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I'm beginning to read the interesting Introduction to Mathematical Logic, by Detlovs and Podnieks, but I'm having some troubles with a few simple concepts.

In an early paragraph, the following theory is described:

Our second example of a formal theory is only a bit more serious. It was proposed by Paul Lorenzen, so let us call this theory L. Propositions of L are all the possible "words" made of letters a, b, for example: a, b, aa, aba, baab. Thus, the set of all these "words" is the language of L. The only axiom of L is the word a, and L has two rules of inference: X |- Xb, and X |- aXa. This means that (in L) from a proposition X we can infer immediately the propositions Xb and aXa. For example, the proposition aababb is a theorem of L: a |- ab |- aaba |- aabab |- aababb rule1 rule2 rule1 rule1 This fact is expressed usually as L |- aababb ( "L proves aababb", |- being a "fallen T").

How do I read a ⊢ ab? Surely not "a proves ab", I suppose.

MJD
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ivarec
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  • Although not a duplicate question, I believe some of the answers here are that for which you're looking. – apnorton Jul 28 '14 at 02:56
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    From $a$ one can deduce $ab$. – André Nicolas Jul 28 '14 at 03:44
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    $X \vdash \varphi$ means that $\varphi$ is derivable from the set $X$ . If we are working in logic (propositional or other), this is exactly : the formula $\varphi$ is provable from the set of axioms or assumptions $X$, by way of the inference rules of the system. But we can use this symbol in a more general context : a formal system. In this case, we say that the expression $\varphi$ is producible (or again : derivable) from the set of expressions $X$, according to the transformation rules of the system. – Mauro ALLEGRANZA Jul 28 '14 at 06:43
  • I'm glad Mauro mentioned the general context. Here, of course, we're dealing with propositions (things that are true or false), so the usual readings of the \vdash all work. In the general context of expressions that aren't necessarily statements, deducibility and provability wouldn't have made much sense, so we'd have to go with producibility/rewritability. – Hunan Rostomyan Jul 28 '14 at 07:19
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    @Mauro Would you please convert your comment to an answer? It's definitely answer-worthy. – Lord_Farin Apr 27 '15 at 10:22

1 Answers1

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$X \vdash \phi$ means that $\phi$ is derivable from the set [of expressions] $X$.

If we are working in logic (propositional or other), this is exactly :

the formula $\phi$ is provable from the set of axioms or assumptions $X$, by way of the inference rules of the system.

But we can use this symbol in a more general context, like that of a formal system.

In this case, we say that :

the expression $\phi$ is producible (or again : derivable) from the set of expressions $X$, according to the transformation rules of the system.

Mauro ALLEGRANZA
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