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.
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 possible forms of such syllogisms, two of which are valid, while two of which are invalid.
The VALID forms are:
I. “Pure” Hypothetical Syllogisms:
(AA) Affirming the Antecedent
In the pure 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, then, is a conditional containing the remaining antecedent as antecedent and the remaining consequent as consequent. consequent. (You might simply think of the middle term – term – the the proposition in common between the two premises – premises – as as being cancelled out.)
If p, then q. p. q
or “Modus Ponens”
(DC) Denying the Consequent
If p, then q. Not q.
It’s not hard to visualize the valid hypothetical syllogism. The following schema illustrate what’s going on:
or “Modus Tollens”
Not p.
And the INVALID forms (or “pretenders”) are: If p, then q.
If p, then not r.
If q, then r.
If not r, then not q.
(So) If p, then r
(So) If p, then not q
(AC) Affirming the Consequent (AC)
If p, then q. q. p.
Other forms are invalid (unless they can be converted into a valid form by the law of contrapositi on – on – see see my notes for categorical syllogisms). (DA) Denying the Antecedent (DA)
If p, then q. Not p.
II. “Mixed” Hypothetical Syllogisms:
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:
i.
P, if not q. q. Not p.
SOLUTION : If not q, then p.
ii.
q. Not p.
1. Affirming the Antecedent (AA)
2. Denying the Antecedent (DA)
P only if Q Wheneve r Q, not R
If q, then not r.
Not r, given p.
If p, then not r.
Invalid (DA) If Tweety is a bird, then Tweety flies.
If Tweety is a bird, then Tweety flies.
Tweety is a bird.
Tweety is not a bird.
Tweety flies
Tweety doesn’t fly.
1.
Valid (HS) 2.
P only if q.
q. Not p. p. Not q. 3.
3. Affirming the Consequent (AC)
If p, q.
4. Denying the Consequent (DC)
Without p, q.
4.
P, provided that q.
Not p. Not p. q. Not q.
If Tweety is a bird, then Tweety flies.
If Tweety is a bird, then Tweety flies.
Tweety flies.
Tweety doesn’t fly.
Tweety is a bird
Tweety is not a bird.
5.
7. 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.
If p, then not q.
6.
If p, then r.
R, unless q.
If p, then q.
If p, then r.
If r, then q.
Assumin g p, q.
8.
P if q. Not q.
p. Not p. not q. 9.
P only if q.
10 .
P else q. Not q.
III. Exercises: The following is a list of schematized hypothetical syllogisms. First, put them into standard form and then determine their validity by identifying thei r form (HS, AA, AC, DA, or DC)
Examples:
Q only if r. P only if r.
SOLUTION : If p, then q.
Not p.
15. Valid (AA) 11 .
13 .
15 .
P unless q.
12 .
Unless p, q.
p.
Not q.
not q.
p.
Only if p, q
14 .
Given p, not q.
Not p.
Not q.
Not q
p.
P whenever q.
16 .
17. Valid (HS)
CHAPTER 29. PRODUCTION.
HYPOTHETICAL SYLLOGISM AND
There are several kinds of deductive argument involving hypothetical propositions or their derivatives. They are distinguished according to whether they involve only hypotheticals, or hypotheticals mixed with categorical forms. The main kinds are syllogism, production, apodosis and dilemma. Note that the valid moods are not here listed in symbolic terms, as we did with categoricals, to avoid obscuring their impact.
Not p, should it be q.
1. Syllogism. q.
Not p
p.
Q
2. Other Derivatives. 3. Production. 17 .
Not p only if q. Wheneve r q, r. R unless p.
Answers to odd exercises:
Hypothetical syllogism is argument whose premises and conclusion are all hypotheticals. It is mediate inference, with minor (symbol P), middle (M), and major (Q) theses, deployed in figures, as was the case in categorical syllogism. Its most primary valid mood , from which all others may be derived by direct or indirect reduction, is as follows. It tells us, as for the analogue in categorical syllogism, that, as H.W.B. Joseph would say, 'w hatever falls under the condition of a rule, follows the rule'.
