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On Weakening the Law of Excluded Middle
Published online by Cambridge University Press: 01 September 1966
Extract
Let us use ‘false’ and ‘not true’ (and cognates) in such a way that the latter expression covers the broader territory of the two; in other words, a statement's falsity implies its non-truth but not vice versa. For example, ‘John is ill’ cannot be false without being nontrue; but it can be non-true without being false, since it may not be true when ‘John is not ill’ is also not true, a situation we could describe by saying ‘It is neither the case that John is ill nor the case that John is not ill.’
- Type
- Discussions/Notes
- Information
- Dialogue: Canadian Philosophical Review / Revue canadienne de philosophie , Volume 5 , Issue 2 , September 1966 , pp. 232 - 236
- Copyright
- Copyright © Canadian Philosophical Association 1966
References
1 This is consistent with allowing that for some statements falsity does mutually entail non-truth: e.g., molecular statements, the A-proposition of syllogistic logic under a Boolean interpretation.
2 Strictly, this should read ‘Any statement of the form “p v ∼ p” is true’, or ‘Any substitution-instance of “p v ∼ p” is true’, or ‘The statement-form “p v ∼ p” is a tautology.’
3 Since ∼ C (p v ∼ p), i.e., ∼ Cp · ∼ Cp.
4 Since ∼ C (Cp v ∼ Cp), i.e., ∼ CCp · ∼ C ∼ Cp.
5 I.e., ∼ p, or ∼ Cp, or ∼ CCp, or ∼ CCCp, and so on.
6 Since ∼ Cp.