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So I'm reading The GUHA Method of Automatic Hypothesis Determination by P. Hajek, and he talks about something called "L-implication". Forgive the stupid question, but what does that mean? I'm not a math major, so please go easy on me.

While I'm on this, what do each of the other terms he uses mean:

  1. L-true
  2. L-false
  3. factual
  4. L-implication
  5. L-equivalence

Here is how the author introduces these notions: guha

As a follow up, there is a theorem that uses L-implication, which I have absolutely no idea what it talks about. Does he just mean "implied" when he says "L-implied"?

theorem5

P.S.: Don't now how to tag this aside from logic. I'm actually reading the paper for my data-mining background research.

Andriy Drozdyuk
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2 Answers2

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To me it looks like the "L" it just an abbreviation of "logical", so the concepts are

  • "logically true" (more often called "logically valid", that is, true in all interpretations)
  • "logically false" (i.e. contradictory, false in all interpretations)
  • "logical implication" (that is, $P_1\to P_2$ is true in every interpretation. Or equivalently: $P_2$ is true in every interpretation, if any, that makes $P_1$ true)
  • "logical equivalence"

The prefix is just a way to remind the reader that we're talking about precisely defined technical concepts here, which may or may not align with the reader's intuition about how those words ought to be used.

hmakholm left over Monica
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  • I can believe this :) – BrianO Jan 23 '16 at 10:19
  • Thanks that makes sense, still not sure why he goes through all this trouble - seems like the usual logical concepts... P.S.: I've included a greater slice from the paper with the surrounding context. – Andriy Drozdyuk Jan 23 '16 at 17:39
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I've no access to Hajek's paper, but it seems that the concepts your are asking for are the basic (meta-)logical concept:

L-true means (logically) valid

L-false means contradictory

a formula that is neither L-true nor L-false is factual

L-implication is logical consequence.

See :

let us introduce several concepts which are logical in the sense just indicated. We shall call them L-concepts, and shall form terms for them with the prefix "L-". [...] we say that a sentential formula is L-true just in case [...] it is true for every value-assignment.

A sentential formula is said to be L-false (or logically false, or contradictory) in case [...] it is false for every value-assignment.

And see [ page 19 ] : Ch.A.6. L-IMPLICATION AND L-EQUIVALENCE.

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Mauro ALLEGRANZA
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    The interpretation of "L" as "language" seems to be at odds with the fact that the images use an upright L rather than an italic $L$, which it would be if it were a variable referring to a particular language. It also really doesn't make sense to relativize concepts such as logical validity to a language -- a formula is either logically valid in every language that it is well-formed in, or in none of them, so relativizing it to a language would not add any information. – hmakholm left over Monica Jan 23 '16 at 10:01
  • So factual is just some formula that is true for some values but not for others? Basically what I intuitively think of when I see logical formulas...? – Andriy Drozdyuk Jan 23 '16 at 17:34
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    P.S.: I've also added a bigger excerpt to the image from the book. – Andriy Drozdyuk Jan 23 '16 at 17:37
  • @drozzy - thanks for the Biblio: R.Carnap is there. – Mauro ALLEGRANZA Jan 23 '16 at 20:34
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    @drozzy - yes; factual is a formula like $P \land Q$, that is satisfiable (i.e. truth for some truth-assignement) but not L-true (i.e. always true). – Mauro ALLEGRANZA Jan 23 '16 at 20:35