User:Nayru25/sandbox/A Style Guide for Boolean Connectives

Motivation

Currently (May 2018) the pages for the Logical Connectives are a mess of competing styles and presentations, which hinders comprehension. I propose this style guide to define a common presentation for all the boolean connectives.

Scope

The article Logical connectives, and the articles

Boolean Connective	Logic Gate
Alternative denial	NAND gate
Converse implication
Implication	IMPLY gate
Disjunction	OR gate
Negation	NOT gate
Exclusive or	XOR gate
Biconditional	XNOR gate
Statement
Joint denial	NOR gate
Material nonimplication
Converse nonimplication
Logical conjunction	AND gate

Boolean Connectives

Article Layout

With exception to Negation and Material Conditional, as they have a more general scope.

• The Lead section should
  • Provide the unicode characters for the connective so people can copy and paste it
  • Have a link to the equivalent logic gate, if there is one
  • Include the relevant Venn Diagram
  • Discuss prominent alternative notation
• Sections should include, in this order
  • A section for the definition, including
    • A sub-section on the truth table of the connective.
    • If there is anything more to add on the Venn diagram for the connective, a sub-section for that.
    • A sub-section on equivalent logical formulas
  • Other alternative notation, not noted in the Lead
  • History, if any of note

Typesetting of Formulas

• In general, use <math>...</math> to display logical formulas.
• Use math upper case roman letters, that is $P$ not $p$ for terminal symbols.
• Prefer $P,Q,R,S$ over $A,B,C,D$
  • Current exceptions include
    • Logical_conjunction and Logical_disjunction, which uses $A$ and $B$, because there's too much to change at the moment
    • Material_implication, which uses $p$ and $q$, because of historical reasons
      • This is mostly fine because material implication has a broader reach than the boolean operators
    • Logical_biconditional which uses $P$ and $Q$ until about halfway, then uses $A$ and $B$ because of the images

Connectives

In general,

Sheffer stroke	Prefer $P\uparrow Q$ to $P\mathop {|} Q$
Converse implication	Prefer $P\leftarrow Q$ to $P\subset Q$
Implication	Prefer $P\rightarrow Q$ to $P\supset Q$
Disjunction	Prefer $P\lor Q$ to $P+Q$
Negation	Prefer $\neg P$ to ${\mathord {\sim }}P$, or ${\overline {P}}$
Exclusive or	Prefer $P\oplus Q$ to $P\veebar Q$, or $P\nleftrightarrow Q$
Biconditional	Prefer $P\leftrightarrow Q$ to $P=Q$, or $P\equiv Q$
Joint denial	Use $P\downarrow Q$
Material nonimplication	Prefer $P\nrightarrow Q$ to $P\not \supset Q$
Converse nonimplication	Prefer $P\nleftarrow Q$ to $P\not \subset Q$
Logical conjunction	Prefer $P\wedge Q$ to $P\cdot Q$, $P\times Q$, or $P\mathop {\And } Q$

Truth Tables

For boolean connectives the table should look like

$P$	$Q$	$P\star Q$
T	T	?
T	F	?
F	T	?
F	F	?

using T and F for the truth-values, and without the headings "Input" and "Output", as a boolean connective expresses a relation, not a computation.

Logic Gates

Article Layout

• The Lead section should have a link to the equivalent boolean connective.

Typesetting of Operators

• Terminal symbols should be upper case roman, sans-serif (i.e. A over a)
• Prefer A, B, C to P, Q, R

Truth Tables

For logic gates, the truth table should look like

Input		Output
A	B	A OP B
0	0	?
0	1	?
1	0	?
1	1	?

using 1 and 0 as the truth-values, and starting with "0 0" at the top.

(This appears to be the style used in the articles on logic gates; I would prefer to start with "1 1", but there may be some reason I'm not aware of that they start with "0 0")

Engineering Notation

Engineering notation is recommended when writing formulae for logic gates.

Typesetting

• Use <math>...</math>
• Use math upper case roman letters, that is $A$ not $a$ for terminal symbols
• Prefer $A,B,C,D$ over $P,Q,R,S$

Disjunction	Use $A+B$
Negation	Use ${\overline {A}}$, not $A'$
Exclusive or	Use $A\oplus B$
Logical conjunction	Prefer $AB$ over $A\cdot B$, except when used for emphasis