{"id":4841,"date":"2021-10-19T11:29:37","date_gmt":"2021-10-19T09:29:37","guid":{"rendered":"http:\/\/icar.cnrs.fr\/dicoplantin\/?p=4841"},"modified":"2025-04-01T07:25:02","modified_gmt":"2025-04-01T05:25:02","slug":"deduction-e","status":"publish","type":"post","link":"https:\/\/icar.cnrs.fr\/dicoplantin\/deduction-e\/","title":{"rendered":"Deduction"},"content":{"rendered":"

DEDUCTION<\/span><\/h1>\n

1. In ordinary language<\/span><\/h2>\n

In ordinary language, the word deduction<\/em> is homonymous. As a derived of to deduct<\/em>, deduction means \u201csubtraction\u201d, and does not directly concern argumentation. As a derived of to deduce<\/em>, it can be used as an umbrella term, to refer to any kind of argumentation, that is, the derivation of a conclusion from a set of data taken as premises. Deductions are presented by the arguer to the other participants as valid<\/em> and sound<\/em>.<\/p>\n

The well-known Holmesian \u201cdeductive method\u201d proceeds as follows:<\/p>\n

-Watson visits Sherlock Holmes.
\n<\/em>\u2018In practice again, I observe. You did not tell me you intended to go into harness.\u2019<\/span>
\n\u2018Then how do you know?<\/span>
\n\u2018I see it, I deduce it. How do I know that you have been getting yourself very wet lately, and that you have a most clumsy and careless servant girl?\u2019<\/span>
\n\u2018My dear Holmes, this is too much. You would certainly have been burned, had you lived a few centuries ago. It is true that I had a country walk on Thursday and came home in a dreadful mess, but I have changed my clothes I can’t imagine how you deduce it. As to Mary Jane, she is incorrigible, and my wife has given her notice; but there again, I fail to see how you work it out.\u2019<\/span>
\nHe chuckled to himself and rubbed his long, nervous hands together.<\/span>
\n\u2018It is simplicity itself,\u201d said he, \u201cmy eyes tell me that on the inside of your left shoe, just where the firelight strikes it, the leather is scored by six almost parallel cuts. Obviously they have been caused by someone who has very carelessly scraped round the edges of the sole in order to remove crusted mud from it. Hence, you see my double deduction that you had been out in vile weather, and that you had a particularly malignant boot-slitting specimen of the London slavery.\u2019<\/span>
\nArthur Conan Doyle, Adventures of Sherlock Holmes \u2014 Scandal in Bohemia<\/em>, 1891[1]<\/a>.<\/span><\/p>\n

This \u00ab\u00a0deduction\u00a0\u00bb seems to correspond to an argument from natural signs<\/a>, or if considered as the derivation of an explanatory hypotheses, to an abductive argument, more than to a logical deduction.<\/a><\/p>\n

2. In Cartesian philosophy<\/span><\/h2>\n

A deduction is a series of operations which, according to valid<\/em> rules, links a set of true<\/em> premises (axioms, true propositions) to a conclusion<\/p>\n

Many things are known although not self-evident, so long as they are deduced from principles known to be true by a continuous and uninterrupted movement of thought, with clear intuition of each point. (Descartes [1628], Rule III).<\/span><\/p>\n

In this sense, a well-conducted deduction is a demonstration<\/em>, that produces apodictic<\/em> (incontrovertible) knowledge, defined as \u201cany necessary conclusion from other things known with certainty\u201d (ibid<\/em>.).<\/p>\n

Valid and sound syllogistic reasoning is a type of deductive reasoning, that is sometimes used as a reference for valid argumentation. Argumentation developing the definition of a word<\/a> and its implications, or the various forms of argument from the absurd<\/a>, are examples of deductions in natural language.<\/p>\n

3.\u00a0In logic<\/span><\/h2>\n

According to Kleene, a proof<\/em> is based on axioms<\/em>, while a deduction<\/em> is based on hypotheses<\/em>:<\/p>\n

The proof of theorems, or the deduction of consequences of assumptions, in mathematics typically proceeds \u00e0 la Euclid, by putting sentences in a list called a \u201cproof\u201d or \u201cdeduction\u201d. We use the word \u201cproof\u201d\u00a0(and call the assumptions \u201caxioms\u201d) when the assumptions have a permanent status for a theory under consideration, \u201cdeduction\u201d when we are not thinking of them as permanent\u201d (1967, \u00a79, Proof theory: provability and deducibility, p. 33)<\/span><\/p>\n

In logic, \u201ca (formal) proof (in the propositional calculus)\u201d is defined as \u201ca finite list of (occurrences of) formulas B1<\/sub><\/strong>\u2026\u2026Bl<\/sub> <\/strong>such as each of which is an axiom of the propositional calculus, or comes by the \u2283\u2013rule from a pair of formulas preceding in the list\u201d (id<\/em>. p. 34).<\/p>\n

