Lecture 5

Law of Non-Contradiction

"One cannot say of something that it is and that it is not in the same respect and at the same time." (Aristotle)

It is true that a proposition and its negation cannot both be true.

This principle states that for any proposition, P, the following cannot be true:

'P and not-P'.

If someone says 'It is raining and it is not raining', then we know that what she has said cannot be true. (We don't have to check.)

More informally, when a person says 'It is finished and it is not finished', she is contradicting herself. Contradictions cannot be true. Therefore, it cannot be true that it is finished and it is not finished.

Law of Excluded Middle

Tertium non datur (Latin; "Third not available")

It is true that, for any proposition, either the proposition is true or its negation is true.

This states that for any proposition, P, the following is true: either P or not-P.

For example, take the proposition 'It is raining'. Well, we don't know if that is true. (We have to check.)

However, take the proposition 'Either it is raining or it is not raining'. We know that this is true. There is no third alternative.

More informally, something either is, or isn't, the case. For example, either he went to California, or he didn't.

Principle of Bivalence

It is true that every proposition is either true or false.

This states that for any proposition, P, the following is true: P is true or P is false.

For example, if someone says 'It is raining', then we know, without having to check, that it is true that is raining, or it is false that it is raining.

Ex nihilo, nihil fit

(Latin; "From nothing, nothing comes.")

It states that nothing cannot be the cause of something. This is sometimes identified with the Principle of Sufficient Reason.

Law of Parmenides

"If A knows that P, then P".

This states that knowledge can only be of what is true. It is not possible to know a falsehood.

(One can, of course, believe a falsehood. But, because it is a falsehood, one cannot know it.)