Part I: Lesson

1. Key Definitions (Review)

• Hypothetical Proposition: a type of compound proposition in which two propositions are linked through the assertion that the truth of the antecedent implies the truth of the consequent.
  • The form of this proposition is “If p, then q”.
  • Ex: “If it snows in New York, then the temperature in New York is less than 32 degrees Celsius”.

• Antecedent: the initial proposition of the pair which appears in the hypothetical.
  • In the above example, the antecedent is “it snows in New York”.

• Consequent: the final proposition of the pair which appear in the hypothetical
  • In the above example, the antecedent is “the temperature in New York is less than 32 degrees Fahrenheit”.

2. Modus Ponens and Modus Tollens for Hypothetical Propositions:

(i) Modus Ponens: A rule of inference which states that if a hypothetical, “If p, then q”, is true, and the antecedent “p” is true, then the consequent, “q” is true. In variable notation, the rule appears in the form: 

1. If p, then q
2. p
3. Therefore, q

(ii) Modus Tollens: A rule of inference which states that if a hypothetical, “If p, then q”, is true, and the consequent “q” is false, then the antecedent “p” is false. In variable notation, the rule appears in the form:

1. If p, then q
2. not-q
3. Therefore, not p

Explanatory Note: The hypothetical informs us that the truth of p implies the truth of q. This being the case, if p isn’t true, then q can’t be true either, due to the fact that the hypothetical links the truth of p to that of q (Modus Ponens). On the other hand, if q isn’t true, since the truth of p ensures the truth of q, then p can’t be true (Modus Tollens).

3. Modus Ponens and Modus Tollens again, but with Categorical Propositions

(i) Modus Ponens: This rule occurs when (a) every member of a certain category A is a member of another category B, and (b) that some thing(s) C is/are member(s) of category A. It follows that C is/are member(s) of category B.

1. All A’s are B’s
2. C is an A / All C’s are A’s
3. Therefore, C is a B / All C’s are B’s

(ii) Modus Tollens: This rule occurs when (a) every member of a certain category A is a member of another category B, and (b) that some thing(s) C is/are not members of category B. It follows that C is/are not member(s) of category A.

1. All A’s are B’s
2. C is not a B / No C’s are B’s
3. Therefore, C is not an A / No C’s are A’s.

It becomes clear that these versions of Modus Ponens and Modus Tollens are quite similar to the ones introduced in section 2. “All A’s are B’s,” means “If something is an A, then it is a B.” For Modus Ponens, if something specific, C, is an A, it follows that C is a B. We can also visualize these with Euler circles. Here is Modus Ponens:

As one can see, all A’s are B’s here, and all C’s are A’s. It follows that all C’s are B’s! Now, here is Modus Tollens:

Here, we see that all A’s are B’s, and that no C’s are B’s. It follows that no C’s are A’s!