Introduction to Logic.

In mathematical logic, statements have one of two values: True (T) or False (F). This creates a two value system.

Most children start learning math when they learn to count. The counting sequence, \(1, 2, 3, \dots\) is infinite, and infinity is a difficult concept even for adults. Then they add 0 to the system, and negatives, and irrationals, etc. It is a very technical and complicated system. The system we are studying now only has 2 points: T and F.

Language and Logic

In this section, we take different parts of speech and formally define logical operations on the T, F space.

Negation
Negation will "flip" the value. The symbol for negation is \(\neg\). The statement \(\neg P\) is read "not P". Written as formula: \(\neg T = F\) and \(\neg F = T\).

Conjunction
The symbol for conjunction (and/but) is \(\wedge\). The statement \(P \wedge Q\) is read "P and Q". Written as formula: \(T \wedge T = T\), \(T \wedge F = F\), \(F \wedge T = F\), and \(F \wedge F = F\).

Disjunction
There are two symbols for disjunction.

The symbol for inclusive or is \(\vee\). The statement \(P \vee Q\) is read "P or Q". Written as formula: \(T \vee T = T\), \(T \vee F = T\), \(F \vee T = T\), and \(F \vee F = F\).

The symbol for exclusive or is \(\oplus\). The statement \(P \oplus Q\) is read "P exclusive or Q". Written as formula: \(T \oplus T = F\), \(T \oplus F = T\), \(F \oplus T = T\), and \(F \oplus F = F\).

Truth Tables

Truth tables are similar to addition or multiplication tables. But, when learning addition, we cannot list all possible sums. We need to create algorithms so we can, in theory, compute a sum of any two numbers. In logic, we only have two states. So, we can define an operation by computing for all possible inputs as we did above. A truth table give us an organized way to do this.

The following are the truth tables of the operations listed above.