Call a statement A of a first-order language L logically true in the probabilistic sense if for all probability functions P, P(A) = 1. Where S is a set of statements of L, say that A is logically entailed by S in the probabilistic sense if, for all P, P(A) = 1 if P(B) = 1 for each member B of S. This sense of logical entailment is strongly sound and strongly complete: S fi A iff A is logically entailed by S in the probabilistic sense. And taking S to be ∅, we have fi A iff A is logically true in the probabilistic sense.

This is an abstract from a paper about Logic and Probability. I would really appreciate it if someone could shed some light on what exactly does "fi" represent above and where does it comes from. Is it logical consequence? Thank you.

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    $\begingroup$ My bet is that it is some kind of font problem; your source is somehow typeset or displayed with an "fi" ligature character where there was supposed to be something like $\vdash$ or $\vDash$. This can sometimes happen if one tries copy-pasting from a typeset abstract which uses specially-encoded fonts for mathematical symbols. $\endgroup$ Jun 3, 2017 at 9:01
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    $\begingroup$ Perhaps they mean the reverse of "if". As in "$S$ fi $A$" means "$A$ if $S$", ie "if $S$, then $A$" $\endgroup$
    – Astyx
    Jun 3, 2017 at 9:02
  • $\begingroup$ The same text appears, still with a "fi", in secion 5.1 of philrsss.anu.edu.au/people-defaults/alanh/papers/comp_logic.pdf -- which seems to make @Astyx's interpretation more likely. $\endgroup$ Jun 3, 2017 at 9:06
  • $\begingroup$ Thank you for your answers, yes this is the text I am reading. I don't have the book to double check if is a font error. It is just that I haven't seen anything similar before in a Logic textbook. Why write "S fi A" and not "A if S" or "if S, then A"? $\endgroup$
    – Floyd
    Jun 3, 2017 at 9:14

1 Answer 1


The "fi" seems to represent the entailment relation that mathematical logicians usually write "$\vdash$".

I found the exact text you quote, including "fi", in section 5.1 of a paper by A. Hájek, http://philrsss.anu.edu.au/people-defaults/alanh/papers/comp_logic.pdf, which cites H. Leblanc for the idea. And Leblanc states this particular result -- but now with "$\vdash$" -- as Theorems 100 and 105 of Handbook of Philosophical Logic (ed. Gabbay and Guenthner, Springer, 2013) vol 2, p. 100-102. (This is not exactly what Hájek cites, but was what I could find a Google Books preview of).

It is not clear whether representing $\vdash$ by "fi" is the result of a font problem (copy-pasting from a typeset paragraph where $\vdash$ is encoded as a glyph with the same codepoint that represents an fi-ligature in another font), or if Hájek deliberately decided that spelling "if" backwards would be a better symbol for entailment than the usual $\vdash$.

Looking closer, after the OP confirmed he was quoting Hájek, I would say it is probably a font problem. If we zoom in on the PDF I link above, we see that the symbol is definitely an "fi" ligature, whereas other appearances of the letters "fi" in the paper -- such as in the word "define" -- do not use ligatures.

  • $\begingroup$ Well, I would never have thought of checking it this way. Thank you very much! I guess it is clear now. $\endgroup$
    – Floyd
    Jun 3, 2017 at 9:25

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