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Appendix A: Notes on Truth, Validity, and Soundness

I. Definitions and Rules.

A. Definitions.

1. Argument: any group of propositions of which one is claimed to follow logically from the others.
2. Inference: the reasoning process by which a logical relation such as entailment is perceived.
3. Entailment: a relation between or among propositions such that the truth of one proposition is determined by the truth of another proposition or propositions and such that this determination is a function solely of the meanings of the propositions concerned.
4. Valid Argument: a deductive argument whose conclusion follows necessarily form its premiss or premisses. (Usually an inference is said to be valid if it is permitted by the laws of some logic.)
5. Sound Argument: a valid deductive argument which has true premisses.   (Obviously, the conclusion is true as well.)

B. Rules.

1. A deductive argument is valid only if its conclusion follows necessarily from its premisses.
2. The fact that a deductive argument is valid does not imply that any of the propositions in the argument are true.
3. If the premisses of a valid deductive argument are true, then the conclusion must be true.
4. In an invalid argument any combination of truth values for the various propositions may occur.
5. An argument is sound if and only if it is valid and has true premisses.

II. Problems and Examples

A. The following examples serve to show the possible combinations of truth values in valid categorical syllogisms.

 Case 1 Case 2 Case 3 Case 4 premisses T F T F conclusion T T logically impossible to be F F

Examples:

Case 1:
(T) All cattle are mammals.
(T) All Angus are cattle.
(T) All Angus are mammals.

Case 2:
(F) All plants are animals.
(F) All deer are plants.
(T) All deer are animals.

Case 3:
It's logically impossible to construct an example. (If a valid argument could have true premisses and a false conclusion, then logic could not be used to extend our knowledge.)

Case 4:
(F) No pens are markers.
(F) All pencils are pens.
(F) No pencils are markers.

B. The following examples serve to show the possible combinations of truth values in invalid categorical syllogisms. Note that every combination of truth values is possible in invalid arguments.

 Case 1 Case 2 Case 3 Case 4 premisses T F T F conclusion T T F F

Examples:

Case 1:
(T) Some states are tyrannies.
(T) All dictatorships are tyrannies.
(T) Some dictatorships are states.

Case 2:
(F) No sparrows are birds.
(F) No flying creatures are birds.
(T) Some flying creatures are sparrows.

Case 3:
(T) All acids are chemicals.
(T) Some carbon compounds are not acids.
(F) Some carbon compounds are not chemicals.

Case 4:
(F) All essays are books.
(F) No tomes are books.
(F) All tomes are essays.

C. All of the following statements are true. Study each carefully. Refer to the above outline to see how each statement is true.

1. A sound deductive argument is a deductive argument which is valid and whose premiss(es) are true. (Cf., A: 1 above.)

2. It is possible for a deductive argument to be both valid and unsound. (Cf., A: 2, 4 above.)

3. If a deductive argument is sound, it cannot be invalid. (Cf., A: 1, 3 above).

4. If the premisses of a deductive argument are true, then the argument can be valid or invalid. (Cf., A: 1; B: 1, 3 above.)

5. If the conclusion of a deductive argument is true, then the premisses can be true or false. (Cf., A: 1, 2; B: 1, 2 above.)

6. If a deductive argument is sound, then its conclusion must be true. (Cf., A: 1, 3 above.)

7. If the premisses of a deductive argument are true, then the conclusion can be true or false. (Cf., A: 2, 4; B: 2, 4 above.)

8. If a deductive argument has a false premiss, then it must be unsound. (Cf., A: 2, 4; B: 2, 4 above.)

9. If a deductive argument is valid, then its conclusion can be true or false. (Cf., A: 1, 2, 4 above.)

10. If every proposition in a deductive argument is true, then the argument can be sound or unsound. (Cf., A: 1; B: 1 above.)

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