1. Invalid (AC) 3. Valid (AA) 5. Valid (HS) 7. Invalid (AA [but wrong conclusion!]) 9. Valid (HS)
This primary mood is valid irrespective of whether the hypotheticals involved are of uns pecified base, normal (contingency-based), or abnormal. That is generally true for its primary derivatives, too; but subaltern derivatives are only applicable in cases where both theses are known to be logically contingent (and not just problematic), because the subalterns require eductive processes which depend on this condition for their validity.
11. Invalid (AC) 13. Valid (DC)
If M, then Q
if P, then M
If M, then nonQ if nonP, then M so, if nonP, then nonQ
so if P, then Q
This is a first figure syllogism. Its validity obviously follows from the meaning of the operator 'ifthen' involved. Although the connection in hypotheticality is expressed by modal conjunctive statements, 'if-then' underscores an additional, not-tautologous, sense, occurring on a finer level. This teaches us a purely conjunctive argument, from which many laws for the logic of conjunction may be inferred, that:
If nonM, then nonQ if nonP, then nonM so, if nonP, then nonQ
(ii) Next, from one of the valid, uppercase, perfect moods, we derive the primary, valid, mood, by reductio ad absurdum, as follows. Note that the major premise is uppercase, and the minor premise and conclusion are lowercase.
If M, then Q if P, not-then nonM so, if P, not-then nonQ
contrapose major: deny conclusion: get anti-minor
If nonQ, then nonM if P, then nonQ if P, then nonM
The premises: {M and nonQ} is impossible, and {P and nonM} is impossible, together yield the conclusion: {P and nonQ} is impossible.
This could be written symbolically as , note.
a.
From this primary mood, we can draw up the following full list of valid, lowercase, perfect moods, in the first figure, by substituting antitheses for theses in every possible combination.
If M, then Q if P, not-then nonM so, if P, not-then nonQ
.
If M, then nonQ if P, not-then nonM so, if P, not-then Q (i) From the primary valid mood, we can draw up the following full list of valid, moods, in first figure, by substituting antitheses for theses in every poss ible combination.
If M, then Q if P, then M so, if P, then Q
If M, then nonQ if P, then M so, if P, then nonQ
If M, then Q if nonP, then M so, if nonP, then Q
If nonM, then Q if P, not-then M so, if P, not-then nonQ
If nonM, then nonQ if P, not-then M so, if P, not-then Q
If M, then Q if nonP, not-then nonM so, if nonP, not-then nonQ
If nonM, then Q if nonP, not-then M so, if nonP, not-then nonQ
If M, then nonQ if nonP, not-then nonM so, if nonP, not-then Q
If nonM, then nonQ if nonP, not-then M so, if nonP, not-then Q
If nonM, then Q if P, then nonM so, if P, then Q
If nonM, then nonQ if P, then nonM so, if P, then nonQ
If nonM, then Q if nonP, then nonM so, if nonP, then Q
(iii) Next, from one of the valid, uppercase, perfect moods, we derive the primary, valid, mood, by reductio ad absurdum, as follows. Note the change in polarity of the minor thesis in the conclusion, which defines the moods as imperfect, and the distinct mixed polarity of the middle thesis in the two premises. Note also that the minor premise is uppercase, and the major premise and conclusion are lowercase.
so, if P, not-then nonQ. , not-then Q then nonM f nonP, not-then Q
deny conclusion: contrapose minor: get anti-major:
If nonP, then Q if M, then nonP if M, then Q
From this primary mood, we can draw up the following full list of valid, imperfect moods, in the first figure, by substituting antitheses for theses in every possible combination.
The following sample can be derived from moods of type (i) by obvert-inverting the conclusion, or equally well from moods of type (iii) by replacing the major premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion.
If M, then Q If M, not-then Q if P, then nonM so, if nonP, not-then Q
If nonM, not-then Q if P, then M so, if nonP, not-then Q
if P, then M so, if nonP, not-then Q.
