112 lines
3.6 KiB
Markdown
112 lines
3.6 KiB
Markdown
---
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tags:
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- propositional-logic
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- logic
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---
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# Validity and entailment
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### Informal definition
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In order to say whether an argument is 'good' or 'bad' we must have criteria of
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evaluation. in logic there are different criteria of evaluation:
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- **Deductive validity**
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An **argument is deductively valid if and only if it is not possible for the
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premises to be true and the conclusion false**. Linking to consistency: it is
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not possible to consistently assert all of the premises but deny the
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conclusion.
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- **Inductive strength**
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We do not say that inductive arguments have 'validity' because despite
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inductive premises being true, the conclusion may be falsifiable. Therefore we
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say inductive 'strength' rather than 'validity'. An argument is inductively
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strong if and only if the conclusion is probably true given the premises.
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#### Demonstration
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The Socrates demonstration above is an example of deductive validity.
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The following is an example of an argument that is inductively strong:
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```
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99% of deaf persons have no musical talent.
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Beethoven was deaf.
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___________________________________________
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Beethoven had no musical talent.
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```
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The test for a strong inductive argument is not whether the conclusion is true,
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rather it concerns the evidence the premises provide in support of the
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conclusion.
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> In propositional logic we are concerned solely with deductive validity or
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> invalidity.
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### Formal definition
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> An argument is truth-functionally valid if and only if there is no
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> truth-assignment on which all the premises are true and the conclusion is
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> false.
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Linking this to derivation, we say:
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> In a system of derivation in propositional logic, an argument is valid if the
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> conclusion of the argument is derivable within the system of derivation from
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> the set consisting of the premises, and invalid otherwise.
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### Demonstration
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The inference from the set ${P, P \rightarrow Q}$ to $Q$ is valid
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### Truth-table
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| $P$ | $Q$ | $P \rightarrow Q$ | $P$ | $Q$ | Assessment |
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| --- | --- | ----------------- | --- | --- | ---------- |
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| T | T | T | T | T | Valid |
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| T | F | F | T | F | |
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| F | T | T | F | T | |
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## Entailment
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### Informal definition
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Entailment as a concept is almost identical to validity. We say that a
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proposition is entailed by a set of propositions if it is not possible for every
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member of this set to be true and the proposition to be false.
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The difference with validity resides in the fact that the propositions are
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distinguished in terms of whether they are premises or a conclusion. So,
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technically, validity is a subclass of entailment. A case of entailment where we
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distinguish propositions in terms of whether they are premises or conclusions. A
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proposition may be entailed by a given set without that proposition being the
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_conclusion_ of the set and where the set is a syllogism.
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### Formal definition
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> A finite set of sentences $\Gamma$ $\vdash$ $P$ if and only if there is no
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> truth-assignment in which every member of $\Gamma$ is true and $P$ is false.
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#### Informal demonstration
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```
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It is raining.
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If it is raining then the pavement will be wet.
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The pavement is wet.
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```
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#### Formal demonstration
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$$
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\{ P, P\rightarrow Q \} \vdash Q
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$$
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#### Truth-table
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| $P$ | $Q$ | $P \rightarrow Q$ | $P$ | $Q$ | Assessment |
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| --- | --- | ----------------- | --- | --- | ---------- |
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| T | T | T | T | T | Valid |
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| T | F | F | T | F | |
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| F | T | T | F | T | |
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