The \u2283\u2013rule is \u201cthe modus ponens or rule of detachment\u201d, defined as \u201cthe operation of passing from two formulas of the respective form A<\/strong> and A<\/strong> \u2283 B<\/strong> to the formula B<\/strong>, for any choice of A<\/strong> and B<\/strong> [\u2026]. In an inference by this rule, the formulas A<\/strong> and A<\/strong> \u2283 B<\/strong> are the premises and B<\/strong> is the conclusion\u201d (ibid<\/em>.).<\/p>\n

3.1\u00a0Validity and Soundness<\/span><\/h3>\n

Under such a definition, a deduction is taken to be a valid<\/em> and sound<\/em> deduction. Now, a series of propositions can be advanced by a speaker as a valid and sound deduction without actually being so.
\nTo be valid, the deduction must be carried out according to the laws of (a well-defined system of) logic. For example, the inference\/deduction from a false proposition to a true one \u201cP<\/strong>(F) \u2192 Q<\/strong>(T)\u201d is valid, but not sound: to be sound, the deduction must start from axioms or, more generally, from true propositions.<\/p>\n

The implication<\/em> (conditional) is a binary logical connective<\/a>. A deduction<\/em> is a chain of operations connecting well-formed expressions by a rule. For example, the rule of modus ponens<\/em> (\u2283\u2013rule, see above) makes it possible to deduce \u201cB<\/strong>\u201d from the two premises \u201cA <\/strong>\u2192<\/strong>B<\/strong>\u201d and \u201cA<\/strong>\u201d (hypothetical syllogism), by a three-step deduction:<\/p>\n

A \u202f\u2192\u00a0B<\/span>
\nA<\/span>
\nso<\/em>, B<\/span><\/p>\n

The same reasoning can be expressed as an implication which expresses a logical law:<\/p>\n

\u201cIf the implication is true and the antecedent is true, then the consequent is true<\/em>\u201d<\/span>
\n[(A\u00a0\u2192\u00a0B) & A]\u00a0\u2192\u00a0B<\/span><\/p>\n

Let’s consider a true conditional \u201cR<\/strong> \u2192 W<\/strong>\u201d, \u201cIf it rains, the lawn is wet<\/em>\u201d:
\nW<\/strong> is a necessary<\/em> condition for R<\/strong>; R<\/strong> is a sufficient<\/em> condition for W<\/strong>.<\/p>\n

3.2 If a sufficient condition for W is satisfied, then W<\/span><\/h3>\n

If the antecedent of a true conditional is true, then its consequent is true.<\/p>\n\n\n\n\n\n
\u00a0R<\/strong> \u2192\u00a0W<\/strong><\/td>\nR<\/strong> is a sufficient condition for W<\/strong><\/td>\nIf it rains, the grass is wet<\/em><\/td>\n<\/tr>\n
\u00a0R<\/strong><\/td>\nthis sufficient condition is met<\/td>\nIt is raining<\/em><\/td>\n<\/tr>\n
\u00a0so, W<\/strong><\/td>\nso,<\/em> W<\/strong> is met<\/td>\nso, the grass is wet<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

This rule is based on the affirmation of the antecedent<\/em> of a true implication. It is also known as the modus (ponendo) ponens <\/em>rule: the deduction posits (ponendo<\/em>) the truth of the antecedent R, <\/strong>in order to affirm (ponens<\/em>) the truth of the consequent W<\/strong>.<\/p>\n

The idea of sufficient condition is also expressed as:<\/p>\n

not-(A & not-B)<\/span><\/p>\n

In the ordinary world and in natural language, a situation in which it could rain without the grass getting wet is unthinkable.<\/p>\n

3.3 If a necessary condition for R is not true, then R is not true<\/span><\/h3>\n

If the consequent of a true conditional is not true, then its antecedent is not true.<\/p>\n\n\n\n\n\n
R<\/strong> \u2192\u00a0W<\/strong><\/td>\nW<\/strong> is a necessary condition for R<\/strong><\/td>\nIf it rains, the grass is wet<\/em><\/td>\n<\/tr>\n
not-W<\/strong><\/td>\nthis sufficient condition is not met<\/td>\nThe grass is not wet<\/em><\/td>\n<\/tr>\n
so,<\/em> not-R<\/strong><\/td>\nso,<\/em> R<\/strong> is not met<\/td>\nSo, <\/em>it is not raining<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

This rule is based on the negation of the consequent<\/em> of a true implication, also known as the modus (tollendo) tollens<\/em> rule, the mode that, by denying (the consequent), denies (the antecedent).<\/p>\n