If M, not-then nonQ if P, then nonM so, if nonP, not-then nonQ
If nonM, not-then nonQ if P, then M so, if nonP, not-then nonQ
If M, not-then Q if nonP, then nonM so, if P, not-then Q
If nonM, not-then Q if nonP, then M so, if P, not-then Q
If M, not-then nonQ if nonP, then nonM so, if P, not-then nonQ
If nonM, not-then nonQ if nonP, then M so, if P, not-then nonQ
In summary, we thus have a total of 3X8 = 24 primary valid moods in the first figure, plus 2X8 = 16 subaltern valid moods. Or a total of 40 valid moods, out of 8X8X8 = 512 possibilities.
b.
(iv) . These are valid only with normal hypotheticals, unlike the preceding, because they are derived from the latter by subalternating a lowercase premise or being subalternated by an uppercase conclusion. Their premises are always both uppercase, If Q, then M and their conclusion lowercase. if P, then nonM so, if P, then nonQ The following sample can be derived from moods of type (i) by obverting the conclusion, or equally well from moods of type (ii) by replacing the minor premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature.
.
(i) From one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, mood, of the second figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the major premise, through a valid, uppercase, perfect mood, of the first figure.
with same major: deny conclusion: get anti-minor:
If Q, then M if P, not-then nonQ so, if P, not-then nonM
From this primary, valid mood, we can draw up the following full list of valid, uppercase, perfect moods, in the second figure, by substituting antitheses for theses in every possible combination.
If M, then Q if P, then M
If Q, then M if P, then nonM so, if P, then nonQ
If Q, then nonM if P, then M so, if P, then nonQ
If nonQ, then M if P, then nonM so, if P, then Q
If nonQ, then nonM If nonQ, then M if P, then M if nonP, not-then M so, if P, then Q so, if nonP, not-then nonQ
If Q, then M if nonP, then nonM so, if nonP, then nonQ
If Q, then nonM if nonP, then M so, if nonP, then nonQ
If nonQ, then M if nonP, then nonM so, if nonP, then Q
If nonQ, then nonM if nonP, then M so, if nonP, then Q
(ii) Next, from one of the valid, uppercase, perfect moods, of the first figure, we derive the primary, valid, mood, of the second figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the major premise, through a valid, lowercase, perfect mood, of the first figure. Note that the major premise is uppercase, and the minor premise and conclusion are lowercase.
If nonQ, then nonM if nonP, not-then nonM so, if nonP, not-then nonQ
(iii) . These are valid only with normal hypotheticals, unlike the preceding, because they are derived from the latter by subalternating a lowercase premise or being subalternated by an uppercase conclusion. Their premises are always both uppercase, and their conclusion lowercase. The following sample can be derived from moods of type (i) by obverting the conclusion, or equally well from moods of type (ii) by replacing the minor premise with its obvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature.
If Q, then M if P, then nonM so, if P, not-then Q.
, then M not-then M f P, not-then Q
with same major: deny conclusion: get anti-minor:
If Q, then M if P, then Q if P, then M
From this primary mood, we can draw up the following full list of valid, lowercase, perfect moods, in the second figure, by substituting antitheses for theses in every possible combination.
The following sample can be derived from moods of type (i) by obvert-inverting the conclusion. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion.
If Q, then M if P, then nonM
If Q, then M if P, not-then M so, if P, not-then Q
If Q, then nonM if P, not-then nonM so, if P, not-then Q
If nonQ, then M if P, not-then M so, if P, not-then nonQ
If nonQ, then nonM if P, not-then nonM so, if P, not-then nonQ
If Q, then M if nonP, not-then M so, if nonP, not-then Q
If Q, then nonM if nonP, not-then nonM so, if nonP, not-then Q
so, if nonP, not-then nonQ.
The following sample can be derived from moods of type (ii) by replacing the minor premise with its obvertinvertend. On this basis, 8 subaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion. Note the distinct uniform polarity of the middle thesis in the two premises.
If Q, then M
if P, then M so, if nonP, not-then Q.
