80-110 The Nature of Mathematical Reasoning
July 10, 2000
Statement/proposition. A sentence that can be true or
false. E.g., `Dogs are more intelligent than cats', `3+4=7'.
Argument/proof. List of statements that are connected by
Premise. Statement that is assumed to be true at the beginning
of a proof.
Conclusion. Statement at the end of a proof.
Inference rule/justification. Truth preserving rule that connects two or more statements within a proof. `Truth preserving' means that whenever the antecedents are true, then the conclusion is also true. (Thus, an inference rule might be understood as being a `little proof' by itself -- with premises and conclusion, but no intermediate steps.)
Kinds of arguments
- Deductively valid argument.
An argument is deductively valid if it is impossible for the premises to be true and the conclusion false at the same time.
- Sound argument.
An argument is sound if it is deductively valid, and it has true premises.
- Inductive argument.
An argument is inductive if the truth of the premises and the intermediate statements make it more likely that the conclusion is also true, but they do not guarantee its truth.
- Formal argument.
An argument is formal if the validity of the argument does not
depend on the meaning of the symbols employed.
- Irrelevant argument.
An argument is irrelevant if the validity of the argument does not
depend on the statements used in the argument.
- Fallacious argument.
An argument is fallacious if one of the inference rules used is not admissible.