All reasoning from natural signs involves this kind of deduction.<\/p>\n

4. Paralogisms of deduction<\/span><\/h2>\n

4.1\u00a0Denying the antecedent<\/span><\/h3>\n

It is not possible to deny the existence of a phenomenon on the basis of the absence of a sufficient condition for the given phenomenon. The following deduction is invalid:<\/p>\n\n\n\n\n\n
\u00a0R<\/strong> \u2192\u00a0W<\/strong><\/td>\nR<\/strong> is a sufficient condition for W<\/strong><\/td>\nIf it rains, the lawn is wet<\/em><\/td>\n<\/tr>\n
\u00a0not-R<\/strong><\/td>\nthis sufficient condition is not met<\/td>\nIt does not rain<\/em><\/td>\n<\/tr>\n
\u00a0*so,<\/em> not-W<\/strong><\/td>\n*so,<\/em> W<\/strong> is not met<\/td>\n*So, <\/em>the lawn is not wet<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Raining, a sufficient condition for the lawn to be wet, was incorrectly assumed to be necessary.<\/p>\n

4.2\u00a0Affirming the consequent<\/span><\/h3>\n

It is not possible to infer the existence of a phenomenon from the existence of a necessary condition of that phenomenon. The following deduction is invalid:<\/p>\n\n\n\n\n\n
\u00a0R<\/strong> \u2192\u00a0W<\/strong><\/td>\n\u00a0W<\/strong> is a necessary condition for R<\/strong><\/td>\nIf it rains, the lawn is wet<\/em><\/td>\n<\/tr>\n
\u00a0W<\/strong><\/td>\n\u00a0this necessary condition is satisfied<\/td>\nThe lawn is wet<\/em><\/td>\n<\/tr>\n
\u00a0*so,<\/em> R<\/strong><\/td>\n\u00a0*so,<\/em> R is met<\/strong><\/td>\n*So, <\/em>it is raining<\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

Observing that the grass is wet is not a sufficient basis to conclude that it is raining.<\/p>\n

5. Pragmatics of deduction<\/span><\/h2>\n

The rules of deduction are defined within the framework of a logical system in which all the components of the argument are explicit and well defined.<\/p>\n

Ordinary situations are different; ordinary reasoning is not about formal systems, but about causes and effects in the empirical world, see Causality<\/a>. This world is represented by the body of shared knowledge; it follows that only relevant knowledge needs to be made explicit.<\/p>\n

Suppose that the lawn might be wet because it has rained, because the lawn has been watered, because a pipe has leaked, or simply because of a heavy dew. If it is contextually obvious that the lawn has not been watered (I know what I have done), that there is no leak (for the simple reason that there is no water pipe in the garden), and that there is no dew (at this time of the day), then I can safely say that if the grass is wet, it is because it has rained, or is raining.<\/p>\n

Only the superficial form of reasoning is fallacious. Full evaluation must take the context into account and reconstruct the argument explicitly, case by case, thereby eliminating the other sufficient conditions and transforming the latter into a necessary and sufficient condition. This is a direct application of Grice’s cooperation principle.<\/a><\/p>\n

\n

[1]<\/a> Quoted from Arthur Conan Doyle, The Penguin Complete Sherlock Holmes<\/em>. London: Penguin Books, 1981. P. 162.<\/span><\/p>\n

\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

DEDUCTION 1. In ordinary language In ordinary language, the word deduction is homonymous. As a derived of to deduct, deduction means \u201csubtraction\u201d, and does not directly concern argumentation. As a derived of to deduce, it can be used as an umbrella term, to refer to any kind of argumentation, that is, the derivation of a […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-4841","post","type-post","status-publish","format-standard","hentry","category-non-classe"],"_links":{"self":[{"href":"https:\/\/icar.cnrs.fr\/dicoplantin\/wp-json\/wp\/v2\/posts\/4841","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/icar.cnrs.fr\/dicoplantin\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/icar.cnrs.fr\/dicoplantin\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/icar.cnrs.fr\/dicoplantin\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/icar.cnrs.fr\/dicoplantin\/wp-json\/wp\/v2\/comments?post=4841"}],"version-history":[{"count":14,"href":"https:\/\/icar.cnrs.fr\/dicoplantin\/wp-json\/wp\/v2\/posts\/4841\/revisions"}],"predecessor-version":[{"id":13903,"href":"https:\/\/icar.cnrs.fr\/dicoplantin\/wp-json\/wp\/v2\/posts\/4841\/revisions\/13903"}],"wp:attachment":[{"href":"https:\/\/icar.cnrs.fr\/dicoplantin\/wp-json\/wp\/v2\/media?parent=4841"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/icar.cnrs.fr\/dicoplantin\/wp-json\/wp\/v2\/categories?post=4841"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/icar.cnrs.fr\/dicoplantin\/wp-json\/wp\/v2\/tags?post=4841"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}