The Logic of Inquiry: Deduction

A deductive argument is also called a “syllogism.” It is an argument that is structured in such a way that, if the terms are valid and if the syllogism is validly constructed, then the conclusion must necessarily be valid. Deductions are “arguments by implication”—the conclusion is already implied in the terms of the argument; all that the argument does is make the conclusion apparent. For example, consider the implied syllogism in Red’s retort to his brother:

Red’s argument is in the form of “Terriers have beards. You are not a terrier. Therefore, you do not have a beard.” (The argument is false, by the way, but we will get to that later.)

A deduction links the initial term to its conclusion through a “middle” term. This is the classical syllogism, of the form “All men are mortal. Socrates is a man. Therefore Socrates is mortal.” There are rules for this linking

· Terminology: “Universal” means a statement of the form “All x are ….” “Particular” means a statement of the form “At least one x is….” (or, in the example above, “Socrates is…”)

· Distribution: The middle term must be used in universal form in at least 1 of the premises, or else the conclusion may not be expressed in universal form. In the example above, “man” is the middle term, linking Socrates (a particular premises) with mortality (a universal premise).

· Negatives: The use of negation must be carefully handled

o May not have 2 negative premises (“All men are not mortal. Socrates is not a man. Therefore…. What?)

o If either premise is negative, the conclusion must be couched in the negative (“All men are not mortal. Socrates is a man. Therefore Socrates is not mortal” or “All men are mortal. Socrates is not a man. Therefore Socrates is not a mortal man.”).

Exploration of the implications of deduction led to the development of “symbolic logic,” which converts the terms of a syllogism to purely symbolic form (as is done in algebra). By the way, sentences in what philosophers of logic call “common language” (what we call “English”) can be translated into symbolic logic, and symbolic logic (through Boolean algebra) can be converted into mathematical expressions. The rules of symbolic logic are few:

Negation: symbolized by a ‘tilde’ (~)
Conjunction: A and B (A∧B)
Disjunction: A or B (A∨B)
Implication: If A, then B (A⇒B) –or, “A implies B”
Material Equivalence: A⇒B if and only if A (A≡B) –or, “A is equivalent to B”

These five operations can be used to generate 9 “Rules of Inference”:

Modus ponens: A implies B; A, so therefore B (A⇒B, A, \B)
Modus tollens: A implies B; not B, therefore not A (A⇒B, ~B, \~A)
Hypothetical syllogism: A implies B; B implies C, therefore A implies C (A⇒B, B⇒C, \A⇒C)
Disjunctive Syllogism: A or B, not A, therefore B (A∨B, ~A, \B)
Constructive Dilemma: A implies B and C implies D, A or C, therefore B or D (A⇒B∧C⇒D, A∨C \B∨D)
Destructive Dilemma: A implies B and C implies D, not B or not D, therefore not A or not C (A⇒B∧C⇒D, ~B∨~D \~A∨~C)
Simplification: A and B, therefore A (A∧B, \A)
Conjunction: A, B, therefore A and B (A, B, \A∧B)
Addition: A, therefore A or B (A, \A∨B)

Finally, from the five operations and the 9 rules of inference, 10 “Rules of Replacement” can be developed:

De Morgan’s Theorem: “Not A and B” is equivalent to “not A or not B” (~[A∧B]≡[~A∨~B])

“Not A or B” is equivalent to “not A and not B” (~[A∨B]≡[~A∧~B])

Commutation: A or B is equivalent to B or A (A∨B≡B∨A)

A and B is equivalent to B and A (A∧B≡B∧A)

Association: A or “B or C” is equivalent to “A or B” or C (A∨[B∨C]≡[A∨B]∨C)

A and “B and C” is equivalent to “A and B” and C (A∧[B∧C]≡[A∧B]∧C)

Distribution: A and “B or C” is equivalent to “A and B” or “A and C” (A∧[B∨C]≡[A∧B] ∨[A∧C])

A or “B and C” is equivalent to “A or B” and “A or C” (A∨[B∧C]≡[A∨B]∧[A∨C])

Double Negation: A is equivalent to not not A (A≡~~A)
Transposition: A implies B is equivalent to not B implies not A (A⇒B≡~B⇒~A)
Material Implication: A implies B is equivalent to not A or B (A⇒B≡~A∨B)
Material Equivalence: “A is equivalent to B” is equivalent to “A implies B and B implies A” ([A≡B]≡[{ A⇒B}∧{B⇒A}])

“A is equivalent to B” is equivalent to “A and B or not A and not B” ([A≡B]≡[{ A∧B}∨{~A∧~B}])

Exportation: “A and B implies C” is equivalent to “A implies that B implies C” ([A∧B]⇒C≡ A⇒[B⇒C])
Tautology: A is equivalent to “A or A” (A≡ A∨A)

A is equivalent to “A and A” (A≡ A∧A)

Using these rules, it is possible to examine the formal structure of any statement or combination of statements to determine their “truth value” (as the logicians like to call it)—in other words, to determine whether the argument is properly structured. To go back to Red’s argument in the cartoon, if he had properly structured the syllogism, it might have been, “All of the animals that have beards are terriers. You are not a terrier. Therefore you do not have a beard.” This is formally true (that is, it is a properly structured syllogism) but empirically false, since not all the animals that have beards are terriers (billy goats, for example, have beards; and so do many college professors). Always keep in mind that a correctly formed syllogism does not guarantee the truth of the conclusion—that also depends on the empirical adequacy of the terms used in the syllogism.

There is a nifty website for reviewing the principles of deductive logic and practicing your skill at catching fallacies. It’s at http://www.duniho.com/fergus/sillysyllogisms.html Have fun!

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