Supervaluationism and Vagueness

1. The Core Idea: Vagueness as Semantic Indecision

Supervaluationism begins with an intuitive idea: vagueness arises because we haven't made up our minds about where to draw the line. For a vague predicate like 'tall', there are clear positive cases (a 2.5m person) and clear negative cases (a 1.2m person), but we have left it open where the boundary lies for the cases in between.

The supervaluationist formalises this idea using the concept of precisifications (also called 'sharpenings'). A precisification of a vague language is a way of making it completely precise by drawing a sharp boundary for every vague term. An admissible precisification is one that respects the clear cases. For example, any admissible precisification of 'tall' must count the 2.5m person as tall and the 1.2m person as not tall, but it can draw the line anywhere in the borderline region.

From this, we get three key concepts:

• A statement is supertrue if it is true on every admissible precisification.
• A statement is superfalse if it is false on every admissible precisification.
• A statement has a truth-value gap if it is true on some admissible precisifications and false on others. These are the borderline cases.

Let's ground this with a reading.

A crucial feature of supervaluationism—and a major part of its appeal—is that it preserves all the theorems of classical logic. For example, consider the statement "Either the person is tall or the person is not tall," an instance of the Law of Excluded Middle ($P\vee \neg P$). Even if the person is a borderline case, this disjunction is supertrue. Why? Because on any given precisification, the person will be classified as either 'tall' or 'not tall'. So, on any precisification, one of the disjuncts will be true, making the entire disjunction true. Since it's true on every precisification, it is supertrue.

Supervaluationism thus rejects the principle of bivalence (not every statement is true or false) but saves the Law of Excluded Middle (every statement of the form $P\vee \neg P$ is a logical truth).

2. Solving the Sorites Paradox

How does this framework solve the Sorites paradox? Let's revisit the inductive form of the argument:

1. Base Case: A person with 0 hairs is bald. ($\Phi {\alpha }_0$)
2. Inductive Step: For any $n$, if a person with $n$ hairs is bald, then a person with $n+1$ hairs is bald. ($\forall n(\Phi {\alpha }_n\to \Phi {\alpha }_{n+1})$)
3. Conclusion: A person with 500,000 hairs is bald.

The supervaluationist analysis is as follows:

• The base case is supertrue. It's true on all admissible precisifications.
• The conclusion is superfalse. It's false on all admissible precisifications.
• The crucial inductive step is not supertrue. In fact, it's superfalse.

To see why, recall from our last lesson that the negation of the inductive step is $\exists n(\Phi {\alpha }_n\wedge \neg \Phi {\alpha }_{n+1})$. This statement asserts that there is a sharp cutoff.
For a supervaluationist, this existential statement is supertrue. On any single admissible precisification, a sharp line is drawn somewhere—say, at 15,000 hairs. For that precisification, it is true that there exists an $n$ (namely, 15,000) where the person is bald but is not bald with one more hair. Since this holds for every precisification (each will have its own cutoff point), the existential statement $\exists n(\Phi {\alpha }_n\wedge \neg \Phi {\alpha }_{n+1})$ is supertrue.

Since the negation of the inductive step is supertrue, the inductive step itself must be superfalse. The paradox is blocked.

However, supervaluationism achieves this with a subtle move. While it asserts that $\exists n(\Phi {\alpha }_n\wedge \neg \Phi {\alpha }_{n+1})$ is supertrue, it denies that any specific instance of this statement is supertrue. For any particular number of hairs $k$ in the borderline region, the statement "$\Phi {\alpha }_k\wedge \neg \Phi {\alpha }_{k+1}$" is true on some precisifications (those that draw the line at $k$) and false on others. Therefore, it is neither supertrue nor superfalse; it has a truth-value gap.

This allows the supervaluationist to claim there is a sharp boundary without being committed to there being any particular place where the boundary is.

3. Evaluation: Strengths and Weaknesses

Supervaluationism presents an elegant solution, but it is not without significant philosophical costs. A full evaluation requires weighing its strengths against its challenging consequences.

Strengths:

• Preserves Classical Tautologies: As we saw, it validates all theorems of classical logic, like the Law of Excluded Middle. This is a conservative and powerful feature.
• Intuitive Semantic Model: It captures a natural intuition that vagueness is a kind of linguistic or semantic indecision.
• Avoids Epistemic Mystery: Unlike epistemicism, it doesn't postulate unknowable facts about where a sharp boundary lies. The boundary doesn't exist in reality; its "existence" is merely a shared feature of our ways of making things precise.

