Deductive Arguments

A deductive argument is one whose premises are claimed to provide conclusive grounds for the truth of its conclusion.

If a deductive argument is valid, it is impossible for its premises to be true without its conclusion also being true.

Example 1 Everything made of copper conducts electricity. [P] This wire is made of copper. [P] Therefore, this wire will conduct electricity. [C]

Example 2 If Joe signed the contract, then the contract is binding. [P] If the contract is binding, then Joe owes Jed $100. [P] Therefore, if Joe signed the contract, Joe owes Jed $100. [C]

Take Note… Remember that the validity of an argument has nothing to do with whether its premises are factual or true.

Take Note… Validity has to do with the nature of the connection between premises and conclusion.

Example 3 All green cheese smells like daisies. [P] The moon is made of green cheese. [P] Therefore, the moon smells like daisies. [C]

Deductive Argument Forms Disjunctive Syllogisms Hypothetical Syllogisms Categorical Syllogisms

1. Disjunctive Syllogism Contains a compound, disjunctive (alternative) premise asserting the truth of at least one of its alternatives, and a premise that asserts the falsity of one of those alternatives.

Disjunctive Proposition Either A or B is true. A and B are called the disjuncts or alternatives.

Disjunctive Syllogism Either A or B is true. A is not true. (or B is not true.) Therefore B is true. (or A is true.)

Example Either the next Olympics will be held in Atlanta or in Athens. It won't be held in Athens. It will be held in Atlanta.

2. Hypothetical Syllogism Contains one or more compound, hypothetical (or conditional) propositions, affirming that if one of its components (the antecedent) is true then the other (the consequent) is also true.

Hypothetical Proposition If A, then B. A is called the antecedent. B is called the consequent.

2.A. Pure Hypothetical Syllogism Contains conditional propositions only. If A is true, then B is true. If B is true, then C is true. Therefore, if A is true, then C is true.

Example If I win the lotto, then I will have money. If I have money, then I can pay my debts. Therefore, if I win the lotto, then I can pay my debts.

2.B. Mixed Hypothetical Syllogism Contains both a conditional premise and a categorical premise. Subdivided into two types: modus ponens and modus tollens.

If A is true, then B is true. A is true. Therefore B is true. 2.B.i. Modus Ponens If A is true, then B is true. A is true. Therefore B is true. From Latin ponere, “to affirm.”

Example If today is Tuesday, then the garbage truck will arrive. Today is Tuesday. Therefore the garbage truck will arrive.

If A is true, then B is true. B is not true. Therefore A is not true. 2.B.ii. Modus Tollens If A is true, then B is true. B is not true. Therefore A is not true. From Latin tollere, “to deny.”

Example If the dog detects an intruder, then the dog will bark. The dog did not bark. Therefore the dog did not detect an intruder.

3. Categorical Syllogism Contains two premises (a major and a minor premise) and a conclusion, all of which are in the form of categorical propositions.

Categorical Syllogism All A are B. All C are A. Therefore all C are B. There are 15 valid forms of the categorical syllogism.

Example All men are patriots. All boxers are men. Therefore all boxers are patriots.

Three Formal Fallacies Caveat Three Formal Fallacies

Fallacies are errors in reasoning Fallacies are errors in reasoning. Formal fallacies are errors in the structure of an argument.

1. Affirming a Disjunct Either A or B is true. A is true. Therefore B is not true. The truth of one disjunct does not mean that the other is false. Both of the disjuncts may be true.

Example Either he will buy the drinks or she will make the popcorn. Therefore she will not make the popcorn.

2. Affirming the Consequent If A, then B. B. Therefore A. The conditional statement “If A, then B” claims that if A happens, B follows. It does not claim that if B happened, A preceded it.

Example If Francis Bacon wrote Hamlet, then he was a great writer. Francis Bacon was a great writer. Therefore he wrote Hamlet.

3. Denying the Antecedent If A, then B. Not A. Therefore not B. The conditional statement “If A, then B” claims that if A happens, B follows. It does not claim that if A does not happen, B will not happen.

Example If it rains, then the school grounds will be wet. The school grounds are wet. Therefore it rained.