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Notes on Truth, Validity, and Soundness


I. These important definitions and rules form the basis of our study of logic.  More information is available in the lecture on Truth, Validity, and Soundness.


  1. Some key definitions of the terms are important in order to understand "the way logic works."

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

    2. Inference (a psychological process):  the reasoning process by which a logical relation such as entailment is perceived.

    3. Entailment (a logical process):  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
    5. the laws of some logic.)

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


  1. A thorough understanding of these rules is helpful in understanding the basis of logic.

    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. Some problems and examples will serve to show how these terms and rules are applied.

  1. 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.


  1.  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.


  1. All of the following statements are true. Study each carefully. Refer to the above outline to see how each statement is true.  For more practice on exercises like these, see the practice quiz on Truth, Validity, and Soundness.

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

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

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

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

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

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

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

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

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

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

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