# Equality of booleans in Emacs

October 14, 2018
(emacs)

There is a fun story about booleans in Emacs Lisp - there are no booleans in Emacs Lisp. Sort of. Because we have a symbol nil, which means an empty list. You can write it as nil or () - they both stand for the same object, the symbol nil.

(eq nil ())                             ; => t

Since LISP is all about list processing, empty list is something very false. So false that we don’t have special symbol for false values, as empty list serves this purpose well.

Everything that is not an empty list has a meaning of true. However, there is a symbol t which is the preferred way to represent the truth value true.

So nil and t are considered canonical boolean values. There is a function booleanp that returns t if the argument is a canonical boolean value and nil otherwise.

The fun begins when you need to check if two boolean values are equal. Since non-nil (or not an empty list) can mean many different things (like "Emacs is the only true editor") you can’t just do regular equality check.

(equal t "Emacs is the only true editor") ; => nil

There are, however, several tricks to get it working. The most obvious solution is to convert value to a canonical boolean value.

# if/when

We can directly use if function.

(if "Some truth" t nil)                 ; => t
(if 42 t nil)                           ; => t
(if t t nil)                            ; => t
(if nil t nil)                          ; => nil

(let ((a t)
(b "Emacs is the only true editor"))
(equal (if a t nil) (if b t nil)))    ; => t

(defun boolean-eq (a b)
(equal (if a t nil)
(if b t nil)))

(let ((a t)
(b "Emacs is the only true editor"))
(boolean-eq a b))                     ; => t

Directly using if is a little bit cumbersome, but when we hide it inside of a helper function it’s not that bad, actually.

The same result can be achieved by using when.

(when "Some truth" t)                   ; => t
(when 42 t)                             ; => t
(when t t)                              ; => t
(when nil t)                            ; => nil

(let ((a t)
(b "Emacs is the only true editor"))
(equal (when a t) (when b t)))        ; => t

(defun boolean-eq (a b)
(equal (when a t)
(when b t)))

(let ((a t)
(b "Emacs is the only true editor"))
(boolean-eq a b))                     ; => t

# not

There is another function we can use - not, which returns t if the argument is nil, and returns nil otherwise. Yes, it negates the value, but the result is one of the canonical booleans, so we are good.

Since $$a \equiv b$$ is equivalent to $$\neg a \equiv \neg b$$, we can just compare negated values.

$$a$$ $$b$$ $$\neg a$$ $$\neg b$$ $$a \equiv b$$ $$\neg a \equiv \neg b$$
0 0 1 1 1 1
0 1 1 0 0 0
1 0 0 1 0 0
1 1 0 0 1 1
(not "Some truth")                      ; => nil
(not 42)                                ; => nil
(not t)                                 ; => nil
(not nil)                               ; => t

(let ((a t)
(b "Emacs is the only true editor"))
(equal (not a) (not b)))              ; => t

(defun boolean-eq (a b)
(equal (not a) (not b)))

(let ((a t)
(b "Emacs is the only true editor"))
(boolean-eq a b))                     ; => t

This one looks a little bit better when used without a helper function, at least in my opinion.

# xor/or/and

Sometimes you want to do something when two ‘boolean’ values are not equal.

(let ((a nil)
(b "Emacs is the only true editor"))
(unless (equal (not a) (not b))
(message "Some real work begins"))) ; => Some real work begins

For such situations, there is a xor function, which returns nil when both arguments are equal in the canonical boolean form and t otherwise.

(xor nil nil)                           ; => nil
(xor nil t)                             ; => t
(xor t nil)                             ; => t
(xor t t)                               ; => nil

(xor "Some truth" nil)                  ; => t
(xor "Some truth" t)                    ; => nil

(xor 42 nil)                            ; => t
(xor 42 t)                              ; => nil

(let ((a nil)
(b "Emacs is the only true editor"))
(when (xor a b)
(message "Some real work begins"))) ; => Some real work begins

Other functions (like or, and) also convert values to canonical boolean values. So you can keep it in mind.

# Epilogue

The sole purpose of this post is fun. If you didn’t get your portion of fun, then it’s not funny at all. Please fix it somehow.