately revived a confused metaphysics of "essentialism."

Key Takeaways:

• Modal operators like 'necessarily' ($\square$) are not truth-functional, making modal logic an intensional logic.
• Referential opacity occurs in contexts where the substitution of co-referring terms can change a sentence's truth value. Modal operators create such contexts.
• The de re / de dicto distinction resolves the ambiguity in opaque modal statements by clarifying whether necessity applies to the proposition (dictum) or the object (res).
• De re ascriptions attribute essential properties to objects, a notion Quine found philosophically incoherent and a primary reason to be suspicious of modal logic.

Preview of the Next Lesson:
We have just seen the problem of referential opacity and Quine's skeptical reaction to the de re modality it seems to imply. In the next lesson, we will delve deeper into his objections. Our learning outcome will be to analyse Quine's arguments against the coherence of quantified modal logic, focusing on his critique of essentialism. This will involve a closer look at the logical and philosophical difficulties of "quantifying into" modal contexts.