83 lines
2.6 KiB
Markdown
83 lines
2.6 KiB
Markdown
---
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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
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members of the set to be true at the same time**. A set of propositions is
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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
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seriously.
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The set is inconsistent because not all of them can be true. If (1), (3), (4)
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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
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> only if there is at least one truth-assignment in which all propositions of
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> $\Gamma$ 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
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both propositions are true. In fact there are two (corresponding to each
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disjunct) but one is sufficient.
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| $P$ | $Q$ | $ P \lor Q $ | $Q$ |
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| --- | --- | ------------ | --- |
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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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## Derivation
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> In terms of logical derivation, a finite $\Gamma$ of propositions is
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> **inconsistent** in a system of derivation for propositional logic if and only
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> if a proposition of the form $P \& \lnot P$ is derivable from $\Gamma$. It is
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> **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
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logically inconsistent.
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A
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[contradiction](Logical_truth_and_falsity.md#logical-falsity)
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has very important consequences for reasoning because if a set of propositions
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is inconsistent, any other proposition is derivable from it.
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_A demonstration of the the consequences of deriving a contradiction in a
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sequence of reasoning._
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Here we want to derive some proposition $Q$. If we can derive a contradiction
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from its negation as an assumption then, by the
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[negation elimination](Negation_Elimination.md)) rule, we can
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assert $Q$. This is why contradictions should be avoided in arguments, they
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'prove' everything which, by association, undermines any particular premise you
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are trying to assert.
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