Weaknesses:

This is where the evaluation becomes more technical. Supervaluationism's biggest problems arise from the logical consequences of identifying truth with supertruth.

1. Failure of Standard Inference Rules

While supervaluationism preserves classical theorems, it does not preserve the validity of some classical rules of inference, particularly those that involve discharging assumptions. Your background in constructing mathematical proofs makes you well-placed to appreciate the gravity of this. Rules like Conditional Proof (the Deduction Theorem), reductio ad absurdum, and Contraposition are fundamental to everyday reasoning.

To see how this failure occurs, let's turn to an excellent paper by Timothy Williamson, which discusses a famous example from Delia Graff Fara.

Let's recap the core point from the reading.

• Argument IIB: X is tall. Y is not tall. Therefore, X is more than a millimetre taller than Y. A supervaluationist considers this argument valid (in a material sense). If the premises are supertrue (X is clearly tall, Y is clearly not tall), the vagueness of 'tall' guarantees a buffer zone wider than 1mm, so the conclusion is also supertrue.
• Argument IIA: X is tall. X is no more than a millimetre taller than Y. Therefore, Y is tall. This is the contrapositive of IIB. But this argument is not valid for a supervaluationist. It is a classic Sorites premise: if X is a borderline case of tallness, the premises could be true while the conclusion is not.

Since IIB is considered valid but its contrapositive IIA is not, the rule of contraposition fails. Similar examples can be constructed to show failures of conditional proof and reductio ad absurdum. For a theory that sought to preserve classical logic, giving up its core inference rules is a very high price to pay.

2. The Problem of Truth and Disquotation

Supervaluationism struggles with the seemingly obvious principle that the statement "'P' is true" should be equivalent to the statement "P". This is known as the disquotational schema, famously discussed by Tarski: 'Snow is white' is true if and only if snow is white.

If we identify 'true' with 'supertrue', this schema fails. Let $P$ be "Harry is bald," where Harry is a borderline case.

• "Harry is bald" ($P$) has a truth-value gap.
• "'Harry is bald' is true" is false, because $P$ is not supertrue.

This creates a mismatch between truth as a predicate of sentences and the sentences themselves.

3. Higher-Order Vagueness

Finally, the theory seems to merely push the problem of vagueness up a level. Supervaluationism relies on a sharp distinction between admissible and inadmissible precisifications. But isn't the notion of what counts as a 'reasonable' or 'admissible' way to draw a line itself vague? This problem is known as higher-order vagueness. If the set of admissible precisifications has fuzzy boundaries, the entire supervaluationist machinery is threatened with a new Sorites paradox at the meta-level.

Conclusion

Today we have evaluated supervaluationism as a sophisticated and influential response to the Sorites paradox. Its elegant solution—preserving classical theorems while allowing for truth-value gaps—comes at a significant cost.

Key Takeaways:

• Supervaluationism treats vagueness as semantic indecision. It defines supertruth as truth on all admissible precisifications.
• It resolves the Sorites paradox by showing that the existential claim that a sharp boundary exists ($\exists n(\Phi {\alpha }_n\wedge \neg \Phi {\alpha }_{n+1})$) is supertrue, while every specific instance of that boundary is not, thus having a truth-value gap.
• The primary strength of supervaluationism is its preservation of all classical logical truths, including the Law of Excluded Middle.
• Its primary weakness is the failure of core classical inference rules like contraposition and conditional proof, making it far more revisionary of logical practice than it first appears.
• Further objections include its break with the disquotational nature of truth and the problem of higher-order vagueness.

Ultimately, the choice between epistemicism and supervaluationism is a choice between different philosophical costs: the epistemicist accepts mysterious unknowable facts, while the supervaluationist must abandon fundamental patterns of logical inference.

Preview of the Next Lesson:

The challenges to classical logic we've seen so far—the Liar and the Sorites—have prompted responses that either posit strange facts (epistemicism) or alter semantics in complex ways (supervaluationism). Our next lesson will analyse the philosophical motivations for relevance logic, a more radical approach that challenges the very definition of logical entailment in classical logic, arguing that premises must be genuinely relevant to the conclusion for an inference to be valid.