| 1. | An efficient way to prove this is to take advantage of bivalence.
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| 2. | This contradicts the Principle of bivalence : every sentence must be either true or false.
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| 3. | Considering this thesis led Aristotle to reject the principle of bivalence for assertions concerning the future.
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| 4. | These are fundamentally consequences of the law of bivalence, which makes all such connectives merely Boolean functions.
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| 5. | The law of bivalence does not hold in intuitionistic logic, only the law of non-contradiction.
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| 6. | In formal logic, the principle of bivalence becomes a property that a semantics may or may not possess.
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| 7. | The argument relies heavily on the principle of bivalence : the idea that any proposition is either true or false.
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| 8. | Under this system, any theorem necessarily dependent on classical logic's principle of bivalence would fail to be valid.
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| 9. | In other words, P and not-P . This violates the law of noncontradiction and, by extension, bivalence.
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| 10. | Aristotle to embrace bivalence for such future contingents; Chrysippus, the Stoic logician, did embrace bivalence for this and all other propositions.
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