Exploring Logical Space

Philosophers, mathematicians, computer scientists and logicians study the conditions under which things are identical or distinct. This website illustrates the logical space of identity principles, which imply which and which are inconsistent, using the ancient Greek thought experiment of Theseus's ship.

The Athenians kept the ship of Theseus for centuries, replacing its planks one at a time until none of the originals was left: call the result the repaired ship. Suppose that someone kept the old planks and put them back together: call that the rebuilt ship. Which one, if either, is the very ship that Theseus sailed?

Here are six principles about this situation which you might want to endorse.

Not all of these principles can be true together, however: Repaired, Rebuilt, Distinctness and Exactly One are jointly inconsistent, so endorsing any three of them requires rejecting the other. Exploring logical space is working out exactly which combinations can be true, and what each commits you to.

The ship of Theseus is a self-contained illustration of a much larger research project about the contours of logical space. That larger project, which relies heavily on AI computations, uses advanced formal languages to explore logical relationships among different principles and views of identity.

Since each principle can be true or false, the number of different combinations, for making each of the principles true or false individually, is 2, or in other words:

Theorems, checked by AI in Lean, can be used to show which combinations are possible.

Some interesting results

1. Given Leibniz's law, the bundle theory (each thing is identical to the bundle of its properties) and haecceitism (each thing is identical to its own thisness) are inconsistent. Lean ✓

2. Given Leibniz's law, if properties with matching instances are identical, whatever their types, then every truth is necessary. Lean ✓

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