The article explores the connection between different views concerning what makes a universal generalization true and the choice between classical and (semi)intuitionistic logic.

The article first articulates two different philosophical conceptions of what explains the truth of a universal generalization, and the motivations behind them. It then offers a formal tool to model these different conceptions. Finally, it proves some theorems that clarify how the two different conceptions relate to classical and intuitionistic logic, respectively. The article combines philosophical insight and technical precision. The final result is illuminating.

At the core of the article lies the distinction between instance-based and generic explanations of the truth of a universal generalization.

It is tempting to think that the reason why a universal generalization is true, if it is, is that each instance of the generalization is true. To use one of Linnebo’s examples, the natural explanation of why it is true that all his students are born on a Monday is that his students are $ a_1, \dots a_n$ and each of them is born on a Monday. This is an example of an instance-based explanation.

Instance-based explanations seem completely plausible in cases like this one. In other cases, though, the explanation of why a generalization is true seems to work differently. The standard proof that for all prime numbers there is a larger prime, for instance, offers an alternative to the instance-based style of explanation. The proof does not run through all the primes and check that for each of them there is a larger one. Rather, the proof is based on the existence of a purely general procedure to find, given any prime, a larger one. This is a case of generic explanation.

Instance-based and generic explanations differ in some important respects. Generic explanations are uniform in a way that instance-based ones are not. The reason why a student is born on a Monday is different for different students. The proof that there are infinitely many primes, on the other hand, provides a uniform method to find, given a prime number, a larger one. The generic explanation is also dispositional: given a prime number, the procedure delivers a larger one. It does not deliver infinitely many primes all at once.

Linnebo reviews different reasons to be interested in generic explanations. One is that instance-based explanations might not always be available. If certain domains of objects are indefinitely extensible, then the truth of a universal generalization about that domain cannot be instance-based, because the instances are never all jointly available. The view that certain domains of discourse are indefinitely extensible is controversial, but not unintelligible. The formal machinery presented in the article helps make sense of it. In any case, it is worth noticing that instance-based generalizations might not be the best explanations even when they are available: explaining why all bachelors are unmarried by running a poll seems like a waste of resources.

The formal model used by Linnebo to make precise sense of the distinction between generic and instance-based generality is a version of truthmaker semantics. The key difference with possible world semantics is that formulas are verified by states which, unlike possible worlds, can be partial in the sense that they can fail to verify or falsify a formula. States are ordered by a parthood relation, $\leq $ , which is meant to represent the relation holding between two states when the latter preserves all the information contained in the former.

The partial order has a top element, , and a bottom element, . is the only inconsistent state in the model, and verifies every formula. is the state where no information at all is available. Any pair of states $ \{s,s^\prime \}$ has a least upper bound, their fusion, denoted by $s \sqcup s^\prime $ . States are incompatible if their fusion is , i.e., if fusing them yields an impossible state. Maximally consistent states are those which are not proper parts of any consistent state.

The semantics specifies how the truthmakers of a formula depend on those of its components. In particular, the clause for the universal quantifier is the that a state verifies ( $\Vdash $ ) a universal generalization if and only if, for any object, when fused with any state that has that object in its domain, the states verifies the instance of the generalization concerned with that object. Readers familiar with Kleene’s realizability semantics will recognize a similar idea. In symbols:

$$\begin{align*}\begin{aligned} s \Vdash \forall x\,\phi(x) &\;\text{iff}\; s \sqcup t \Vdash \phi(a) \text{ for every } t \text{ and every } a \in D(t). \end{aligned} \end{align*}$$

The key result of the article is that the set of formulas verified by in every model coincides with the set of theorems of intuitionistic predicate logic. On the other hand, the set of formulas verified, in every model, by all maximally consistent states coincides with the set of theorems of classical logic.

The correct formulation of the theorem needs to add the qualification free to intuitionistic logic, given that $\exists x\left (x=x\right )$ , for instance, is a theorem of intuitionistic logic which is not verified by , given the plausible assumption that the domain of be empty. A similar qualification should be applied to the part concerning classical logic.

The article further explores how the addition of certain principles expressed in the language of plural logic yields a semi-intuitionistic logic intermediate in strength between intuitionistic and classical logic.