Notes on Truth, Validity, and Soundness

Definitions

Argument:

any group of propositions of which one is claimed to follow logically from the others.

Inference:

the reasoning process by which a logical relation such as entailment is perceived.

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.

Valid Argument:

a deductive argument whose conclusion follows necessarily from its premiss or premisses. (Usually an inference is said to be valid if it is permitted by the laws of some logic.)

Sound Argument:

a valid deductive argument which has true premisses. (Obviously, the conclusion is true as well.)

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.

Problems and Examples

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	F	logically impossible	F
			to be false

Examples:

Case 1A

(T) All cattle are mammals.

(T) All Angus are cattle.

(T) All angus are mammals.

Case 2A

(F) All plants are animals.

(F) All deer are plants.

(T) All deer are animals.

Case 3A

An example is logically impossible to construct. (If a valid argument could have true premisses and a false conclusion, then logic could not be used to extend our knowledge.)

Case 4A

(F) No pens are markers.

(F) All pencils are pens.

(F) No pencils are markers.

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 1B

(T) Some states are tyrannies.

(T) All dictatorship are tyrannies.

(T) Some dictatorships are states.

Case 2B

(F) No sparrows are birds.

(F) No flying creatures are birds.

(T) Some flying creatures are sparrows.

Case 3B

(T) All acids are chemicals.

(T) Some carbon compounds are not acids.

(F) Some carbon compounds are not chemicals.

Case 4B

(F) All essays are books.

(F) No tomes are books.

(F) All tomes are essays.

All of the following statements are true. Study each carefully. Refer to the cases mentioned in order to see how each statement is true.

1. A sound deductive argument is a deductive argument which is valid and whose premiss or premisses are true. (Cf., Case 1A above.)
2. It is possible for a deductive argument to be both valid and unsound. (Cf., Cases 2A and 3A above.)
3. If a deductive argument is sound, it cannot be invalid. (Cf., Cases 1A and 3A above.)
4. If the premisses of a deductive argument are true, then the argument can be valid or invalid. (Cf., Cases 1A, 1B, and 3B above.)
5. If the conclusion of a deductive argument is true, then the premisses can be true or false. (Cf., Cases 1A, 2A, 1B, and 2B above.)
6. If a deductive argument is sound, then its conclusion must be true. (Cf., Cases 1A and 3A above.)
7. If the premisses of a deductive argument are true, then the conclusion can be true or false. (Cf., Cases 2A, 4A, 2B, and 4B above.)
8. If a deductive argument has a false premiss, then the argument must be unsound. (Cf., Cases 2A, 4A, 2B, and 4B above.)
9. If a deductive argument is valid, then its conclusion can be true or it can be false. (Cf., Cases 1A, 2A, and 4A above.)
10. if every proposition in a deductive arguent is true, then the argument can be either sound or unsound. (Cf., Cases 1A, and 1B above.)

See http://philosophy.lander.edu/logic/tvs_quiz.html for more examples of true-false questions on the topic of "Truth, Validity, and Soundness." Also, see http://philosophy.lander.edu/logic/tvs.html for lecture notes on this topic.