Hypothetical Syllogisms

Hypothetical syllogisms are short, two-premise deductive arguments, in which at least one of the premises is a conditional, the antecedent or consequent of which also appears in the other premise.

I.  “Pure” Hypothetical Syllogisms:

In the pure hypothetical syllogism (abbreviated HS), both of the premises as well as the conclusion are conditionals.  For such a conditional to be valid the antecedent of one premise must match the consequent of the other.  What one may validly conclude, then, is a conditional containing the remaining antecedent as antecedent and the remaining consequent as consequent.  (You might simply think of the middle term – the proposition in common between the two premises – as being cancelled out.)

It’s not hard to visualize the valid hypothetical syllogism.  The following schema illustrate what’s going on:

 If p, then q. If q, then r. (So) If p, then r If p, then not r. If not r, then not q. (So) If p, then not q

Other forms are invalid (unless they can be converted into a valid form by the law of contraposition – see my notes for categorical syllogisms).

II.  “Mixed” Hypothetical Syllogisms:

In mixed hypothetical syllogisms, one of the premises is a conditional while the other serves to register agreement (affirmation) or disagreement (denial) with either the antecedent or consequent of that conditional.  There are thus four possible forms of such syllogisms, two of which are valid, while two of which are invalid.

The VALID forms are:

 (AA) Affirming the Antecedent   or “Modus Ponens” If p, then q. p. q (DC)  Denying the Consequent   or “Modus Tollens” If p, then q. Not q. Not p.

And the INVALID forms (or “pretenders”) are:

 (AC)  Affirming the Consequent (AC) If p, then q. q. p. (DA)  Denying the Antecedent (DA) If p, then q. Not p. Not q.

You will want to remember these rules for validity!!!

You can perhaps see why these forms are valid or invalid by considering a very simple example.  Think of the following four syllogisms:

 1.  Affirming the Antecedent (AA)   If Tweety is a bird, then Tweety flies. Tweety is a bird. Tweety flies 2.  Denying the Antecedent (DA)   If Tweety is a bird, then Tweety flies. Tweety is not a bird. Tweety doesn’t fly. 3.  Affirming the Consequent (AC)   If Tweety is a bird, then Tweety flies. Tweety flies. Tweety is a bird 4.  Denying the Consequent (DC)   If Tweety is a bird, then Tweety flies. Tweety doesn’t fly. Tweety is not a bird.

While syllogisms 1. and 4. above seem to follow logically, it’s clear that 2. and 3. do not, and for precisely the same reason – that there are things that fly other than birds (bats, for instance).  And Tweety might just happen to be one of those.  AA and DC are thus considered valid, while AC and DA are considered invalid.

III.  Exercises:  The following is a list of schematized hypothetical syllogisms.  First, put them into standard form and then determine their validity by identifying their form (HS, AA, AC, DA, or DC)

Examples:

 i. P, if not q. q. Not p. SOLUTION: If not q, then p. q. Not p.   Invalid (DA) ii. P only if Q Whenever Q, not R Not r, given p. SOLUTION: If p, then q. If q, then not r. If p, then not r.   Valid (HS) 1. If p, q. q. p. 2. P only if q. Not p. Not q. 3. Without p, q. Not p. q. 4. P, provided that q. Not p. Not q. 5. If p, then not q. R, unless q. If p, then r. 6. If p, then r. If p, then q. If r, then q. 7. Assuming p, q. p. not q. 8. P if q. Not q. Not p. 9. P only if q. Q only if r. P only if r. 10. P else q. Not q. Not p. 11. P unless q. p. not q. 12. Unless p, q. Not q. p. 13. Only if p, q Not p. Not q 14. Given p, not q. Not q. p. 15. P whenever q. q. p. 16. Not p, should it be q. Not p Q 17. Not p only if q. Whenever q, r. R unless p.

1.  Invalid (AC)

3.  Valid (AA)

5.  Valid (HS)

7.  Invalid (AA [but wrong conclusion!])

9.  Valid (HS)

11.  Invalid (AC)

13.  Valid (DC)

15.  Valid (AA)

17.  Valid (HS)