Logic, Simple and Compound Statements, Logical operations and Truth Tables, Conditional Statements and Proofs

CONTENT

-Meaning of Simple and Compound Statements.

–  Logical Operations and the Truth Tables.

-Conditional Statements and Indirect Proofs.

SIMPLE AND COMPOUND PROPOSITIONS

A preposition is a statement or a sentence that is either true or false but not both. We shall use upper case letters of English alphabets such as A, B, C, D, P, Q, R, S, …, to stand for simple statements or prepositions. A simple statement or proposition is a statement containing no connectives. In other words a proposition is considered simple if it cannot be broken up into sub-propositions. On the other hand, a compound proposition is made up of two or more propositions joined by the connectives. These connectives are and, or, if …then, if and only if. They are also called logic operators. The table below shows the logic operators and their symbols.

Figure 1

1. The statement ∼P is known as the negation of P. thus ∼P means not P or ‘it is false that P…’ or ‘it is not true that P…’
2. If P and Q are two statements (or propositions), then:
3. The statement P ^ Q is called the conjunction of P and Q. thus, P ^Q means P and Q.
4. The statement P V Q is called the disjunction of P and Q. thus, P V Q means either P or Q or both P and Q. notice that the inclusive or is used.
5. The statement P ⇒ Q is called the conditional of P and Q. a conditional is also known as implication P ⇒  Q means if P then Q or P implies Q.
6. The statement P ⇔  Q is called the biconditional of P and Q, where the symbol ⇔ means if and only if (or iff for short). Thus P ⇔  Q means P ⇔ Q and Q ⇔  P.

The Truth Tables

The truth or falsity of a proposition is its truth value, ie. A proposition that is true has a truth value T and a proposition that is false has a truth value F. the truth tables for the logical operators are given below.

Figure 2

If P is true (T), then ~P is false and if P is false, then ~P is true.

Converse, Inverse, & Contrapositive – Conditional & Biconditional Statements, Logic, Geometry

The converse of the conditional statement ‘if P then Q’ is the conditional statement ‘if Q then P’, i.e. the converse of P⇒Q is Q ⇒ P.

Inverse statement

The inverse of the conditional statement ‘if P then Q’ is the conditional statement ‘if not P then not Q’.

i.e. the inverse of P ⇒ Q is ∼P ⇒∼Q.

Contrapositive statement

The converse of the conditional statement ‘if P then Q’ is the conditional statement ‘if not Q then not P’.

i.e. the contrapositive of P ⇒ Q is ∼P⇒∼P

LOGICAL OPERATIONS AND TRUTH TABLES

Tautology and Contradiction

When a compound proposition is always true for every combination of values of its constituent statements, it is called a tautology. On the other hand, when the proposition is always false it is called a contradiction.

THE CHAIN RULE

The chain rule states that if X, Y and Z are statements such that X ⇒Y and Y ⇒ Z, then X ⇒ Z. a chain of statements can have as many ‘links’ as necessary. Example 5 is an example of the chain rule.

When using chain rule. It is essential that the implication arrows point in the same direction. It is not of much value, for example, to have something like X ⇒Q ⇐ R because no useful deductions can be made from it.

Example

In the following argument, determine whether or not the conclusion necessarily follows from the given premises.

All drivers are careful. (1^st premise)

Careful people are patient (2^nd premise)

Therefore all drivers are patient (conclusion)

If         D: people who are drivers

            C: people who are careful

            P: people who are patient 

Then    D ⇒ C             (1st premise)

And     C⇒P                (2^nd premise)

If         D ⇒ C and      C ⇒ P

Then    D ⇒ P              (chain rule)

The conclusion follows from the premises.

CONDITIONAL STATEMENTS AND INDIRECT PROOFS.

Another method we can use to determine the validity of arguments especially the more complex ones is to construct the truth tables as will be seen in the following examples.