Quantificationalism

Idea

Consider:

1. Gavin Newsom is the governor of California.
2. Gavin Newsom was not the governor of California in 2018.
3. Gavin Newsom would not have been the governor of California, had the 2021 recall election gone another way.

These sentences are all express truths. Here is a straightforward explanation of why. There are temporary truths: propositions that are true, but not always true. There are also contingent truths: propositions that are true, but not necessarily true. The proposition that Newsom is the governor of California is both a temporary and a contingent truth: it is true, but was not true in 2018, and would not have been true had the 2021 recall election gone another way.

Let Temporalism the view that there are temporary truths and let Modalism be the view that there are contingent truths. Quantificationalism is a view that is to quantification what Temporalism and Modalism are to time and modality respectively. It says that there are quantificationally relative truths: propositions that are true, but not true at or relative to all domains of quantification.

Let me elaborate further. Consider:

4. Every G7 country guarantees universal healthcare.
5. With the exception of the United States, every G7 country guarantees universal healthcare.

Now, 4 is false, because the United States is a G7 country that does not guarantee universal healthcare. However, it is the only exception, so 5 is true. The Quantificationalist explains this by saying that the proposition that every G7 country guarantees universal healthcare is a quantificationally relative falsehood: it is in fact false, but it is true relative to the restricted domain of G7 countries other than the United States.

Regimentation

The statements of Temporalism and Modalism contain quantification over propositions. I do not intend this to be interpreted as first-order quantification over a special sort of abstract objects called propositions. Instead, I use it as a shorthand for higher-order quantification in sentence position.

Thus Temporalism implies the thesis that tense operators are non-trivial. An operator is non-trivial if it changes the truth value of at least one of its arguments. Likewise, Modalism implies the thesis that modal operators are non-trivial.

In a higher-order language equipped with sentential operators and standing respectively for ‘sometimes’ and ‘possibly,’ Temporalism and Modalism can be expressed as follows.

Temporalism

Modalism

See Temporalism and Modalism on these formulations and for comments on how they differ from other views falling under the same label in the literature.

I understand Quantificationalism to be analogous to Temporalism and Modalism as just stated.

these formulations of  \ref{temp} and \ref{mod}. That is, the quantificationalist posits a non-trivial sentential operator, , expressing the notion of being true at, or relative to, some domain of quantification.

Quantificationalism

The logic of can be partially characterized by studying the logics of more specific expressions called domain specifiers, which regiment natural language expressions like ‘with the exception of the United States’ as used in 5 above. Domain specifiers are constructed by means of a syncategorematic expression , which combines with a predicate of any type of the form and a term of any type to produce a term of type .¹ A domain specifier is the syncategorematic expression resulting from filling in the first argument place of .

Intendedly, the meaning of is derived from the meaning of by forcing all quantification over entities of type ”involved” in  to range over a domain determined by the meaning of . For example, if expresses the property is a country other than the U.S., then then can be likened to the English expression ‘with the exception of the U.S.’ Likewise, expresses the property is a prime number, then can be likened to the English expression ‘among prime numbers,’ whose semantic function is to force all quantification within its scope to range over prime numbers only. We can give a model-theoretic representation of the meaning of domain specifiers using quantificational substitution structures.

Let be the dual of . In the correct logic for Quantificationalism, is a master modality for all domain specifiers: is provable precisely when is provable for all domain specifiers .² Importantly, we allow domain specifiers of the form where is a variable: free variables are better than quantifiers at expressing generality in free logic, and Quantificationalists should theorize in a free logic.

Footnotes

1. Officially, I allow any finite sequence of predicates of pairwise distinct types to occur in the first argument place of . This ensures domains of quantification at different types can be changed simultaneously. See Languages for Quantificationalism for more details. ↩

2. Officially: iff is provable for all domain specifiers, where is any sequence of predicates of pairwise distinct types. ↩