 To clarify things a bit, we consider, briefly, the logicist view in connection with arbitrary logical systems  \(\mathcal{L}_{X}\).[27]  We obtain a particular logical system by setting \(X\) in the appropriate way. Some examples: If \(X=I\), then we have a system at the level of FOL [following the standard notation from model theory; see e.g. (Ebbinghaus et al. 1984)]. \(\mathcal{L}_{II}\) is second-order logic, and \(\mathcal{L}_{\omega_I\omega}\) is a “small system” of infinitary logic (countably infinite conjunctions and disjunctions are permitted). These logical systems are all extensional, but there are intensional ones as well. For example, we can have logical systems corresponding to those seen in standard propositional modal logic (Chellas 1980). One possibility, familiar to many philosophers, would be propositional KT45, or  \(\mathcal{L}_{KT45}\).[28]  In each case, the system in question includes a relevant alphabet from which well-formed formulae are constructed by way of a formal grammar, a reasoning (or proof) theory, a formal semantics, and at least some meta-theoretical results (soundness, completeness, etc.). Taking off from standard notation, we can thus say that a set of formulas in some particular logical system \(\mathcal{L}_X\), \(\Phi_{\mathcal{L}_X}\), can be used, in conjunction with some reasoning theory, to infer some particular formula \(\phi_{\mathcal{L}_X}\). (The reasoning may be deductive, inductive, abductive, and so on. Logicist AI isn’t in the least restricted to any particular mode of reasoning.) To say that such a situation holds, we write       \[     \Phi_{\mathcal{L}_X} \vdash_{\mathcal{L}_X} \phi_{\mathcal{L}_X}     \]   