Sentential Calculus for IT 101

This article is a little introduction to sentential calculus for IT engineering students. I will try to explain it as correct and simple as possible.

Sentential calculus is the foundation for discrete and computational mathematics. It teaches you what is logical in a mathematical sense. Let’s begin with some terms.

True or False

Every predicate must either equal true or false. Always. There is no null, like it should not exist for booleans.


  • 3+1 = 4 –> True
  • I study IT –> True
  • Unicorns exist –> False

Sentential Connective

The sentential connectives connect two or more predicates as mentioned above. Let’s say we have the example

  • A(x) := “x >2”


  • B(x) := “x<10”

If we want to say x is between 2 and 10 you can do it with the AND connector. But more in this table.

Math Informatics
¬A if(!A)
A^B if(A&&B)
AvB if(A||B)
A=>B if(!A||B), if(a) then b=true
A<=>B if((!A||B)(A||!B))

In this table I tried to match the connectives with a matchin programming pattern. Of course it’s only pseudo code.


A couple of rules regarding equal predicates:

  • double negation leads to no negation: a != !false –> a = true
  • A^B <=> B^A
  • AvB <=> BvA
  • (A^B)^C <=> A^(B^C)
  • etc…

De Morgan

The de morgan rules are actually pretty simple:

Every time you negate a sentential connective, you negate every predicate and change the connective sign.


  • ¬(A^B) <=> ¬Av¬B
  • ¬(AvB) <=> ¬A^¬B


Quantifiers give you the ability to generate a predicate from other (multiple) predicates. Let’s say we have two predicates:

  • A(x) := “x is a prime number”
  • B := “There is a prime number that is a divisor of 24”


  • B:= “there is (at least) one x where A(x)” which is true


  • C:= ” every number x fulfills A(x)” which is wrong


Here’s a simpe list of quantifiers using natural numbers:

  • ∀xA(x) means: For each x in N -> A(x) == true
  • ∃xA(x) means: There is at least one x in N where A(x) == true