How to define semantic logic “implies” and “not” operators for use in LyX

logiclyxmath-operatorsstacking-symbolssymbols

In Schechter's Classical and nonclassical logics, he distinguishes symbols at the semantic level, from those at the syntactic level, by encircling the operators. For example, formal logic uses "∧" (\wedge) for conjunction, while semantic logic uses an encircled version, which can be implemented by \owedge from the stmaryrd package:

But semantic operators corresponding to implication ("→", \rightarrow) and negation ("¬", \neg) don't exist among the usual symbols. How can these be generated in such a way that they can be used from LyX?

Best Answer

Personally, I keep all of my math macros in a separate LyX file that I can then include from other LyX documents (via Insert > File > Child Document). The solution requires placing some of the LaTeX code in a LyX preamble (because LyX macros can't handle the \mathpalette macro's structure). So I also have a separate LaTeX file for the preamble code, which I can then \input{…} in the "LaTeX preamble" setting of both the child and parent LyX documents.

In math-macros-preamble.tex, I place:

\newcommand{\semanticlogic}[2]{\ooalign{$\m@th#1\ocircle$\cr\scalebox{0.9}[1]{$\m@th#1#2$}\cr}}

\newcommand{\encircled}[1]{\mathpalette\semanticlogic{#1}}

\semanticlogic is a macro that takes as its arguments (1) the style macro (provided by \mathpalette) and (2) the operator to be encircled. The \encircled macro wraps \mathpalette so that it's easy to use inside a LyX macro.

In math-macros.lyx, I add \input{math-macros-preamble} to its preamble, then define a macro from \newcommand{\oimplies}{\mathbin{\encircled{\shortrightarrow}}}, giving it a LyX representation like \mathbin{$\ocircle\shortrightarrow$} so that it looks nicer when inlined into the parent document. Similarly, I create \onot as \encircled{\mkern1mu \neg}.

In the parent document, where I use the logic operators, I also add \input{math-macros-preamble} to its preamble, and include math-macros.lyx as a child document.

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[Tex/LaTex] How to stack limits for maths operators

I like it better when the i and j are aligned:

This is done by \mathrlap and \mathclap (see this TuGboat article). The article describes the use of some new macro's, including:

\def\mathrlap{\mathpalette\mathrlapinternal} 
\def\mathclap{\mathpalette\mathclapinternal}
\def\mathllapinternal#1#2{\llap{$\mathsurround=0pt#1{#2}$}}
\def\mathrlapinternal#1#2{\rlap{$\mathsurround=0pt#1{#2}$}}

add these to your preamble (or load the mathtools package) and you can do something like this:

\[
\sum\limits_{\substack{\mathllap{1\le} i \mathrlap{\le 10} \\ \mathllap{1 \le} j \mathrlap{\le 5}}}^\infty x^i y^j
\]

which produces, in my opinion, a far superior result:

example

Note that I've added the upper limit \inftyas an example, although I don't see how i and j would be smaller than 10 of 5 and still run up to ininity (ah well, it's just an example).

[Tex/LaTex] \Rightarrow vs. \implies, and “does not imply” symbol

To answer your first question, you should use \implies, not \Rightarrow. \Rightarrow is far too small to give a readable result and is not spaced properly. Knuth specially defined \iff to be used for equivalence and \implies is the same but for implication (from the amsmath package). An implication is not a relation like > and, therefore, needs to be spaced according to how it is used. Two thick spaces (which are about an en-space) precede and follow an implication because it's more important than a relation. (Basic rule of math spacing: the more important an operator, the wider the space around it). Compare the readability of the following formulas:

Rightarrow vs. Longrightarrow vs. implies

The first line uses \Rightarrow and is the least legible because the main part of the formula (the implication) is difficult to identify. The second line uses \Longrightarrow and is better, but there's not enough space to set the arrow apart from the surrounding symbols (the two inequalities are spaced as much as the arrow). Finally, in the third line (which uses \implies), the additional space highlights the arrow from the rest and so improves the readability of the formula.

Concerning your second question, the simplest way to negate nearly any symbol whatever its length is to use \centernot from the centernot package:

centernot effect on implies

\documentclass{article}

\usepackage{amsmath}
\usepackage{centernot}

\begin{document}

$A \centernot\implies B$

\end{document}

Of course, wrapping \centernot\implies inside a \notimplies macro like Werner did is a good idea.

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[Tex/LaTex] \Rightarrow vs. \implies, and “does not imply” symbol

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