- All Men are Mortal;
- Socrates is a Man;
- .:. Socrates is Mortal.
A proposition is a declarative sentence in which the predicate is affirmed or denied of the subject. A proposition is the meaning of a declarative sentence. Propositions are either true or false. We indicated above that propositions (and only propositions) are the elements (premises and conclusions) of an argument. Sentences that express commands, pose questions, or convey exhortations are neither true nor false, and therefore are never elements of an argument. Gordon Clark puts it this way:
Of course in English Rhetoric there are questions that are intended as propositions. They are called rhetorical questions. But logically they are propositions. A question that is intended as a question is neither true nor false. It can play no part in an argument.,(Logic, p. 30 PB)
- How are you today?
- Should you eschew speaking falsehood?
- The witness committed perjury.
Arguments divide into two classes: deductive arguments and inductive arguments. This classification amounts to two different claims. The premises of Inductive Arguments claim to provide incomplete or partial reasons in support of a conclusion. The premises of Deductive Arguments claim to provide conclusive reasons for a conclusion. With Inductive Argument the conclusion is said to be either probable or improbable. In Deductive Argument, the conclusion follows necessarily or it does not. That is to say, the conclusion is either a necessary consequence of the premises or it is not a necessary consequence of the premises. Another way of stating the same thing: a Deductive Argument consists of a conclusion deduced from premises. The deduction of conclusions from premises is at the heart of logic.
The phrases, necessary consequence and necessary implication, mean necessary inference. The use of one or the other phrase depends on the emphasis. If one stresses that the premises imply a conclusion, one speaks of necessary implication. If one stresses the conclusion resulting from premises, one speaks of necessary consequence. Either phrase points to a claim of necessary inference between premises and conclusion in a Deductive Argument. If the conclusion of a Deductive Argument is a necessary consequence of the premises, then the argument is valid; otherwise, invalid. Using other words: If the premises of a Deductive Argument necessarily imply the conclusion, then the argument is valid; otherwise, invalid.
Logic is the study of the relation between premises and conclusion in Deductive Arguments. If the conclusion follows from premises necessarily (that is, the conclusion is unavoidable), then the argument enjoys valid status; if not (that is, the conclusion can be avoided), then the argument is invalid. By this definition, every deductive argument is either valid or invalid. If a deductive argument is valid and all of its propositions are True, then the argument is said to Sound; otherwise Unsound.