Deductions#
In propositional logics, sometimes we deduct one logical expression from another; we call them deductions. Deductions form the backbone of logical reasoning. Deductions are used to derive conclusions from premises. They follow rules that guarantee the truth of the conclusion if the premises are true. This concept is important in mathematical proofs in computer algorithms, and reasoning systems.
A deduction is a logical process where, starting from a set of premises (assumptions), we find a conclusion that logically follows. For example, if we know that,
If Edith eats her vegetables, she gets a cookie.
Edith ate her vegetables.
Here, we can deduce that “Edith gets a cookie”. This simple example shows a rule called modus ponens, one of the most common deduction rules.
This notation means: if \(P\) is true, and we also know that \(P → Q\), then \(Q\) must be true.
Modus Ponens#
Modus ponens (Latin for “method of affirming”) is a fundamental rule of inference which states that if a conditional statement (\(P → Q\)) and its antecedent (\(P\)) are both true, then its consequent (\(Q\)) must also be true.
In propositional logic, the rule is expressed as:
Example
Premise 1: “If it rains, the grass will be wet.”
Premise 2: “It is raining.”
Conclusion: “The grass wil be wet.”
To prove that modus ponens is a valid rule, we can use a truth table. Let us construct one for \(P → Q\):
\(P\) |
\(Q\) |
\(P → Q\) |
|---|---|---|
true |
true |
true |
true |
false |
false |
false |
true |
true |
false |
false |
true |
From the table, we can see that whenever \(P\) is true and \(P → Q\) is true, \(Q\) must also be true. This confirms that modus ponens is a valid form of deduction.