66 lines
3.9 KiB
Markdown
66 lines
3.9 KiB
Markdown
---
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tags:
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- Logic
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- propositional-logic
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- proofs
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---
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When we construct a formal proof in logic we are seeking to show that a certain proposition is **derivable** from other propositions. We use the words *derivation* and *proof* interchangeably.
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>
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> A sentence $P$ is derivable in a system of propositional logic from a finite set of sentences if and only if there is a derivation in this system in which all and only the members of the set are **primary assumptions** and $P$ is the sentence on the last line.
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We express the above symbolically as $\Gamma \vdash P$ . (Be careful not to confuse *derivable* ($\vdash$) from *entails* ($\vDash$).
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Derivability is therefore a property that a proposition can possess relative to a set.
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For instance to demonstrate derivability for:
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$$
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{\sim F \lor D, F, D \supset (G & H)} \vdash G &H
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$$
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We would establish $\sim F \lor D, F, D \supset (G & H)$ as primary assumptions and then, using the derivation rules of the system conclude with $G&H$. Every sentence in the derivation is either a **primary assumption** or an **auxiliary** assumption or justified by the rules of the derivation. An auxiliary assumption is one belonging to a sub-derivation. The primary assumptions belong to the main derivation.
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For any given derivation of the form $\Gamma \vdash P$ there may be a number of ways of demonstrating the derivation (more than one application of the rules governing the system) but only one is sufficient to establish derivability.
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>
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> We will tend to use the terms *derivation* and *proof* interchangeably but we should note that there is a technical distinction in that a **proof is a derivation in which all of the assumptions have been discharged**
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## Constructing proofs
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We place assumptions above derivations and mark them *A* in the right-hand margin. We use a shorthand for the derivation rules and also place these in the right-hand margin.
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We divide assumptions from derivations with a horizontal line. We number each line and use this to refer to the line we are applying the derivation to. Sub-proofs follow this structure recursively.
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This is known as *Fitch notation*
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*Schematically*
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*Applied example*
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## Sub-proofs
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When a sub-proof is terminated, the assumption with which it starts is said to be *discharged*. It's conclusion can then be drawn upon and invoked within the main proof by reference to its start and end line number. However statements within the sub-proof cannot be referred to again from within the main proof, only its result.
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## Derivation rules
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Derivation rules are [syntactic](Syntax%20of%20sentential%20logic.md) rather than semantic. They are applied on the basis of their form rather than on the basis of the truth conditions of the sentences they are applied to.
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>
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> Derivation rules can be applied without having an interpretation of the symbols in mind. A derivation rule tells us that: given a group of symbols with a certain structure, we can write down another group of symbols with a certain structure.
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Each of the main logical connectives has an associated derivation rule. The binary connectives have pairs of rules, one for the introduction of the connective and one for its elimination.
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The main derivation rules:
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* [Negation Introduction](Negation%20Introduction.md)
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* [Negation Elimination](Negation%20Elimination.md)
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* [Conjunction Introduction](Conjunction%20Introduction.md)
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* [Conjunction Elimination](Conjunction%20Elimination.md)
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* [Disjunction Introduction](Disjunction%20Introduction.md)
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* [Disjunction Elimination](Disjunction%20Elimination.md)
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* [Conditional Introduction](Conditional%20Introduction.md)
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* [Disjunction Elimination](Disjunction%20Elimination.md)
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* [Biconditional Introduction](Biconditional%20Introduction.md)
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* [Biconditional Elimination](Biconditional%20Elimination.md)
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