37 lines
1.1 KiB
Markdown
37 lines
1.1 KiB
Markdown
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---
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tags:
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- propositional-logic
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- logic
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---
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# Theorems and empty sets
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We know that when we construct a
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[derivation](Formal_proofs_in_propositional_logic.md#derivation-rules) we start
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from a set of assumptions and then attempt to reach a proposition that is a
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consequence of the starting assumptions. However it does not always have to be
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the case that the starting set contains members. The set can in fact be empty.
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_Demonstration_
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We see in this example that there is no starting set and thus no primary
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assumptions. Instead we start with nothing other than the proposition we wish to
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derive. The proposition is effectively derived from itself. In these scenarios
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we say that we are constructing a derivation from an **empty set**.
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Propositions which possess this property are called theorems:
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> A proposition $P$ or a system of propositions in propositional logic is a
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> theorem in a system of derivation for that logic if $P$ is derivable from the
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> empty set.
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We represent a theorem as:
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$$
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\vdash P
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$$
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(There is no preceding $\Gamma$ as the set is empty. )
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