Propositional Logic

Propositional logic is a mathematical technique for analysing logical statements. A statement such as "If it is raining I will take an umbrella." is reduced to a mathematical expression and analysed using Boolean Algebra and axioms.

A proposition is constructed from a series of statements (identifiers) which can be either true or false, and operators.
The operators are:

negation not b ¬b
conjunction b and c b∩c
disjunction b or c b∪c
implication b implies c b=>c
equality b equals c b=c

If b and c are propositions that can be either true or false we have the following results for all the operations on b and c:

b c ¬b b∩c b∪c b=>c b=c
F F T F F T T
F T T F T T F
T F F F T F F
T T F T T T T

To illustrate this consider the following logic to decide if I am wet.
b is "it is raining" and this can be either true or false
c is "I am outside" which can also be true or false.
the operator I will use is and.
If raining and outside are both true then I will get wet. (T∩T is T)
If raining is true but outside is false then I will not get wet. (T∩F is F)
If raining is false but outside is true then I will not get wet. (F∩T is F)
If raining is false and outside is false then I will not get wet. (F∩F is F)

Note that the or operator is an inclusive or, which means that T and T is true rather then F.

The implies operator is a difficult one to understand. Part of the difficulty arises from the fact that there is no causality associated with it. In other words if we take the English statement "if it rains there will be no picnic" we can write it as

rain => no picnic

We might be tempted to think that if it doesn't rain there will be a picnic

not rain => picnic

but this would not be true in logic.

Order of operators

There is an order of operators just as there is in arithmetic. The rules are:

  1. Sequences of the same operator are calculated from left to right
  2. The order of evaluation for different operators is not, and, or, imp, equals.

So for example b∩¬c=>d is equivalent to (b∩(¬c))=>d

To avoid confusion it is best to make liberal use of brackets rather than rely on the order of operators, just as we do in arithmetic.

Transforming English to propositional form

English statements can be transformed into propositional form for logical analysis. To do this we break the English into "atomic parts" and assign these to identifiers and describe the relationship between them with Boolean operators.

As an example consider the sentence "If it rains but I stay at home, I won't be wet."

The atomic parts might be

  • it rains: r
  • picnic is cancelled: pc
  • be wet: wet
  • stay at home: s

So the sentence be written as (r∩s)=>¬wet

I'll be wet if it rains becomes r=>wet

Note that English is sometimes ambiguous, and a sentence that appears perfectly clear may result in more than one logical expression. The translator also defines the atomic parts, and this can also result in different perfectly valid propositions for the same sentence.

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