# Appendix D Notes on accessibility

Special attention has been paid to make this HTML version as accessible as possible, especially to readers using Assistive Technology (AT), such as screen readers. It has, however, not been extensively tested. If you are using it with a screen reader or Braille terminal, or are helping a student who relies on such tools (e.g., as instructor or accessibility advisor), and you have feedback please get in touch!

We have adopted the HTML style provided by BookML, developed for the Leeds mathematics department by Vincenzo Mantova. It provides a collapsible navigation menu, buttons at the top of each page to select the font size, switch between serif and sans-serif font, and black-on-white, dark-on-sepia, or light-on-dark display options.

## Symbols and formulas

All logical symbols and formulas in this book are converted to MathML and displayed using MathJax. MathJax provides additional accessibility features for formulas, which are found in the MathJax context menu—activate/right click on any formula to activate it. If your screen reader does not read out formulas such as $\forall x(A(x)\land B(x))$ you may need to activate Speech Output in the Speech submenu of the MathJax accessibility menu.

For reference, here is a table of all the symbols used in the text, how they are (probably?) pronounced, and what they are called in the text. In the rightmost column we provide a suggested way to enter them using ASCII characters, if inserting special symbols (in a homework assignment or email to your instructor, say) is not an option.

Symbol Pronounciation Meaning ASCII equivalent
$\neg$ not sign logical not ~ or -
$\vee$ or logical or \/
$\wedge$ and logical and /\ or &
$\rightarrow$ right arrow conditional -> or >
$\leftrightarrow$ left right arrow biconditional <-> or <>
$\bot$ up tack contradiction _|_ or !?
$\forall$ for all universal quantifier A or @
$\exists$ there exists existential quantifier E or 3
$\therefore$ therefore therefore :.
$\vdash$ right tack proves |-
$\nvdash$ does not prove does not prove |/-
$\vDash$ true entails |=
$\nvDash$ not true does not entail |/=
$\Box$ white square necessary []
$\Diamond$ white diamond possible <>

Subscripts should be pronounced by screen readers, although if the subscript is a number, they may not be. They can be represented by an underline, e.g., $A_{2}$ as A_2.

The expressions ‘blank’ and ‘if⁠f’ are used throughout the textbook. ‘Iff’ is short for ‘if and only if’. The HTML versions of both are provided with ARIA labels to help screen readers pronounce them properly (i.e., as ‘blank’ and ‘if-eff’).

## Proofs

The natural deduction proofs in parts IV, VII and VIII use vertical lines to indicate where subproofs start and end. Such vertical lines extend from the assumption line of the subproof to its last line and are displayed between the line numbers and the formulas in any given line. This makes proofs a special challenge for users with low vision or complete loss of vision.

To make these proofs accessible in this HTML version, proofs are coded as tables. Each table line has four columns: the line number, a subproof level indicator, a formula, and a justification. The subproof level indicator is a number recording how many nested subproofs the current line is contained in. It is 0 if the line is not contained in a subproof, 1 if it is in a subproof, 2 if it is in a subproof nested within another subproof, and so on. When reading out a subproof level indicator, screen readers should also announce if a subproof has just been closed on the previous line, and when a new subproof starts. The table header rows and subproof level indicators are hidden so that proofs visually appear as in the printed text.

Here is an example of such a proof:

Line number
Subproof level
Formula
Justification
$1$
0
$(W\vee X)\vee(Y\vee Z)$
PR
$2$
0
$X\rightarrow Y$
PR
$3$
0
$\neg Z$
PR
$4$
open subproof, 1
$W\vee X$
AS
$5$
open subproof, 2
$W$
AS
$6$
2
$W\vee Y$
$\vee$I $5$
$7$
close subproof, open subproof, 2
$X$
AS
$8$
2
$Y$
$\rightarrow$E $2$, $7$
$9$
2
$W\vee Y$
$\vee$I $8$
$10$
close subproof, 1
$W\vee Y$
$\vee$E $4$, $5$$6$, $7$$9$
$11$
close subproof, open subproof, 1
$Y\vee Z$
AS
$12$
1
$Y$
DS $11$, $3$
$13$
1
$W\vee Y$
$\vee$I $12$
$14$
close subproof, 0
$W\vee Y$
$\vee$E $1$, $4$$10$, $11$$13$

It has 14 lines, with 3 premises, 2 levels of subproof nesting, and two pairs of adjacent subproofs. For instance, the subproof beginning on line 5 ends at line 6, and line 7 starts another subproof. So the subproof levels of lines 6 and 7 is the same, but lines 6 and 7 are in different subproofs. If you cannot see the subproof lines, you have to pay special attention to how the subproof level numbers change and when a formula is an assumption. A screen reader should announce line 7 as “7, close subproof, 2, open subproof, $X$, AS.”