2.8 Methods of Proof PHIL 012 1/26/2001

Outline Announcements Homework questions (14-17) Arguments, truth, and validity Proving validity Assignment

Argument An argument is a collection of sentences in which some of the sentences (the premises) are meant to support another of the sentences (the conclusion). The conclusion is the claim you want to set forth. The premises are the grounds of that claim. In other words, the conclusion is what you want people to believe and the premises are the reasons given for believing it.

Validity and Truth Statements, sentences, and claims are either true or false. Arguments are either valid or invalid. A valid argument is one whose conclusion is a logical consequence of its premises. This means that it is impossible for the premises to be true and the conclusion false.

A Valid Argument All men are mortal Socrates is a man Socrates is mortal Assuming that the premises are true, it is impossible for the conclusion to be false. This is based upon the structure or form of the argument. Validity and invalidity are a function of form, not content.

An Invalid Argument Lucretius is mortal All men are mortal Lucretius is a man Note that despite the fact that the premises and conclusion are true, the argument is still invalid. It is invalid because its conclusion does not follow from its premises. It has a bad form.

A valid argument with a false conclusion All penguins are ducks Chilly Willy is a penguin Chilly Willy is a duck The argument is valid because it has a good form. That is, its structure is such that its conclusion follows from its premises. Its conclusion is false, however, because one of its premises is false.

Truth and validity You are guaranteed of a true conclusion If and Only If (IFF) It has a valid form, and All of its premises are true.

Proving Validity Whether a premise is true is a question of content. Whether an argument is valid is a question of form. There are many straightforward ways of checking the truth or falsehood of a premise. We need a method of making sure that the form of an argument is valid.

Which of the following argument forms are valid or invalid? All S are M No P are M No P are S Some P are M Some M are S Some S are not P Some S are M Some M are P Some S are P All S are M All M are P All S are P Some M are P Some P are M Some S are M Some S are P No S are M No P are M No S are P Some S are M Some P are not M Some S are not P All S are M Some M are P

Proof A proof is a step by step demonstration that a given conclusion follows from the premises of an argument. In constructing a proof, we take small steps, transforming the premises according to a set of given rules into the conclusion.

Example Aristotle was a Platonist. (1) All Platonists believe in forms. (2) No one who believes in forms understands evolution. (3) Everyone who does not understand evolution believes that species are constant. (4) Aristotle believes that species are constant.

Example Continued Aristotle was a Platonist. (1, premise) All Platonists believe in forms. (2, premise) Aristotle believes in forms. (5, from 1 & 2) No one who believes in forms understands evolution. (3, premise) Aristotle believes in forms. (5, see above) Aristotle doesn’t understand evolution. (6, from 3 & 5)

Example Continued Everyone who does not understand evolution believes that species are constant. (4, premise) Aristotle does not understand evolution. (6, see above) Aristotle believes that species are constant. (conclusion, from 4 & 6) We have shown in a step by step fashion that it is impossible for premises 1-4 to be true without the conclusion (6) also being true.

Summary Sentences are true or false. Arguments are valid or invalid. A valid argument is one whose form is such that IF the premises are true, the conclusion must also be true. A proof is a step-by-step demonstration which shows that the conclusion must be true in any circumstances in which the premises are all true.

Assignment Due Midnight Sunday Night: Problems 14-17 For Monday, Read 2.8 Due Midnight Tuesday Night: Problems 18-20