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Logical argument

In philosophy and logic, an argument is a series of statements typically used to persuade someone of something or to present reasons for accepting a conclusion. The general form of an argument in a natural language is that of premises (typically in the form of propositions, statements or sentences) in support of a claim: the conclusion. The structure of some arguments can also be set out in a formal language, and formally defined "arguments" can be made independently of natural language arguments, as in math, logic, and computer science.

In a typical deductive argument, the premises guarantee the truth of the conclusion, while in an inductive argument, they are thought to provide reasons supporting the conclusion's probable truth. The standards for evaluating non-deductive arguments may rest on different or additional criteria than truth, for example, the persuasiveness of so-called "indispensability claims" in transcendental arguments, the quality of hypotheses in retroduction, or even the disclosure of new possibilities for thinking and acting.

The standards and criteria used in evaluating arguments and their forms of reasoning are studied in logic. Ways of formulating arguments effectively are studied in rhetoric (see also: argumentation theory). An argument in a formal language shows the logical form of the symbolically represented or natural language arguments obtained by its interpretations.

Informal arguments as 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 rational structure – the relationship of claims, premises, warrants, relations of implication, and conclusion – is not always spelled out and immediately visible and must sometimes be made explicit by analysis.

Major Premise: Source E is an expert in subject domain S containing proposition A.
Minor Premise: E asserts that proposition A is true (false).
Conclusion: A is true (false).
CQ1: Expertise Question. How credible is E as an expert source?
CQ2: Field Question. Is E an expert in the field that A is in?
CQ3: Opinion Question. What did E assert that implies A?
CQ4: Trustworthiness Question. Is E personally reliable as a source?
CQ5: Consistency Question. Is A consistent with what other experts assert?
CQ6: Backup Evidence Question. Is E's assertion based on evidence?

Tweedy is a bird.
Birds generally fly.
Therefore, Tweedy (probably) flies.
Tweedy is a bird.
Birds generally fly.
Therefore, Tweedy (probably) flies.
Elliptical arguments
  • A deductive argument asserts that the truth of the conclusion is a logical consequence of the premises. Based on the premises, the conclusion follows necessarily (with certainty). For example, given premises that A=B and B=C, then the conclusion follows necessarily that A=C. Deductive arguments are sometimes referred to as "truth-preserving" arguments.
  • A deductive argument is said to be valid or invalid. If one assumes the premises to be true (ignoring their actual truth values), would the conclusion follow with certainty? If yes, the argument is valid. Otherwise, it is invalid. In determining validity, the structure of the argument is essential to the determination, not the actual truth values. For example, consider the argument that because bats can fly (premise=true), and all flying creatures are birds (premise=false), therefore bats are birds (conclusion=false). If we assume the premises are true, the conclusion follows necessarily, and thus it is a valid argument.
  • If a deductive argument is valid and its premises are all true, then it is also referred to as sound. Otherwise, it is unsound, as in the "bats are birds" example.
  • An inductive argument, on the other hand, asserts that the truth of the conclusion is supported to some degree of probability by the premises. For example, given that the U.S. military budget is the largest in the world (premise=true), then it is probable that it will remain so for the next 10 years (conclusion=true). Arguments that involve predictions are inductive, as the future is uncertain.
  • An inductive argument is said to be strong or weak. If the premises of an inductive argument are assumed true, is it probable the conclusion is also true? If so, the argument is strong. Otherwise, it is weak.
  • A strong argument is said to be cogent if it has all true premises. Otherwise, the argument is uncogent. The military budget argument example above is a strong, cogent argument.
  • 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 logicians are tiresome; therefore, some Greeks are tiresome. Invalid argument: the tiresome logicians might all be Romans (for example).
  • 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 this does not mean the conclusion has to be true; it is only true if the premises are true, which they may not be!)
  • Some men are hawkers. Some hawkers are rich. Therefore, some men are rich. Invalid argument. This can be easier seen by giving a counter-example with the same argument form:
    • Some people are herbivores. Some herbivores are zebras. Therefore, some people are zebras. Invalid argument, as it is possible that the premises be true and the conclusion false.
  • Some people are herbivores. Some herbivores are zebras. Therefore, some people are zebras. Invalid argument, as it is possible that the premises be true and the conclusion false.
  • People often are not themselves clear on whether they are arguing for or explaining something.
  • The same types of words and phrases are used in presenting explanations and arguments.
  • The terms 'explain' or 'explanation,' et cetera are frequently used in arguments.
  • Explanations are often used within arguments and presented so as to serve as arguments.
  • Likewise, "...arguments are essential to the process of justifying the validity of any explanation as there are often multiple explanations for any given phenomenon."
  • Shaw, Warren Choate (1922). The Art of Debate. Allyn and Bacon. p. 74. 
  • Robert 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.
  • Vincent F. Hendricks, Thought 2 Talk: A Crash Course in Reflection and Expression, New York: Automatic Press / VIP, 2005,
  • 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.
  • Ch. Perelman and L. Olbrechts-Tyteca, The New Rhetoric, Notre Dame, 1970. This classic was originally published in French in 1958.
  • Henri Poincaré, Science and Hypothesis, Dover Publications, 1952
  • Frans 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. S. Stebbing, A Modern Introduction 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.
  • Walton, Douglas; Christopher Reed; Fabrizio Macagno, Argumentation Schemes, New York: Cambridge University Press, 2008.
  • Carlos Chesñevar, Ana Maguitman and Ronald Loui, Logical Models of Argument, ACM Computing Surveys, vol. 32, num. 4, pp. 337–383, 2000.
  • T. Edward Damer. Attacking Faulty Reasoning, 5th Edition, Wadsworth, 2005.
  • Charles Arthur Willard, A Theory of Argumentation. 1989.
  • Charles Arthur Willard, Argumentation and the Social Grounds of Knowledge. 1982.
  • Salmon, Wesley C. Logic. New Jersey: Prentice-Hall (1963). Library of Congress Catalog Card no. 63-10528.
  • Aristotle, Prior and Posterior Analytics. Ed. and trans. John Warrington. London: Dent (1964)
  • Mates, Benson. Elementary Logic. New York: OUP (1972). Library of Congress Catalog Card no. 74-166004.
  • Mendelson, Elliot. Introduction to Mathematical Logic. New York: Van Nostran Reinholds Company (1964).
  • Frege, Gottlob. The Foundations of Arithmetic. Evanston, IL: Northwestern University Press (1980).
  • Martin, Brian. The Controversy Manual (Sparsnäs, Sweden: Irene Publishing, 2014).


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