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proof_theory [2014/03/28 00:19] nikolaj |
proof_theory [2014/03/28 10:42] nikolaj |
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In the proof theoretic logic, we use a variable which represents a collection of object language formulae, called //context//, and generally denoted $\Gamma$. Moreover, we then must deal with several notion of "and": We possibly have a notion of conjunction in the object language (generally written $\land$), but we also need two conjunctions in the meta language: A gap in the top line denotes a conjunction of premises as introduces in [[Logic]], and a comma between formulae denotes a conjunction which arises from such a gap in a rule of the object language. | In the proof theoretic logic, we use a variable which represents a collection of object language formulae, called //context//, and generally denoted $\Gamma$. Moreover, we then must deal with several notion of "and": We possibly have a notion of conjunction in the object language (generally written $\land$), but we also need two conjunctions in the meta language: A gap in the top line denotes a conjunction of premises as introduces in [[Logic]], and a comma between formulae denotes a conjunction which arises from such a gap in a rule of the object language. | ||
- | There are dozens of proof theories, but in the following we present some rules of a traditional one. The rules are all pretty intuitive, here the simplest ones: | + | There are dozens of proof theories, but in the following we present some rules of a traditional one. The rules are all pretty simple to interpret, proof theoretically |
^ ${\large\frac{}{\phi\vdash\phi}}(identity)$ ^ | ^ ${\large\frac{}{\phi\vdash\phi}}(identity)$ ^ | ||
^ ${\large\frac{\Gamma,\Psi\vdash\vartheta}{\Gamma,\alpha,\Psi\vdash\vartheta}}(weaken)$ ^ | ^ ${\large\frac{\Gamma,\Psi\vdash\vartheta}{\Gamma,\alpha,\Psi\vdash\vartheta}}(weaken)$ ^ | ||
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- | The second two are also easy to interpret, but it's worth pointing out that they give $\Gamma$ the property of a set over a list: | ||
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^ ${\large\frac{\Gamma,\alpha,\alpha,\Psi\vdash\vartheta}{\Gamma,\alpha,\Psi\vdash\vartheta}}(contract)$ ^ | ^ ${\large\frac{\Gamma,\alpha,\alpha,\Psi\vdash\vartheta}{\Gamma,\alpha,\Psi\vdash\vartheta}}(contract)$ ^ | ||
^ ${\large\frac{\Gamma,\alpha,\beta,\Psi\vdash\vartheta}{\Gamma,\beta,\alpha,\Psi\vdash\vartheta}}(exchange)$ ^ | ^ ${\large\frac{\Gamma,\alpha,\beta,\Psi\vdash\vartheta}{\Gamma,\beta,\alpha,\Psi\vdash\vartheta}}(exchange)$ ^ | ||
+ | ^ ${\large\frac{\Gamma\vdash\alpha\hspace{.5cm}\alpha,\Psi\vdash\vartheta}{\Gamma,\Psi\vdash\vartheta}}(cut)$ ^ | ||
- | From a computational point of view, the cut rule has a little more going on: | + | It's worth pointing out that $contract$ and $exchange$ last two rules give $\Gamma$ the property of a set over a list. The rules $contract$ and $weaken$ are modified in Linear logic, which, roughly speaking, considers arguments to be a bounded resource. It has applications in programming. From a computational point of view, the $cut$ rule has a little more going on and is focal point of interesting theorems about proof theory, see e.g. [[http://en.wikipedia.org/wiki/Cut-elimination_theorem|Cut-elimination_theorem]]. |
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- | ^ ${\large\frac{\Gamma\vdash\alpha\hspace{.5cm}\alpha,\Psi\vdash\vartheta}{\Gamma,\Psi\vdash\vartheta}}(cut)$ ^ | + | |
=== Reference === | === Reference === | ||
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[[http://en.wikipedia.org/wiki/Proof_theory|Proof theory]], | [[http://en.wikipedia.org/wiki/Proof_theory|Proof theory]], | ||
[[http://en.wikipedia.org/wiki/Sequent_calculus|Sequent_calculus]], | [[http://en.wikipedia.org/wiki/Sequent_calculus|Sequent_calculus]], | ||
+ | [[http://en.wikipedia.org/wiki/Linear_logic|Linear logic]], | ||
[[http://en.wikipedia.org/wiki/Cut-elimination_theorem|Cut-elimination_theorem]] | [[http://en.wikipedia.org/wiki/Cut-elimination_theorem|Cut-elimination_theorem]] | ||