Barcan Both Ways
| Abstract | The Barcan formula made its first appearance in [1] as ♦(∃x)Φ 3 (∃x)♦Φ, in the logic S2. This logic, S2, despite interesting properties, is not much seen these days, and the Barcan formula itself has been simplified to ♦(∃x)Φ ⊃ (∃x)♦Φ. The paper [1] was one of the earliest serious investigations of first-order modal logic and, of necessity, it was axiomatic since a first-order modal semantics was still some years away. In the system as formulated, the converse of the Barcan formula was provable, but the Barcan formula itself was not. Consequently the Barcan formula was added as an axiom schema because it enabled an interesting development that otherwise seemed impossible. | |||||||||
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