A Logical Connective (also called a logical operator) is a symbol or a word which is used to connect two or more sentences. Each logical connective can be expressed as a truth function.
- NOT (Negation)
- AND (Conjunction)
- EITHER OR (Disjunction)
- IF-THEN (Material Implication)
In logical reasoning, we deal with statements that are essentially sentences in English language. However, factual correctness is not important. We are only interested in logical truthfulness of the statements. We can represent simple statements using symbols like p and q. When simple statements are combines using logical connectives, compound statements are formed.
Negation - NOT
Negation is the opposite of a statement. For example,
- Statement: It is raining.
- Negation: It is NOT raining.
Disjunction - EITHER OR
When two statements are connected using OR, at least one of them is true. For example,
- Either p or q: p alone is true; q alone is true; both are true
In such situation, valid inference is If p did not happen, then q must happen. And If p did not happen, then p must happen.
Conjunction - AND
When two statements are connected using AND, both statements have to be true for compound statement to be true.
- p and q: p should be true as well as q should be true
Material Implication - IF THEN
If p, then q (p --> q): It is read as p implies q. It means that if we know p has occured, we can conclude that q has occured. In such situations, only valid inference is "If ~q, then ~p"; If q did not happen, then p did not happen.
Negation of Compound Statements
- Negation (p OR q) is same as Negation p AND Negation q
- Negation (p AND q) is same as Negation p OR Negation q
- Negation (p --> q) is same as Negation p --> Negation q
Logical Connectives Summary Table
|Given||Similar as||Valid Inference|
|If p, Then q||If ~q, Then ~p|
|Only If p, Then q||If q, Then p||If ~p, Then ~q|
|Unless p, Then q||If ~p, Then q||If ~q, Then p|
|Either p or q||If ~p, Then q|
|If ~q, Then p|