Inference rules

As we have seen, many arguments have a form (logical structure) that can be represented using truth-functional connectives. Some argument forms are used so frequently in human reasoning that it is worthwhile to memorize them. Memorizing a few simple forms can enable you to evaluate the validity of an argument quickly, in your head, without writing anything down. Familiarity with these common forms also makes it easier to construct complex arguments and proofs, as we’ll see later in this chapter.

An inference rule is a valid argument form used as a sort of template for evaluating arguments or constructing formal proofs. An inference rule can be represented using lowercase letters as variables (place-holders or “blanks”) into which we can insert any well-formed formula (WFF). Any argument that can be constructed by replacing each variable in an inference rule with a WFF is an instance of that inference rule.

Any valid argument form could be used as an inference rule. For now, we’ll focus on four of the most commonly employed inference rules:

• Modus ponens (MP), Latin for “method of affirming,” is a valid argument form in which one premise is a conditional and the other premise affirms the antecedent of that conditional. The conclusion is the consequent of the conditional:

  (x ⊃ y)

  x

  ∴ y

  Here is an instance of modus ponens, replacing the variables x and y with the formulas ‘A’ and ‘(B • C)’, respectively:

  (A ⊃ (B • C))

  A

  ∴ (B • C)

  Here’s another instance of MP, where x = (P • ~Q) and y = ((R ≡ ~S) ∨ ~R):

  ((P • ~Q) ⊃ ((R ≡ ~S) ∨ ~R))

  (P • ~Q)

  ∴ ((R ≡ ~S) ∨ ~R)

  Since modus ponens is a valid argument form, both of the above instances are also valid. If we recognize any argument as an instance of MP, we don’t even have to test it for validity: we already know it must be valid.


• Modus tollens (MT), Latin for “method of denying,” also has a conditional premise. Its other premise denies the consequent of that conditional—in other words, the other premise is the negation of the consequent—and the conclusion is the negation of the antecedent:

  (x ⊃ y)

  ~y

  ∴ ~x

  Here is an instance of modus tollens, where x = (P ∨ Q) and y = ~R:

  ((P ∨ Q) ⊃ ~R)

  ~~R

  ∴ ~(P ∨ Q)



• Disjunctive syllogism (DS) is so called because its longest premise is a disjunction (and the word “syllogism” derives from an ancient Greek word meaning “inference” or “conclusion”). The other premise of a disjunctive syllogism negates one of the two disjuncts, and the conclusion is the other disjunct:

  (x ∨ y)

  ~x

  ∴ y

  Alternatively:

  (x ∨ y)

  ~y

  ∴ x

  Here is an instance of disjunctive syllogism, where x = (A ⊃ B) and y = C:

  ((A ⊃ B) ∨ C)

  ~(A ⊃ B)

  ∴ C

  Here’s another instance of disjunctive syllogism, replacing the variables x and y with the same formulas used in the previous example. This time, however, the second disjunct is negated instead of the first:

  ((A ⊃ B) ∨ C)

  ~C

  ∴ (A ⊃ B)



• Hypothetical syllogism (HS) involves two conditional (or “hypothetical”) premises, and the conclusion is also a conditional:

  (x ⊃ y)

  (y ⊃ z)

  ∴ (x ⊃ z)

  Here’s an instance of hypothetical syllogism, where x = ~A, y = (B • C), and z = D:

  (~A ⊃ (B • C))

  ((B • C) ⊃ D)

  ∴ (~A ⊃ D)

All of the instances of an inference rule are valid, since the form (logical structure) of the rule guarantees that the conclusion must be true whenever the premises are true. So, if you recognize an argument as an instance MP, MT, DS, or HS, you immediately know that it is valid, without bothering to test it with truth tables or the truth assignment method.

Here are two tips to help you recognize instances of the above rules more easily:

1. Focus on the main connectives in each premise and conclusion, and imagine replacing its component propositions with variables like x and y.
2. The order of the premises doesn’t matter; they can be switched around. So, for example, the first modus ponens argument given above is still an instance of MP when the order of its premises is switched, like this:

   A

   (A ⊃ (B • C))

   ∴ (B • C)