In logic, an argument is a set of one or more declarative sentences (or "propositions") known as the premises along with another declarative sentence (or "proposition") known as the conclusion. A deductive argument asserts that the truth of the conclusion is a logical consequence of the premises; an inductive argument asserts that the truth of the conclusion is supported by the premises.
Each premise and the conclusion are only either true or false, not ambiguous. The sentences composing an argument are referred to as being either true or false, not as being valid or invalid; arguments are referred to as being valid or invalid, not as being true or false. Some 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. Whichever term is used, each premise and the conclusion must be capable of being true or false and nothing else: they are truthbearers.
Informal arguments are studied in informal logic, are presented in ordinary language and are intended for everyday discourse. Conversely, formal arguments are studied in formal logic (historically called symbolic logic, more commonly referred to as mathematical logic today) and are expressed in a formal language. Informal logic may be said to emphasize the study of argumentation, whereas formal logic emphasizes implication and inference. Informal arguments are sometimes implicit. That is, the logical structurethe relationship of claims, premises, warrants, relations of implication, and conclusionis not always spelled out and immediately visible and must sometimes be made explicit by analysis.
A deductive argument is one which, if valid, has a conclusion that is entailed by its premises. In other words, the truth of the conclusion is a logical consequence of the premises--if the premises are true, then the conclusion must be true. It would be self-contradictory to assert the premises and deny the conclusion, because the negation of the conclusion is contradictory to the truth of the premises.
Arguments may be either valid or invalid. If an argument is valid, and its premises are true, the conclusion must be true: a valid argument cannot have true premises and a false conclusion.
The validity of an argument depends, however, not on the actual truth or falsity of its premises and conclusions, but solely on whether or not the argument has a valid logical form. The validity of an argument is not a guarantee of the truth of its conclusion. A valid argument may have false premises and a false conclusion.
Logic seeks to discover the valid forms, the forms that make arguments valid arguments. An argument form is valid if and only if all arguments of that form are valid. Since the validity of an argument depends on its form, an argument can be shown to be invalid by showing that its form is invalid, and this can be done by giving another argument of the same form that has true premises but a false conclusion. In informal logic this is called a counter argument.
The form of argument can be shown by the use of symbols. For each argument form, there is a corresponding statement form, called 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 a 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.
The corresponding conditional, of a valid argument is a necessary truth (true in all possible worlds) and so we might say that the conclusion necessarily follows from the premises, or follows of logical necessity. The conclusion of a valid argument is not necessarily true, it depends on whether the premises are true. The conclusion of a valid argument need not be a necessary truth: if it were so, it would be so independently of the premises.
For example: Some Greeks are logicians, therefore some logicians are Greeks: Valid argument; it would be self-contradictory to admit that some Greeks are logicians but deny that some (any) logicans are Greeks.
All Greeks are human and All humans are mortal therefore All Greeks are mortal. : Valid argument; if the premises are true the conclusion must be true.
Some Greeks are logicians and some logician are tiresome therefore some Greeks are tiresome. Invalid argument: the tiresome logicians might all be Romans!
Either we are all doomed or we are all saved; we are not all saved therefore we are all doomed. Valid argument; the premises entail the conclusion. (Remember that does not mean the conclusion has to be true, only if the premisses are true, and perhaps they are not, perhaps some people are saved and some people are doomed, and perhaps some neither saved nor doomed!)
Arguments can be invalid for a variety of reasons. There are well-established patterns of reasoning that render arguments that follow them invalid; these patterns are known as logical fallacies.
A sound argument is a valid argument with true premises. A sound argument, being both valid and having true premises, must have a true conclusion. Some authors (especially in earlier literature) use the term sound as synonymous with valid.
Inductive logic is the process of reasoning in which the premises of an argument are believed to support the conclusion but do not entail it. Induction is a form of reasoning that makes generalizations based on individual instances.
Mathematical induction should not be misconstrued as a form of inductive reasoning, which is considered non-rigorous in mathematics. (See Problem of induction.) In spite of the name, mathematical induction is a form of deductive reasoning and is fully rigorous.
