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Introduction to Logic
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Tools for Thought
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A sentence φ is consistent with a sentence ψ if and only if there is a truth assignment that satisfies both φ and ψ. A sentence ψ is consistent with a set of sentences Δ if and only if there is a truth assignment that satisfies both Δ and ψ.
For example, the sentence (p ∨ q) is consistent with the sentence (¬p ∨ ¬q). However, it is not consistent with (¬p ∧ ¬q).
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As with logical equivalence and logical entailment, we can use the truth table method to determine logical consistency. The following truth table shows all truth assignments for the propositional constants in the examples just mentioned. The third column shows the truth values for the first sentence; the fourth column shows the truth values for the second sentence, and the fifth column shows the truth values for the third sentence. The second and third truth assignments here make (p ∨ q) true and also (¬p ∨ ¬q); hence (p ∨ q) and (¬p ∨ ¬q) are consistent. By contrast, none of the truth assignments that makes (p ∨ q) true makes (¬p ∧ ¬q) true; hence, they are not consistent.
p |
q |
p ∨ q |
¬p ∨ ¬q |
¬p ∧ ¬q |
1 |
1 |
1 |
0 |
0 |
1 |
0 |
1 |
1 |
0 |
0 |
1 |
1 |
1 |
0 |
0 |
0 |
0 |
1 |
1 |
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The distinction between entailment and consistency is a subtle one and deserves some attention. Just because two sentences are consistent does not mean that they are logically equivalent or that either sentence logically entails the other.
Consider the sentences in the previous example. As we have seen, the first sentence and the second sentence are logically consistent, but they are clearly not logically equivalent and neither sentence logically entails the other.
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Conversely, if one sentence logically entails another this does not necessarily mean that the sentences are consistent. This situation occurs when one of the sentences is unsatisfiable. If a sentence is unsatisfiable, there are no truth assignments that satisfy it. So, by definition, every truth assignment that satisfies the sentence (there are none) trivially satisfies the other sentence.
An interesting consequence of this fact is that any unsatisfiable sentence or set of sentences logically entails everything. Weird fact, but it follows directly from our definitions. And it makes clear why we want to avoid unsatisfiable sets of sentences in logical reasoning.
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