Logical argument

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An argument is an attempt to demonstrate the truth of an assertion called a conclusion, based on the truth of a set of assertions called premises. The process of demonstration in an argument, presumes some kind of communication, which could be part of a written text, a speech or a conversation. Arguments can be valid or invalid, although how arguments are determined to be in either of these two categories can often itself be an object of much discussion. Informally one should expect that a valid argument should be compelling in the sense that it is capable of convincing someone about the truth of the conclusion. This validity criterion, however, is inadequate or even misleading since it depends more on the skill of the person constructing the argument to manipulate the person who is being convinced and less on the argument itself. Less subjective criteria for validity of arguments are clearly desirable, and in some cases we should even expect an argument to be rigorous, that is adhere to precise rules of validity. This is the case for arguments used in mathematical proofs. Note that a rigorous proof does not have to be a formal proof.

In ordinary language, people refer to the logic of an argument or use terminology that suggests that an argument is based on logic. Though arguments do use inferences that are indisputably purely logical (such as syllogisms), other kinds of inferences are almost always used in practical arguments. For example, arguments commonly deal with causality, probability and statistics or even specialized areas such as economics.

Argument validity

In evaluating an argument, we consider separately the validity of the premises and the validity of the logical relationships between the premises, any intermediate assertions and the conclusion. The main logical property of an argument that is of concern to us here is whether it is validity preserving, that is if the premises are valid, then so is the conclusion. We will usually abbreviate this property by saying simply that argument is valid. Moreover, in this article we use the term validity of an assertion instead of truth of that assertion, since we regard validity as being dependent on the interpretation of the terms. In other words an assertion may be be valid in one interpretation of its constituent terms, but invalid in another. This is particularly useful in evaluating moral or legal arguments.

If the argument is valid, the premises together entail or imply the conclusion. The ways in which arguments go wrong fall into certain patterns, called logical fallacies. Validity is a semantic characteristic of arguments; independently of this property, and more controversially, arguments should also be scrutinizable, in the sense that the argument be open to public examination and systematic in the sense that the structural components of the argument have public legitimacy.

Arguments are commonly used in mathematics, the physical and social sciences, in philosophy and in law whenever a systematic, scrutinizable method is required for demonstrating the validity of new assertions from other assertions whose validity is assumed and are given as premises.

The mathematical paradigm

In mathematics, an argument can be formalized using symbolic logic. In that case, an argument is seen as an ordered list of statements, each one of which is either one of the premises or derivable from the combination of some subset of the preceding statements and one or more axioms using rules of inference. The last statement in the list is the conclusion. Most arguments used in mathematical proof are rigorous, but not formal. In fact, strictly formal proofs of all but the most trivial assertions are extremely hard to construct and hard to understand without some assistance from a computer. One of the goals of automated theorem proving is to design computer programs to produce and check formal proofs. A study of formal systems of mathematics together with semantic questions such as completeness and validity is often called metamathematics. Of particular note in this direction are the [[G�del's incompleteness theorem]]s for first order theories of arithmetic.

In general usage, however, arguments are rarely formal or even have the rigor of mathematical proofs. The logical relationship between the premises and the conclusion may not be explicitly stated, and sometimes the conclusion itself is left to the reader to supply.

We will assume in this article that valid arguments in mathematics are those that can be recognized as being in principle formalizable in the encompassing formal theory. This is the prevalent belief of mathematical authors. It follows that the theory of valid arguments in mathematics is reducible to the theory of valid inferences in formal mathematical theories. A theory of validity of formal mathematical theories posits two distinct elements: syntax and semantics. Syntax gives the rules for when a formula is correctly constructed. Semantics is essentially a function from formulas to truth values. An expression is said to be valid if the semantic function assigns the value true to it. A rule of inference is valid if and only if it is validity-preserving. An argument is valid if and only if it utilizes valid rules of inference.

Note that in the case of mathematical semantics, both the syntax and semantics are mathematical objects.

Theories of arguments

Theories of arguments are closely related to theories of informal logic. Ideally, a theory of argument should provide some mechanism for explaining validity of arguments.

One natural approach would follow the mathematical paradigm and attempt to define validity in terms of semantics of the assertions in the argument. Though such an approach is appealing in its simplicity, the obstacles to proceeding this way are very difficult for anything other than purely logical arguments. Among other problems, we need to interpret not only entire sentences, but also components of sentences, for example noun phrases such as The best professional soccer player in Rio or The present value of government revenue for the next twelve years.

One major difficulty of pursuing this approach is that determining an appropriate semantic domain is not an easy task, raising numerous thorny ontological issues. It also raises the discouraging prospect of having to work out acceptable semantic theories before being able to say anything useful about understanding and evaluating arguments. For this reason the purely semantic approach is usually replaced with other approaches that are more easily applicable to practical discourse.