An argument is cogent if and only if the truth of the argument's premises would render the truth of the conclusion probable (i.e., the argument is strong), and the argument's premises are, in fact, true. Cogency can be considered inductive logic's analogue to deductive logic's "soundness."
A fallacy is an invalid argument that appears valid, or a valid argument with disguised assumptions. 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 and 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.
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. Example: 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. Example: A witness reasoned: Nobody came out the front door except the milkman therefore the murderer must have left by the back door. (Hidden assumption- the milkman was not the murderer).
Whereas formal arguments are static, such as one might find in a textbook or research article, argumentative dialogue is dynamic. It serves as a published record of justification for an assertion. Arguments can also be interactive, with the proposer and the interlocutor having a symmetrical relationship. The premises are discussed, as well the validity of the intermediate inferences.
Dialectic is controversy, that is, the exchange of arguments and counter-arguments respectively advocating propositions. The outcome of the exercise might not simply be the refutation of one of the relevant points of view, but a synthesis or combination of the opposing assertions, or at least a qualitative transformation in the direction of the dialogue.
Argumentation theory, (or argumentation) embraces the arts and sciences of civil debate, dialogue, conversation, and persuasion. It studies rules of inference, logic, and procedural rules in both artificial and real world settings. Argumentation is concerned primarily with reaching conclusions through logical reasoning, that is, claims based on premises.
Statements are put forward as arguments in all disciplines and all walks of life. Logic is concerned with what consititutes an argument and what are the forms of valid arguments in all interpretations and hence in all disciplines, the subject matter being irrelevant. There are not different valid forms of argument in different subjects.
Arguments as they appear in science and mathematics (and other subjects) do not usually follow strict proof precedures; typically they are elliptical arguments (q.v.) and the rules of inference are implicit rather than explicit. An argument can be loosely said to be valid if it can be shown that, with the supply of the missing premises it has a valid argument form and demonstrateable by an accepted proof procedure.
The basis of mathematical truth has been the subject of long debate. Frege in particular sought to demonstrate (see Gottlob Frege, The Foundations of Arithemetic, 1884, and Logicism in Philosophy of mathematics) that arithmetical truths can be derived from purely logical axioms and therefore are, in the end, logical truths. The project was developed by Russell and Whitehead in their Principia Mathematica. If an argument can be cast in the form of sentences in Symbolic Logic, then it can be tested by the application of accepted proof procedures. This has been carried out for Arithemetic using Peano axioms. Be that as it may, an argument in Mathematics, as in any other discipline, can be considered valid just in case it can be shown to be of a form such that it cannot have true premises and a false conclusion.
Legal arguments (or oral arguments) are spoken presentations to a judge or appellate court by a lawyer (or parties when representing themselves) of the legal reasons why they should prevail. Oral argument at the appellate level accompanies written briefs, which also advance the argument of each party in the legal dispute. A closing argument (or summation) is the concluding statement of each party's counsel (often called an attorney in the United States) reiterating the important arguments for the trier of fact, often the jury, in a court case. A closing argument occurs after the presentation of evidence.
A political argument is an instance of a logical argument applied to politics. Political arguments are used by academics, media pundits, candidates for political office and government officials. Political arguments are also used by citizens in ordinary interactions to comment about and understand political events.
More on Arguments:
Wesley C Salmon, Logic, Prentice-Hall, New Jersey 1963 (Library of Congress Catalog Card no. 63-10528)
More on Logic:
Aristotle, Prior and Posterior Analytics, ed. and trans. John Warrington, Dent: London (everyman Library) 1964
Benson Mates, Elementary Logic, OUP, New York 1
972 (Library of Congress Catalog Card no.74-166004)
Elliot Mendelson, Introduction to Mathematical Logic,, Van Nostran Reinholds Company, New York 1964
More on Logic and Maths:
1884. Die Grundlagen der Arithmetik: eine logisch-mathematische Untersuchung über den Begriff der Zahl. Breslau: W. Koebner. Translation: J. L. Austin, 1974. The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number, 2nd ed. Blackwell. Gottlob Frege, The Foundations of Arithmetic: A logico-mathematical enquiry into the concept of number, 1884, trans Jacquette, Pearson Longman, 2007