1. In ordinary language
In ordinary language, the word deduction is homonymous. As a derivative of to deduct, deduction means “subtraction”, and does not directly concern argumentation. As a derivative of to deduce, it can be used as an umbrella term, to refer to any kind of argumentation, that is of derivation of a conclusion from a set of data taken as premises. Deductions are given as valid and sound by the arguer to the other participants.
The well-known Holmesian “deductive method” proceeds as follows:
-Watson visits Sherlock Holmes.
‘In practice again, I observe. You did not tell me you intended to go into harness.’
‘Then how do you know?
‘I 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?’
‘My 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.’
He chuckled to himself and rubbed his long, nervous hands together.
‘It is simplicity itself,” said he, “my 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 slavey.’
Arthur Conan Doyle, Adventures of Sherlock Holmes — Scandal in Bohemia, 1891.
The reasoning seems to correspond to an argument from natural sign, or if considered as the derivation of an explanatory hypotheses, to an abductive argument, more than to a logical deduction.
2. In Cartesian philosophy
A deduction is a series of operations linking, according to valid rules, a set of true premises (axioms, true propositions) to a conclusion
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 , Rule III).
In this sense, a well-led deduction is a demonstration, producing apodictic (incontestable) knowledge, defined as “any necessary conclusion from other things known with certainty” (ibid.).
Valid and sound syllogistic reasoning is a kind of deductive reasoning, sometimes taken as the reference for valid argumentation. Argumentation developing the definition of a word and its implications, or the various forms of argumentation from the absurd, are examples of deductions in natural language.
3. In logic
According to Kleene, a proof is based on axioms, while a deduction is based on hypothesis:
The proof of theorems, or the deduction of consequences of assumptions, in mathematics typically proceeds à la Euclid, by putting sentences in a list called a “proof” or “deduction”. We use the word “proof” (and call the assumptions “axioms”) when the assumptions have a permanent status for a theory under consideration, “deduction” when we are not thinking of them as permanent” (1967, §9, Proof theory: provability and deducibility, p. 33)
In logic, “a (formal) proof (in the propositional calculus)” is defined as “a finite list of (occurrences of) formulas B1……Bl such as each of which is an axiom of the propositional calculus, or comes by the ⊃–rule from a pair of formulas preceding in the list” (id. p. 34).
The ⊃–rule is “the modus ponens or rule of detachment”, defined as “the operation of passing from two formulas of the respective form A and A ⊃ B to the formula B, for any choice of A and B […]. In an inference by this rule, the formulas A and A ⊃ B are the premises and B is the conclusion” (ibid.).
3.1 Validity and Soundness
Under such a definition, deduction is taken as a valid and sound deduction. Now, a string of propositions can be advanced by as speaker as a valid and sound deduction without being really so.
To be valid, the deduction has to be led according to the laws of (a well-defined system of) logic. For example, the inference from a false proposition to a true one “P(F) → Q(T)” is valid, but not sound: to be sound, the reasoning has to start from axioms or, generally speaking, from true propositions.
The implication (conditional) is a binary logical connective. A deduction is a chain of operations linking well-formed expressions by means of a rule. For example, the rule of modus ponens (⊃–rule, cf. supra) makes it possible to deduce “B” from the two premises “A →B” and “A” (hypothetical syllogism), by a three-step deduction:
A → B
The same reasoning can be expressed as an implication expressing a logical law, S. Connective:
“If the implication is true and the antecedent true, then the consequent is true”
[(A → B) & A] → B
Let’s consider a true conditional “R → W”, “If it rains, the lawn is wet”.
W is a necessary condition for R; R is a sufficient condition for W.
3.2 If a sufficient condition for W is met, then W
If the antecedent of a true conditional is true, then its consequent is true.
|R → W||R is a sufficient condition for W||If it rains, the grass is wet|
|R||this sufficient condition is met||It is raining|
|so W||so W is met||so the grass is wet|
This rule proceeds from the affirmation of the antecedent of a true implication. It is also known as the modus (ponendo) ponens rule: the deduction poses (ponendo) the truth of the antecedent R, in order to affirm (ponens) the truth of the consequent W.
The idea of sufficient condition is also expressed as:
not-(A & not-B)
In the ordinary world and natural language, a situation in which it might rain without the grass becoming wet is unthinkable.
3.3 If a necessary condition for R is not met, then R is not met
If the consequent of a true conditional is not true, then its antecedent is not true.
|R → W||W is a necessary condition for R||If it rains, the grass is wet|
|not-W||this sufficient condition is not met||The grass is not wet|
|so not-R||so R is not met||So it is not raining|
This rule proceeds from the negation of the consequent of a true implication, also known as the modus (tollendo) tollens rule, the mode that, by denying (the consequent), denies (the antecedent).
All reasoning from natural signs involves this kind of deduction.
4. Paralogisms of deduction
4.1 Denying the antecedent
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:
|R → W||R is a sufficient condition for W||If it rains, the lawn is wet|
|not-R||this sufficient condition is not met||It does not rain|
|*so not-W||*so W is not met||*So the lawn is not wet|
Raining, a sufficient condition for the grass to be wet, has been incorrectly considered as necessary.
4.2 Affirming the consequent
It is not possible to infer the existence a phenomenon in view of the prevalence of a necessary condition of this phenomenon. The following deduction is invalid:
|R → W||W is a necessary condition for R||If it rains, the lawn is wet|
|W||this necessary condition is met||The lawn is wet|
|*so R||*so R is met||*So it is raining|
To find that the grass is wet is not a sufficient basis to conclude that it is raining.
5. Pragmatic of deduction
The rules of deduction are defined within the framework of a logical system in which all the components of reasoning are explicit and well defined.
Ordinary situations are different; in particular, and ordinary reasoning only makes relevant knowledge explicit. Let us suppose that the lawn could be wet because it has rained, because the lawn has been watered, because a pipe has leaked, or due simply to a heavy dew. If it is contextually evident that the lawn has not been watered (I know what I have done), that there is no water leaking (for the simple reason that there is no water pipe in the garden), and there is no dew (at that time of the day), then I can safely say that if the grass is wet, it is because it rained, or is raining.
Only the superficial form of reasoning is fallacious. Full evaluation must take the context into account and re-build the argument explicitly, on a case-by-case basis thereby eliminating the other sufficient conditions, transforming the latter into a necessary and sufficient condition. This is a direct application of Grice’s cooperation principle.
 Quoted after Arthur Conan Doyle, The Penguin Complete Sherlock Holmes. London: Penguin Books, 1981. P. 162.