Logic

Jon Reades - j.reades@ucl.ac.uk

1st October 2025

“The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do.” - Ted Nelson

Do You Want?

Consider the following three things I’ve said to my toddler:

Do you want to go to the park and have ice cream?
Do you want cereal or toast for breakfast?
Do not touch that again or else!

Unlike my toddler, the computer does listen, but my toddler has more common sense!

In Code Now

(x,y) = True,False
if x:
  print("x")

if x and y:
  print("x and y")

if x or y:
  print("x or y")

if x and not y:
  print("x and not y")

if not(x and y):
  print("not x and y")

if x is the simplest: if True then print "x". There is no logic, we just do it or we don’t depending on whether x is True.

if x and y is the ‘park and ice cream question’: there’s no ice cream without going to the park. So both things have to be True for this to work out happily and print("x and y").

if x or y is the ‘cereal or toast for breakfast’ question: my daughter is having breakfast either way, so if she answers ‘Yes’ (True) to either then it doesn’t matter that the other one is False, she’s still had breakfast, or we’re still printing “x or y”. And note, if she was really hungry and said yes to both then that would also be breakfast. So if they are both True that’s also fine.

if x and not y: is the negation of one term. So ‘if you behave and don’t touch that again then you will get a treat’ is what we’re looking for here. But you can also negate the entire thing: not(x and y) is also valid logic. So this turns a False on if x and y into a True.

Combining Logic With Operators

Remember that operators like <= and == also produce True/False answers:

x = y = 5
z = 3
if x==y:
  print("x==y")

if x==y and x==z:
  print("x==y and x==z")

if x==y or x==z:
  print("x==y or x==z")

if x==y and not x==z:
  print("x==y and not x==z")

A Special Case

There is a second set of logical operators that apply in very specific circumstances. These are called ‘bitwise’ operators and apply to data specified in bits.

Regular Operator	Bitwise Equivalent
`and`	`&`
`or`	`\|`
`not`	`~`

Let’s see (briefly) how these work…

Working With Bits

x,y = 38,3
print(f"{x:b}") # `:b` means byte-format
print(f"{y:b}")

This gives us that x is '100110' and y is '11', so now:

print(f"{x & y:b}")  # 10
print(f"{x | y:b}")  # 100111
print(f"{x & ~y:b}") # 100100

Perhaps Easier to See This Way?

Operator	1	2	3	4	5	6
x	1	0	0	1	1	0
y	0	0	0	0	1	1
x & y	0	0	0	0	1	0
x \| y	1	0	0	1	1	1
~y	1	1	1	1	0	0
x & ~y	1	0	0	1	0	0

Bitwise operations are very, very fast and so are a good way to, say, find things in large data sets. You’ve been warned.

Nulls: None vs. NaN

Beware of using logic with things that are not what they appear:

None is Python’s way of saying that something has no value at all (not 0 or ""… but None). It is a class.
NaN (Not a Number) is a special numeric data type provided by the numpy package to deal with things that aren’t numbers. -ve and +ve infinity are handled by a similarly special np.inf.

np.nan should be used whenever you are dealing with errors/nulls in data (e.g. see Pandas!).

None vs. NaN

import numpy as np
print(type(np.nan))     # float
print(type(None))       # NoneType

Critically:

print(""==None)         # False
print(None==None)       # True
print(np.nan==None)     # False
print(np.nan==np.nan)   # False!
print(np.nan is np.nan) # True
print(np.isnan(np.nan)) # True

Membership

In / Not In

We’ve touched on these before:

g = ['Harvey','Rose','Batty','Jefferson']

if 'Batty' in g:
  print("In the group!")

if 'Marx' not in g:
  print("Not in the group!")

The set data type also supports in, and not in together with all of the set maths (union, intersect, etc.).

Sets

Membership maths:

s1 = {'cherry','orange','banana','tomato'} # Or s1(...)
s2 = {'potato','celery','carrot','tomato'} # Or s2(...)
print('potato' in s1)      # False
print(s1.difference(s2))   # {'banana', ...}
print(s1.intersection(s2)) # {'tomato'}
print(s1.union(s2))        # {'orange', ...}