For specialized areas of research, some of the semantic problems can be conveniently shoved under the rug if we can avail ourselves of an abstract model. In this case, we can establish a limited semantic interpretation using the terms of the model and the validity of the argument is reduced to that of the abstract model. This kind of reduction is used in the natural sciences generally, and would be particularly helpful in arguing about social issues if the parties can agree on a model. Unfortunately, this prior reduction seldom occurs, with the result that arguments about social policy rarely have a satisfactory resolution.

Another approach is to develop a theory of argument pragmatics, at least in certain cases where argument and social interaction are closely related. This is most useful when the goal of logical argument is to establish a mutually satisfactory resolution of a difference of opinion between individuals.

Argumentative dialogue

Arguments as discussed in the preceding paragraphs are static, such as one might find in a textbook or research article. They serve as a published record of justification for an assertion. Arguments can also be interactive, in which the proposer and the interlocutor have a more symmetrical relationship. The premises are discussed, as well the validity of the intermediate inferences. For example, consider the following exchange, illustrated by the No true Scotsman fallacy:

Argument: "No Scotsman puts sugar on his porridge."

Reply: "But my friend Angus likes sugar with his porridge."

Rebuttal: "Ah yes, but no true Scotsman puts sugar on his porridge."

In this dialogue, the proposer first offers a premise, the premise is challenged by the interlocutor, and finally the proposer offers a modification of the premise. This exchange could be part of a larger discussion, for example a murder trial, in which the defendant is a Scotsman, and it had been established earlier that the murderer was eating sugared porridge when he or she committed the murder.

In argumentative dialogue, the rules of interaction may be negotiated by the parties to the dialogue, although in many cases the rules are already determined by social mores. In the most symmetrical case, argumentative dialogue can be regarded as a process of discovery more than one of justification of a conclusion. Ideally, the goal of argumentative dialogue is for participants to arrive jointly at a conclusion by mutually accepted inferences. In some cases however, the validity of the conclusion is secondary: Emotional outlet, scoring points with an audience, wearing down an opponent, lowering the sale price of an item may be the actual goals of the dialogue. Walton distinguishes several types of argumentative dialogue which illustrate these various goals:

Personal quarrel.
Forensic debate.
Persuasion dialogue.
Bargaining dialogue.
Action seeking dialogue.
Educational dialogue.

Van Eemeren and Grootendorst identify various stages of argumentative dialogue. These stages can be regarded as an argument protocol. In a somewhat loose interpretation, the stages are as follows:

Confrontation: Presentation of the problem, such as a debate question or a political disagreement
Opening: Agreement on rules, such as for example, how evidence is to be presented, which sources of facts are to be used, how to handle divergent interpretations, determination of closing conditions.
Argumentation: Application of logical principles according to the agreed-upon rules
Closing: This occurs when the termination conditions are met. Among these could be for example, a time limitation or the determination of an arbiter.

Van Eemeren and Grootendorst provide a detailed list of rules that must be applied at each stage of the protocol. Moreover, in the account of argumentation given by these authors, there are specified roles of protagonist and antagonist in the protocol which are determined by the conditions which set up the need for argument.

It should be noted that many cases of argument are highly unsymmetrical, although in some sense they are dialogues. A particularly important case of this is political argument.

Much of the recent work on argument theory has considered argumention as an integral part of language and perhaps the most important function of language (Grice, Searle, Austin, Popper). This tendency has removed argumentation theory away from the realm of pure formal logic.

Other theories

In recent decades one of the more influential discussions of philosophical arguments is that by Nicholas Rescher in his book The Strife of Systems. Rescher models philosophical problems on what he calls aporia or an aporetic cluster: a set of statements, each of which has initial plausibility but which are jointly inconsistent. The only way to solve the problem, then, is to reject one of the statements. If this is correct, it constrains how philosophical arguments are formulated.

References

Rober Audi, Epistemology, Routledge, 1998. Particularly relevant is Chapter 6, which explores the relationship between knowledge, inference and argument.
J. L. Austin How to Do things with Words, Oxford University Press, 1976.
H. P. Grice, Logic and Conversation in The Logic of Grammar, Dickenson, 1975.
R. A. DeMillo, R. J. Lipton and A. J. Perlis, Social Processes and Proofs of Theorems and Programs, Communications of the ACM, Vol. 22, No. 5, 1979. A classic article on the social process of acceptance of proofs in mathematics.
Yu Manin, A Course in Mathematical Logic, Springer Verlag, 1977. A mathematical view of logic. This book is different from most books on mathematical logic in that it emphasizes the mathematics of logic, as opposed to the formal structure of logic.
Henri Poincar�, Science and Hypothesis, Dover Publications, 1952
Franz van Eemeren and Rob Grootendorst, Speech Acts in Argumentative Discussions, Foris Publications, 1984.
K. R. Popper Objective Knowledge; An Evolutionary Approach, Oxford: Clarendon Press, 1972.
L. Stebbing, A Modern Introdcution to Logic, Methuen and Co., 1948. An account of logic that covers the classic topics of logic and argument while carefully considering modern developments in logic.
Douglas Walton, Informal Logic: A Handbook for Critical Argumentation, Cambridge, 1998