1) Equality is a foundational concept in functional programming but proving equality can be complicated due to issues with substitution, floating point numbers, and object references. 2) Boolean logic and the law of excluded middle may be too simplistic for modeling real-world scenarios and natural language, where statements can be unknown, doubtful, or ambiguous. 3) Different types of logic relax or modify classical principles like non-contradiction and excluded middle to better model uncertainty, inconsistency, or partial knowledge. Representing truth as a single bit may not always be the most robust approach.

1. Equality is Hard
paul phillips
lambdaconf 2016

2. Too hard. It's not even 10AM.

3. If A = B, does F[A] = F[B] ?
• A ma&er of overriding interest to func4onal programmers
• Subs4tu4on is the founda4on of equa4onal reasoning
• Without subs4tu4on, we are lost

4. Boolean values
in a nutshell

5. The meaning of a
ques/on is the method
of answering it. Tell me
how you are searching,
and I will tell you what
you are searching for.

6. Two possible truth values?
"Are you s*ll abusing animals?"
"This statement is false."
"The program will halt."

7. • I don't know.
• I don't know yet.
• I forgot.
• Pre2y sure.
• Doub7ul.
• The ques<on is nonsense.

8. Category Theory
in a nutshell

9. It depends on what the
meaning of the word
'is' is.

10. Classical logic
in a nutshell

11. Two truth values ought
to be enough for
anybody.

12. Nega%on: not without its problems
Classical logic: p <=> !!p
Intuitionist: p => !!p
Paraconsistent: p <= !!p
Intui&onist: even if p is not false, that doesn't make it true
Paraconsistent: even if p is true, that doesn't mean it isn't false

13. Is nega(on inver(ble?
• He's certainly a/rac0ve.
• He's certainly una/rac0ve.
• He's certainly not una/rac0ve...

14. Is nega(on inver(ble?
• The boss has accepted my plan.
• The boss has rejected my plan.
• The boss has not rejected my plan.

15. Is nega(on inver(ble?
• There is evidence he is at fault.
• There is no evidence he is at fault.
• There is no evidence he is not at fault.

16. I'm exploi+ng natural language.
Maybe it's chea-ng.
But what do Boolean values encode?

17. Taking the principle of
excluded middle from
the mathema5cian
would be like
proscribing the boxer
the use of his fists.

18. Object-oriented equality
in a nutshell

19. All instances are equal,
but some instances are
more equal than
others.

20. Equality depends desperately on symmetry.
Um.
trait Any {
def ==(x: Any): Bool
def !=(x: Any): Bool = !(this == x)
}
trait AnyRef extends Any {
def eq(x: AnyRef): Bool
def ne(x: AnyRef): Bool = !(this eq x)
}

21. Floa%ng point equality
in a nutshell

22. The worst form of
inequality is to try to
make unequal things
equal.

23. FP equality is fish in a barrel, sure.
Why then is it s*ll to be found?

24. def s1(n: Int) = 0.1 * n
def s2(n: Int) = Stream continually 0.1 take n reduceLeft (_ + _)
def s3(n: Int) = BigDecimal("0." + n)
1 to 9 map (n => Seq(
s1(n) == s2(n),
s2(n) == s3(n),
s3(n) == s1(n)
))

25. TTT
TTT
TFF
TTT
TTT
FTF
FTF
FFT
FFT

26. TTT
TTT
TFF
TTT
TTT
FTF
FTF
FFT
FFT

27. Booleans without totality
in a nutshell

28. He said to his friend, "If the Bri2sh march
By land or sea from the town to-night,
Hang a lantern alo+ in the belfry arch
Of the North Church tower as a signal light,
Zero if by land, and one if by sea;
And I on the opposite shore will be...

29. Overspecifica+on
in a nutshell

30. The difficulty in
philosophy is to say no
more than we know.

31. We habitually work with values
which convey total certainty boiled
down to a single bit.
Is this the most robust approach?

32. A typical scala compiler method.
def isMethodWithExtension = (
isMethod
&& owner.isDerivedValueClass
&& !isParamAccessor
&& !isConstructor
&& !hasFlag(SUPERACCESSOR)
&& !isMacro
&& !isSpecialized
)

33. ex falso quodlibet
in a nutshell

34. You get tragedy where
the tree, instead of
bending, breaks.

35. The principle of explosion
Rejected by paraconsistent "inconsistency
tolerant" logics

36. You can have it all
True or False? Why stop at one?

37. Logic is a system for preserving some property
across logical opera3ons.
• Classical logic: truth
• Intui0onist logic: jus'fica'on
• Probability logic: likelihood

38. Laws to consider repealing
• Law of non-contradic/on: !(p && !p)
"Contradictory statements cannot both be true."
• Law of the excluded middle: p || !p
"Either a proposi/on is true or its nega/on is true."

39. Ari$es of truthiness.
_0 = Void // The void stares back at you
_1 = Unit // Indifference logic
_2 = Bool // You Are Here
_3 = Opt[Bool] // billion dollar mistake?
_4 = Bool x Bool // Is it true? Y/N Is it false? Y/N
R = [0,1) // Fuzzy, possibility, probability logics

40. Setoids
In a nutshell

41. Whereof one cannot
speak, thereof one
must be silent.

42. A setoid is a set equipped with an
equivalence rela1on.
A construc(ve setoid is equipped
with an apartness rela1on.

43. Boolean Blindness
To make use of a Boolean you have to know its provenance so that
you can know what it means.

44. Haskell
class Eq a where
(==) :: a -> a -> Bool

45. I know, "performance"
data Compared a =
Equal a a
| DisEqual a a
| Incomparable
class Eq a where
(==) :: a -> a -> Compared a

47. Laws: do not confuse with fallacies

48. PhilosphyProgramming
is a ba0le against the
bewitchment of our
intelligence by means
of our language.

49. The purpose of
abstrac/on is not to be
vague, but to create a
new seman/c level in
which one can be
absolutely precise.

50. The real founda-ons
of his enquiry do not
strike a man at all.
Unless that fact has at
some -me struck him.

51. [Wi$genstein] says
every morning he
begins his work with
hope, and every
evening he ends in
despair.

52. If a lion could talk, we
could not understand
him.

53. If you want to learn
more about the law of
the excluded middle,
take a voyage down to
your public library. It's
all in books!