124 lines
3.4 KiB
Markdown
124 lines
3.4 KiB
Markdown
---
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tags: [propositional-logic]
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---
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# Logical truth and falsity
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We say of certain propositions that they are logically true or logically false.
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## Logical falsity
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### Informal definition
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A proposition is logically false if and only if **it is not possible for the
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proposition to be true**. The proposition itself cannot be consistently
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asserted.
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**Demonstration**
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```
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There is a country that is not a country.
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Apples are fruits and apples are not fruits
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```
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Neither proposition can be true because the truth of the first clause is
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contradicted by the second. By the principle of
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[consistency](Logical_consistency.md), it is not
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possible for both clauses to be true at once therefore the proposition, overall
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has the truth value of false.
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```
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It is raining and it is not raining.
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```
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### Formal definition
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> A proposition P is truth-functionally false if and only if P is false on every
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> truth-value assignment
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### Formal expression
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$$ P \land \lnot P $$
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### Truth-table
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| $P$ | $P \land \lnot P$ |
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| --- | ----------------- |
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| T | F |
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| T | F |
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## Logical truth
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### Informal definition
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A proposition is logically true if and only if it is not possible for the
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proposition to be false. The proposition itself cannot be consistently denied.
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**Demonstration**
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```
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A rose is a rose.
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Today is Tuesday unless today is not Tuesday.
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```
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Regardless of any facts obtaining in the world, these propositions cannot be
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false.should be avoided in arguments, they 'prove' everything whi
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As with logically false propositions, logical truth can also apply to compound
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propositions:
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```
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A rose is a rose and a shoe is a shoe
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```
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### Formal definition
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> A proposition P is truth-functionally true if and only if P is true on every
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> truth-value assignment
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$$ P \lor \lnot P$$
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### Truth-table
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| $P$ | $P \lor \lnot P$ |
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| --- | ---------------- |
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| T | T |
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| F | T |
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### Consequences
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The existence of logically false and logically true propositions affects the
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validity and soundness of arguments in which they are used. These are
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technicalities that have philosophically interesting consequences.
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- If an argument contains premises which are logically false than this argument
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will perforce be valid. This is because one cannot consistently assert the
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premises and deny the conclusion which is the definition of
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[validity](Validity_and_entailment.md). However the
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_reason_ why one cannot consistently assert the premises and deny the
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conclusions is because one cannot consistently assert the premises - they
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conflict with each other. Furthermore as the argument contains false premises,
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it cannot be sound.
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```
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(P1) Russia is a country.
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(P2) Russia is not a country
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(P3) All countries have languages.
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____________________________________________
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(C) Russian is a language.
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```
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- Any argument with a logically true conclusion is valid. Because the conclusion
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cannot be consistently denied it follows that we cannot consistently assert
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the premises _and_ deny the conclusion. Whether or not the argument is sound
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remains an open question however. If the premises happen to be true then the
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argument will be sound on the strength of the conclusion being logically true
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but if the premises are false it will be unsound regardless of the truth of
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the conclusion.
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```
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(P1) Horses have legs.
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(P2) Animals with legs can move.
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____________________________________________
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(C) A horse is a horse
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```
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