Argument (Logic)

This article is about arguments in deductive logic.

In Logic, an argument consists of a set of premises, and a conclusion in which it is asserted that the truth of the conclusion follows from (is entailed by) the premises. Each premise (there must be at least one) of an argument and its conclusion must be capable of being true or false and nothing else. In the literature authors refer to the  premises and conclusion using the terms declarative sentence, statement, proposition, sentence, or even indicative utterance. The reason for the variety is concern about the ontological significance of the terms, proposition in particular, but this is the concern of Philosophy of Logic rather than Logic itself. Whichever term is used, each premise and the conclusion must be capable of being true or false and nothing else: they are ‘truth-bearers’. In this article the term statement will be used

Valid Arguments
A valid argument is one in which the premises cannot be true and the conclusion false. It has been long recognised that the validity of an argument depends solely on its form.. The form of argument can be shown by the use of symbols. An argument-form is valid if and only if all arguments of its form are valid. For each argument-form, there is a corresponding statement-form, a corresponding conditional, and an argument-form is valid if and only its corresponding conditional is a logical truth. (A statement-form which is logically true is also said to be valid statement-form. A statement-form is a logical truth if it is true under all interpretations. A statement-form can be shown to be a logical truth by either (a) showing that it is a tautology or (b) by means of a proof procedure. (Every tautology is a necessary truth but not every necessary truth is a tautology).

Logic is concerned with the validity of an argument, not with the truth or falsity of its premises or conclusion. (Some authors however describe a sound argument as a valid argument with true premises; a sound argument being both valid and having true premises must have a true conclusion. Other authors use the term sound as synonymous with valid)

Each premise (there must be at least one) of an argument and its conclusion is a statement, capable of being true or false and nothing else. (Some authors use the term proposition or sentence instead of statement, but which ever term is used, each premise and the conclusion must be capable of being true or false and nothing else.)

Non arguments
First the premises and the conclusion must be statements, capable of being true and false. Secondly it must be asserted that the conclusion follows from the premises. In English the words "therefore", "so", “because” amd “hence” typically separate the premises from the conclusion of an argument, but this is not necessarily so. Thus: "Socrates is a man, all men are mortal therefore Socrates is mortal" is clearly an argument (a valid one at that), because it is clear it is asserted that that "Socrates is mortal" follows from the preceding statements. However "I was thirsty and therefore I drank" is NOT an argument, despite its appearance. It is not being claimed that "I drank" is logically entailed by "I was thirsty". The "therefore" in this sentence indicates "for that reason" not "it follows that".

Validity
We are pychologically tempted to accept an argument as valid if we believe its conclusion, and to reject an argument as invalid if we disagree with its conclusion. (No doubt we are all keen to have our opinions proven and therefore inclind to accept as valid any argument which endorses them. Similarly we hate to think that the other fellow might be right, and are inclined to reject as invalid any arguments that suppport views contrary to our own.)   It should be carefully noted that there is nothing to stop a valid argument having a false conclusion; this may be the case if the premises are false. Nor is the anything to stop an invalid argument from having a true conclusion. In fact all combinations are possible accept one: it is not possible to have a valid argument with true premises and a false conclusion.

Fallacies
A fallacy is an invalid argument that appear valid, or a valid argument with disguised assumptions.

Elliptical Arguments
Often an argument is invalid because there is a missing premise the supply of which would make it valid. Speakers and writers will often leave out a strictly necessary premise in their reasonings if it is widely accepted and the writer does not wish to state the blindingly obvious. Eg. “Iron is a metal therefore it will expand when heated.” (Missing premise: all metals expand when heated). On the other hand a seemingly valid argument may be found to lack a premise – a ‘hidden assumption’ – which if highlighted can show a fault in reasoning. Eg. A witness reasoned: “Nobody came out the front door accept the milkman therefore the murderer must have left by the back door.” (Hidden assumption- the milkman was not the murderer).