What is L-implication? 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:


*

*L-true

*L-false

*factual

*L-implication

*L-equivalence


Here is how the author introduces these notions:

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"?

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

*

*Rudolf Carnap, Introduction to Symbolic Logic and its Applications (1958, German ed.1954 - Dover reprint), page 15 : Ch.A.5. L-CONCEPTS:


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.
