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Truth and validity are two different notions. Truth is predicated of propositions whereas validity is predicated of arguments. Propositions are either true or false.

Deductive arguments are either valid or invalid. We have noted earlier that a deductive argument claims to provide conclusive proof for its conclusion.

A deductive argument is valid if and only if the premises provide conclusive proof for its conclusion. This notion of validity of deductive argument can also be expressed in either of the following two ways.

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(i) If the premises of a valid argument are all true, then its conclusion must also be true.

(ii) It is impossible for the conclusion of a valid argument to be false while its premises are true.

Any deductive argument that is not valid is called invalid. So, a deductive argument is invalid if its preemies are all true but the conclusion is false. Note that in some cases, even if the premises and the conclusion are all true yet the argument may be invalid. In all cases invalid arguments some of our rules of inference are violated.

The above remark on deductive validity shows the connection between validity of an argument and the truth or falsity of its premises and conclusion. But the connection is not a simple one. Of the eight possible combinations of truth or falsity of premises and the conclusion and validity or invalidity of arguments, only one is completely ruled out.

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The only thing that cannot happen is that the premises are all true, the conclusion is false and the argument is deductively valid.

Given below are the other seven combinations of true and false premises and conclusion with example;

(i) There are valid arguments whose premises as well as the conclusions are all true.

Example:

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All men an mortal.

All kings are men.

Therefore, all kings are mortal.

(ii) There are valid arguments whose premises as well as the conclusions are all false.

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Example:

All cats are six-legged.

All dogs are cats.

Therefore, all dogs are six-legged.

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(iii) There are valid arguments where the premises are all false but the conclusion is true.

Example:

All fishes are mammals.

All whales are fishes.

Therefore, all whales are mammals.

(iv) An argument may have true premises and a true conclusion and nevertheless the argument may be invalid.

Example:

All men an mortal.

All kings are mortal.

Therefore, all kings are men.

(v) There are invalid arguments whose premises are false but the conclusion is true.

Example:

All mammals have wings.

All rabbits have wings.

Therefore, all rabbits are mammals.

(vi) There are invalid arguments in which premises and conclusion are all false.

Example:

All cats are biped.

All dogs are biped.

Therefore, all dogs are cats

(vii) Lastly, an argument in which the premises are true and the conclusion is false will be invalid.

Example:

All Telugus are Indians.

Nehru is not a Tamil.

Therefore, Nehru is not an Indian.

We can summarize our findings in the following tabular way.

Premise |
Conclusion |
Validity of the argument |

T |
T |
Valid Invalid |

T |
F |
XXX Invalid |

F |
T |
Valid Invalid |

F |
F |
Valid Invalid |

The above examples show that invalid arguments allow for all possible combinations true or false premises and true or false conclusion. We cited examples of valid arguments with false conclusion as well as invalid arguments with true conclusions. Thus, it can be noticed that the truth or falsity of the conclusion does not by itself determine the validity or invalidity of the argument. So also the validity of an argument does not by itself guarantee the truth of its conclusion.

We also noticed that valid arguments may have only three out of the four possible truth contributions. A valid argument cannot have true premises and a false conclusion. In other words if an argument is valid and its premises are true, then we can be sure that the conclusion is true.