74 lines
2.4 KiB
Markdown
74 lines
2.4 KiB
Markdown
---
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tags: [propositional-logic]
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---
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# Logical indeterminacy
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The vast majority of propositions in natural and formal logical languages are
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**neither
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[logically true](Logical_truth_and_falsity.md#logical-truth)
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or
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[logically false](Logical_truth_and_falsity.md#logical-falsity)**.
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This makes sense because propositions of this form are all either tautologies or
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contradictions and as such do not express information about the state of events
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in the world. We call propositions that are neither logically true or logically
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false, **logically indeterminate** propositions.
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## Informal definition
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A proposition is logically indeterminate if it is neither logically true or
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logically false. This is to say: it can be both [consistently](Consistency.md)
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asserted and consistently denied.
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For example the proposition:
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```
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It is raining.
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```
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May be true or false thus it can it both be asserted and denied quite
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consistently. It is true if it actually is raining and false if it actually is
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not raining. There is no logical contradiction implied by saying it is raining
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when it isn't raining, this assertion is simply false. There is a contradiction
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in saying that both states obtain. Thus the proposition:
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```
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It is raining and it is not raining.
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```
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Cannot be consistently asserted as there is no possibility of the proposition
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being true. It is either raining or it isn't raining. Given the law for
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conjunction, both conjuncts must be true for the proposition as a whole to be
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true. But in the case of this proposition if one conjunct is true, the other
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must be false and vice versa, hence it is not possible for the proposition to be
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true at all. It can _only_ be false.
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Contrariwise the proposition:
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```
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It is raining or it is not raining.
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```
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Cannot be consistently denied as there is no possibility of it being false. It
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is either raining or not raining. Given the law for disjunction, either disjunct
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can be true to make the proposition as a whole true. Given that it is either
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raining or not raining in either scenario, the proposition as a whole will be
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true. Therefore there is no possibility of it being false, it can _only_ be
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true.
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## Formal definition
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> A proposition P is truth-functionally indeterminate if and only if it is
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> neither truth-functionally true or truth-functionally false. should be avoided
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> in arguments, they 'prove' everything whi
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```
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P
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```
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### Truth-table
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| $P$ | $P$ |
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| --- | --- |
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| T | T |
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| F | F |
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