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# deduction

[dih-duhk-shuhn]

In logic, a type of inference or argument that purports to be valid, where a valid argument is one whose conclusion must be true if its premises are true (see validity). Deduction is thus distinguished from induction, where there is no such presumption. Valid deductive arguments may have false premises, as demonstrated by the example: “All men are mortal; Cleopatra is a man; therefore, Cleopatra is mortal.” Invalid deductive arguments sometimes embody formal fallacies (i.e., errors of reasoning based on the structure of the propositions in the argument); an example is “affirming the consequent”: “If A then B; B; therefore, A” (see fallacy; formal and informal).

The deductive-nomological (or D-N) model is a formalized view of scientific explanation in natural language. It characterizes scientific explanations primarily as deductive arguments with at least one natural law statement among its premises. "Nomological" comes from the Greek word "νόμος" (nomos), i.e., "law."

## Background

The D-N model is known by many names, including the covering law model, the subsumption theory, Hempel's model, the Hempel-Oppenheim model, and the Popper-Hempel model of explanation (Niiniluoto, 1995). Its introduction in the philosophical literature is part of a broad general discussion about the nature of scientific explanation (i.e., what it is, what it should be, etc.).

The D-N model is taught implicitly in schools, and approximates our pre-theoretical conception of science, which many non-experts hold. It was initially formalized by Carl Hempel and Paul Oppenheim in their article Studies in the Logic of Explanation (1948). A sketch of it can be found in Karl Popper's Logic of Scientific Discovery (1959).

## Formalization

The model offers the following account of scientific explanation, where an explanation is set out as a formalized argument:

• Let p be the explanandum - the statement that describes the phenomenon or phenomena to be explained.
• Let s1. . . sn be the explanans - the statements that "explain" the statement P.

In the D-N model, at least one of the statements si must be a "law-like" statement--a problematic concept, but initially thought to be captured by universal affirmatives, i.e., statements of the form "all X are Y." The explanans must be appropriately testable or observable--they must have "empirical content." If the premises are all true and if the argument is deductively valid, then the following constitutes a correct deductive-nomological explanation of p:

s1. . . sn, therefore, p

As a very simple illustration, consider the following: we observe that a piece of chalk falls when released. Why does the chalk fall? A D-N explanation might look like this (without attending to all subtleties in the precisely correct statement of the premises and conclusion):

• Massive objects attract each other with a force proportional to their masses and inversely proportional to the square of their distance apart.
• The chalk and Earth are massive objects.
• Holding the chalk overcomes the force of attraction between it and Earth
• Therefore, the chalk falls when released

The model is positivist in tone and implication, devised as a prescriptive form for scientific explanations. Due to the way that the model eschews any account of causality, scientific modelling, or simplification--and the general rejection of logical positivism--it is no longer accepted as dogma.

• Hempel, Carl G., and Oppenheim, Paul (1948). "Studies in the Logic of Explanation". Philosophy of Science 1948:15, 135-75; reproduced in Hempel, Carl G. (1965). Aspects of Scientific Explanation. New York: Free Press.
• Mayes, Randolph G. (2006). "Theories of Explanation". The Internet Encyclopedia of Philosophy, Fieser & Dowdwn (eds.), http://www.iep.utm.edu/e/explanat.htm
• Niiniluoto, Ilkka (1995). "Covering Law Model". The Cambridge Dictionary of Philosophy, Robert Audi (ed.), New York: Cambridge University Press.
• Popper, Karl. (1959). The Logic of Scientific Discovery. London: Hutchinson.
• Salmon, Wesley (1990) Four Decades of Scientific Explanation, University of Minnesota Press.
• Woodward, James. (2003). "Scientific Explanation". The Stanford Encyclopedia of Philosophy, Edward N. Zalta (ed.), http://plato.stanford.edu/entries/scientific-explanation/