Logical Arguments

Definition

An argument (argument form) is a sequence of statements (statement forms). All statements in an argument, except for the final one are called premises (or assumptions or hypothesis). The final statement is called the conclusion The symbol, $\bullet$, which is read “therefore”, is normally placed before the conclusion.

Example Argument

$p\to q\vee \sim r\text{ (premise)}$ $q\to p\wedge r\text{ (premise)}$ $\bullet p\to r\text{ (conclusion)}$ Note: this argument is invalid

Valid or Invalid

To say that an argument is valid means that no matter what statements are substituted for the statement variables in its premises, if the resulting premises are all true, then the conclusion is also true. i.e. For an argument to be valid: IF all premises are true, THEN conclusion must be true

Test Validity with Truth Table

1. Identify all the premises and conclusion of the argument form
2. Construct a truth table showing the truth values of all the premises and the conclusion
3. Identify critical rows: Rows in which all the premises are true
4. There is a critical row in which the conclusion is false $\to$ The argument form is valid
5. The conclusion in every critical row is true $\to$ The argument form is valid

Syllogism

An argument form consisting of two premises and a conclusion

Modus Ponens

This is a commonly used form of syllogism: if $p$ then $q$ $p$ $\bullet q$

Modus Tollens

Also a commonly used form of syllogism: if $p$ then $q$ $\sim q$ $\bullet \sim p$

Rules of Inference

A form of argument that is valid Modus ponens and modus tollens are both rules of inference