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Full Idea
If we identify logic with first-order logic, and mathematics with the collection of first-order theories, then maybe we can continue to maintain the If-thenist position.
Gist of Idea
Perhaps If-thenism survives in mathematics if we stick to first-order logic
Source
Alan Musgrave (Logicism Revisited [1977], §5)
Book Ref
-: 'British Soc for the Philosophy of Science' [-], p.124
A Reaction
The problem is that If-thenism must rely on rules of inference. That seems to mean that what is needed is Soundness, rather than Completeness. That is, inference by the rules must work properly.
| 10049 | Logical truths may contain non-logical notions, as in 'all men are men' [Musgrave] |
| 10050 | A statement is logically true if it comes out true in all interpretations in all (non-empty) domains [Musgrave] |
| 10060 | Logical positivists adopted an If-thenist version of logicism about numbers [Musgrave] |
| 10058 | No two numbers having the same successor relies on the Axiom of Infinity [Musgrave] |
| 10062 | Formalism seems to exclude all creative, growing mathematics [Musgrave] |
| 10063 | Formalism is a bulwark of logical positivism [Musgrave] |
| 10061 | The If-thenist view only seems to work for the axiomatised portions of mathematics [Musgrave] |
| 10065 | Perhaps If-thenism survives in mathematics if we stick to first-order logic [Musgrave] |