Equalities connected by logical operators

Question

Most (but not all) of the math-mode stuff in my book consists of terms and formulas of a clean logic:

• The terms are recursively constructed from constants (such as 0, 1, 2, true, f, g, +, ∈, or, not, … of any arity including 0) and variables (such as S, x, y, …) using applications of terms to other terms (such as 1+x, f(0,g(y)), x∈S, …).

• The atomic formulas are of the form t₁ = t₂, where t₁ and t₂ are terms (such as 3 = x+2, x∈S and not false = true, …).

• The formulas are recursively constructed from atomic formulas using logical negation (¬) of formulas (such as ¬(1=2+3), ¬(f(x)=g(0,y)), …), conjunctions (∧) and disjunctions (∨) (such as v=2+x ∧ y+2=z, v=x ∨ y∈S = true, …), and variables binders (such as ∃ x: x+1=2, ∀ x,y: x+y=y+x, {x|x+1>y}, …).

(Btw., all terms have sorts, and there is also a well-sortedness condition imposed on terms and atomic formulas. We drop this here for simplicity.)

Now, when I typeset my formulas containing equality = and binary logical connectives ∧ and ∨, do I get counterintuitive distances? E.g., in v=x \land y=z, the distances around \land (a binary operator in LaTeX) are smaller than around = (a relation in LaTeX), aren't they?

If so, what would be the cleanest way to repair this in my setup? I don't really know and was thinking of something like

\documentclass{article}
\newcommand{\formulaConjunction}{\mskip1.2mu plus.6mu minus.6mu\land\mskip1.2mu plus.6mu minus.6mu}
\newcommand{\formulaDisjunction}{\mskip1.2mu plus.6mu minus.6mu\lor\mskip1.2mu plus.6mu minus.6mu}
\newcommand{\memberOf}{\mathbin{\in}}
\begin{document}\noindent
\(v\memberOf S = \mathsf{true} \formulaConjunction y=f(z)\)\\
\(v=x+1 \formulaDisjunction y=z\)
\end{document}

This way, the distances around the logical conjunction and disjunction connectives become slightly larger than around the equality symbol of the logic, and the distances around the equality symbol become slightly larger than the distances around infix operators (such as +, ∈), don't they? However, this seems very ad-hoc to me. Before I change hundreds of formulas in my book this way, are there any potential pitfalls? Is there anything better that works for all of [pdf|xe|lua]latex ? My guess is that the aforementioned logic is somewhat standard, so some someone must have already done this (or at least a similar) exercise.

Answer 1

I gather that you would like to change the amount of whitespace TeX inserts around certain symbols of type mathbin -- specifically, \land and \lor, but maybe some others as well. The default amount of whitespace for mathbin objects (governed by the parameter \medmuskip) is 4mu in PlainTeX and in all LaTeX document classes I'm familiar with. You may want to increase it to at least 6mu or 7mu. Why not 5mu, you may ask? Because that's the default value of \thickmuskip, the parameter that governs the amount of whitespace around objects with status mathrel.

Unfortunately, the number of distinct math-type or math-status categories is quite limited, and none of the existing categories lends itself easily to your typesetting needs. E.g., you don't want to modify the properties of objects with status mathbin and mathrel, right?

However, I can suggest that you create two new user macros via, say, the instructions

\newcommand\Nashland{{\mkern7mu{\land}\mkern7mu}}
\newcommand\Nashlor{{\mkern7mu{\lor}\mkern7mu}}

and use \Nashland and \Nashlor instead of \land and \lor. Of course, this setup assumes that \Nashland and \Nashlor will only ever be surrounded by objects with math status mathord. That is the case in the examples you posted. However, I don't know if this assumption is always valid.

A separate comment: You may be expecting a lot -- and quite possibly too much -- from your readers if you need them to notice and fully understand the semantic significance of subtle differences in the amounts of whitespace that TeX places around various math objects. In "ordinary" (non-Nash...) notation, there's really no semantic significance attached to the amounts of whitespace that surrounds mathbin objects -- such as + and - -- and mathrel objects -- such as =, >, and <. I'm pretty sure that one could switch the values of \medmuskip and \thickmuskip in expressions such as 1+1<2+2 and not cause a major disturbance (though quite possibibly some minor irritation) among readers. But that's not permitted in the use cases that require \Nashland and \Nashlor, is it? Hence, I think you may be better off using parentheses judiciously and not rely on your readers to truly appreciate the semantic significance of subtle differences in whitespace amounts.

---

Here's an MWE (minimum working example). Note that I've had to set the amount of whitespace involved in \Nashland and \Nashlor to a whopping 10mu in order to ensure that readers have no choice but to notice that there must be a non-random, semantic meaning to the amount of whitespace that surrounds the \land and \lor symbols. Well, you might be able to get by with 9mu or 8mu instead of 10mu; I certainly wouldn't go below 8mu.

In contrast, using parentheses requires no appreciation at all of subtle differences in whitespace amounts. Parentheses can indeed cause visual clutter. But what's worse: cause some visual clutter, or risk rampant confusion?

\documentclass{article}
\usepackage{amsmath} % for 'gather*' env.
\newcommand{\true}{\mathsf{true}}
%\newcommand{\false}{\mathsf{false}} % not needed for this example

\newcommand\Nashland{{\mkern10mu{\land}\mkern10mu}}
\newcommand\Nashlor{{\mkern10mu{\lor}\mkern10mu}}

\begin{document}

Case 1: \texttt{\string\land} and \texttt{\string\lor}:
\begin{gather*}
v \in S = \true \land y=f(z) \\
v=x+1 \lor y=z
\end{gather*}

\medskip
Case 2: \texttt{\string\Nashland} and \texttt{\string\Nashlor}:
\begin{gather*}
v \in S = \true \Nashland y=f(z) \\
v=x+1 \Nashlor y=z
\end{gather*}

\medskip
Case 3: Parentheses alongside \texttt{\string\land} and \texttt{\string\lor}:
\begin{gather*}
(v \in S = \true) \land (y=f(z)) \\
(v=x+1) \lor (y=z)
\end{gather*}

\end{document}

Answer 2

You may want, in select places, change the nature of = and \in to be binary operations rather than relations.

You possibly need other symbols.

\documentclass{article}

\newcommand{\true}{\mathsf{true}}
\newcommand{\false}{\mathsf{false}}

\newcommand{\logic}[1]{%
  \begingroup
  \expanded{\mathchardef\in = \the\numexpr\in-"1000}%
  \mathcode`= = \numexpr\mathcode`=-"1000\relax
  #1
  \endgroup
}


\begin{document}

\[
\logic{v \in S = \true \land y=f(z)}
\]

\[
\logic{v=x+1 \lor y=z}
\]

\end{document}