CH02_3_CompoundStatement

Section 2.3: Valid and Invalid Arguments

Definition of an Argument

An argument is a sequence of statements aimed at persuading someone of something. It consists of premises, which are statements that offer evidence or reasons, and a conclusion, which is the statement that the premises are intended to support.

Example:

• Premise 1: All birds have feathers.

• Premise 2: A robin is a bird.

• Conclusion: Therefore, a robin has feathers.

Valid vs. Invalid Arguments

A valid argument is one in which the conclusion must be true if all the premises are true. An invalid argument is one in which the conclusion may be false even if all the premises are true.

Truth Tables for Determining Validity

Truth tables are used to determine the validity of an argument by considering all possible truth value combinations of the premises and conclusion.

• If there is a row in the truth table where all premises are true and the conclusion is false, the argument is invalid.

• If there is no such row, the argument is valid.

Example: Consider the argument:

• Premise 1: If it is raining, then the ground is wet.

• Premise 2: The ground is wet.

• Conclusion: Therefore, it is raining.The truth table for this argument would show that there is a row where both premises are true, but the conclusion is false, indicating that the argument is invalid.

Modus Ponens and Modus Tollens

Modus Ponens (Affirming the Antecedent):

• If p, then q.

• p.

• Therefore, q.Example: If I study hard, I'll get a good grade. I studied hard. Therefore, I'll get a good grade.

Modus Tollens (Denying the Consequent):

• If p, then q.

• Not q.

• Therefore, not p.Example: If it's a sunny day, people will be at the beach. People are not at the beach. Therefore, it's not a sunny day.

Other Valid Argument Forms (Rules of Inference)

• Generalization: If p is true, then "p or q" is also true.

• Specialization: If "p and q" is true, then p is true, and q is true.

• Elimination: If "p or q" is true, and you know p is false, then q must be true.

• Transitivity: If p implies q, and q implies r, then p implies r.

• Proof by Division into Cases: If you can prove a conclusion holds true in two separate cases, and those are the only possible cases, then the conclusion is always true.

Fallacies

Fallacies are common errors in reasoning that can make an argument invalid.Some Common Fallacies:

• Ambiguous Premises: Using unclear or misleading language in premises.

• Circular Reasoning: Assuming the conclusion in your premises.

• Jumping to Conclusions: Drawing conclusions without enough evidence.

• Converse Error: Confusing "If p, then q" with "If q, then p."

• Inverse Error: Confusing "If p, then q" with "If not p, then not q."

Rule of Contradiction

The rule of contradiction states that if assuming a statement is false leads to a contradiction, then the statement must be true.

Example: If assuming that the Earth is flat leads to contradictions with observable evidence, then the statement "the Earth is flat" must be false.