Leibniz’s Compossibility and Compatibility

1200px-NaveGreca

Leibniz believed in discovering a suitable logical calculus of concepts enabling its user to solve any rational question. Assuming that it is done he was in power to sketch the full ontological system – from monads and qualities to the real world.

Thus let some logical calculus of concepts (names?, predicates?) be given. Cn is its connected consequence operator, whereas – for any x – Th(x) is the Cn-theory generated by x.

Leibniz defined modal concepts by the following metalogical conditions:

M(x) :↔ ⊥ ∉ Th(x)

x is possible (its theory is consistent)

L(x) :↔ ⊥ ∈ Th(¬x)

x is necessary (its negation is impossible)

C(x,y) :↔ ⊥ ∉ Cn(Th(x) ∪ Th(y))

x and y are compossible (their common theory is consistent).

Immediately we obtain Leibnizian ”soundness” conditions:

C(x, y) ↔ C(y, x) Compossibility relation is symmetric.

M(x) ↔ C(x, x) Possibility means self-compossibility.

C(x, y) → M(x)∧M(y) Compossibility implies possibility.

When can the above implication be reversed?

Onto\logical construction

Observe that in the framework of combination ontology we have already defined M(x) in a way respecting M(x) ↔ C(x, x).

On the other hand, between MP( , ) and C( , ) there is another relation, more fundamental than compossibility. It is so-called compatibility relation. Indeed, putting

CP(x, y) :↔ MP(x, y) ∧ MP(y, x) – for compatibility, and C(x,y) :↔ M(x) ∧ M(y) ∧ CP(x,y) – for compossibility

we obtain a manageable compossibility relation obeying the above Leibniz’s ”soundness” conditions.

Wholes are combinations of compossible collections, whereas possible worlds are obtained by maximalization of wholes.

Observe that we start with one basic ontological making: MP(x, y) – modality more fundamental than Leibnizian compossibility, for it is definable in two steps. Observe also that the above construction can be done for making impossible and to both basic ontological modalities as well (producing quite Hegelian output in this case!).

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