65 lines
2.6 KiB
Markdown
65 lines
2.6 KiB
Markdown
---
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categories:
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- Logic
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tags: [propositional-logic]
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---
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# Logical consistency
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## Informal definition
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A set of propositions is consistent if and only if **it is possible for all the members of the set to be true at the same time**. A set of propositions is inconsistent if and only if it is not consistent.
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### Demonstration
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The following set of propositions form an inconsistent set:
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1. Anyone who takes astrology seriously is a lunatic.
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2. Alice is my sister and no sister of mine has a lunatic for a husband.
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3. David is Alice's husband and he read's the horoscope column every morning.
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4. Anyone who reads the horoscope column every morning takes astrology seriously.
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The set is inconsistent because not all of them can be true. If (1), (3), (4) are true, (2) cannot be. If (2), (3),(4) are true, (1) cannot be.
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## Formal definition
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> A finite set of propositions $\Gamma$ is truth-functionally consistent if and only if there is at least one truth-assignment in which all propositions of $\Gamma$ are true.
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### Informal expression
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```
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The book is blue or the book is brown
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The book is brown
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```
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### Formal expression
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$$
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\{P \lor Q, Q\}
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$$
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### Truth-table
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$ \{P, Q\} $ form a consistent set because there is at least one assignment when both propositions are true. In fact there are two (corresponding to each disjunct) but one is sufficient.
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```
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P Q P ∨ Q Q
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T T T T *
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T F T F
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F T T T *
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F F F F
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```
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## Derivation
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> In terms of logical derivation, a finite $\Gamma$ of propositions is **inconsistent** in a system of derivation for propositional logic if and only if a proposition of the $P & \sim P$ is derivable from $\Gamma$. It is **consistent** just if this is not the case.
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In other terms, if you can derive a contradiction from the set, the set is logically inconsistent.
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A [contradiction](Logical%20truth%20and%20falsity.md#logical-falsity) contradiction has very important consequences for reasoning because if a set of propositions is inconsistent, every and all other propositions are derivable from that set.
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_A demonstration of the the consequences of deriving a contradiction in a sequence of reasoning._
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Here we want to derive some proposition $Q$. If we can derive a contradiction from its negation as an assumption then, by the [negation elimination](Negation%20Elimination.md) rule, we can assert $Q$. This is why contradictions should be avoided in arguments, they 'prove' everything which, by association, undermines any particular premise you are trying to assert.
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