If M, not-then Q if M, then nonP so, if nonP, not-then Q
In summary, we thus have a total of 2X8 = 16 primary valid moods in the second figure, plus 3X8 = 24 subaltern valid moods. Or a total of 40 valid moods, out of 8X8X8 = 512 poss ibilities.
c.
(ii) Next, from one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, mood, with premise, of the third figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the minor premise, through a valid, lowercase, perfect mood, of the first figure. The conclusion is of course lowercase.
.
If M, then Q if M, not-then nonP (i) From one of the valid, uppercase, perfect so, if P, not-then nonQ moods, of the first figure, we derive the primary, valid, mood, with premise, of the third figure, by reductio ad absurdum, as follows. Alternatively, we could have used direct reduction, by contraposing the major premise, and transposing, through a valid, lowercase, perfect mood, of the first figure. The conclusion is of course lowercase.
, not-then nonQ , then P f P, not-then nonQ
If nonM, not-then Q if nonM, then nonP so, if nonP, not-then Q
deny conclusion: with same minor: get anti-major:
deny conclusion: with same minor: get anti-major:
If P, then nonQ if M, not-then nonP if M, not-then Q
From this primary, valid mood, we can draw up the following full list of valid, perfect moods, with lowercase minor premise, in the third figure, by substituting antitheses for theses in every possible combination.
If P, then nonQ if M, then P if M, then nonQ If M, then Q if M, not-then nonP so, if P, not-then nonQ
From this primary, valid mood, we can draw up the following full list of valid, perfect moods, with lowercase major premise, in the third figure, by If M, then nonQ substituting antitheses for theses in every possible if M, not-then nonP combination. so, if P, not-then Q
If nonM, then Q if nonM, not-then nonP so, if P, not-then nonQ
If nonM, then nonQ if nonM, not-then nonP so, if P, not-then Q
If M, not-then nonQ if M, then P so, if P, not-then nonQ
If nonM, not-then nonQ If M, then Q if nonM, then P if M, not-then P so, if P, not-then nonQ so, if nonP, not-then nonQ
If nonM, then Q if nonM, not-then P so, if nonP, not-then nonQ
If M, not-then Q if M, then P so, if P, not-then Q
If nonM, not-then Q If M, then nonQ if nonM, then P if M, not-then P so, if P, not-then Q so, if nonP, not-then Q
If nonM, then nonQ if nonM, not-then P so, if nonP, not-then Q
If M, not-then nonQ if M, then nonP so, if nonP, not-then nonQ
If nonM, not-then nonQ if nonM, then nonP so, if nonP, not-then nonQ
(iii) Next, from one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, mood, of the third figure, by direct reduction, as follows. Note the change in polarity of the minor thesis in the conclusion, which defines the mood as
If M, then Q
imperfect, and the distinct mixed polarity of the middle thesis in the two premises. Note also that both premises and the conclusion are uppercase.
If M, then Q if nonM, then P so, if nonP, then Q
with same major: contrapose minor: get conclusion:
if M, then P
From this primary mood, we can draw up the following full list of valid, imperfect moods, in the third figure, by substituting antitheses for theses in every possible combination.
If M, then Q if nonM, then P so, if nonP, then Q
so, if P, not-then
If M, then Q if nonP, then M so, if nonP, then Q
If nonM, then Q if M, then P so, if nonP, then Q
nonQ.
The following sample can be derived from moods of type (i) by replacing the major premise with its obvertinvertend, or equally well from moods of type (iii) by obvert-inverting the conclusion. On this basis, 8 subaltern moods can be derived in the usual manner. These are perfect in nature, but note the distinct mixed polarity of the middle thesis in the two premises.
If M, then Q if nonM, then P so, if P, not-then Q.
If M, then nonQ if nonM, then P so, if nonP, then nonQ
If nonM, then nonQ if M, then P so, if nonP, then nonQ
If M, then Q if nonM, then nonP so, if P, then Q
If nonM, then Q if M, then nonP so, if P, then Q
If M, then nonQ if nonM, then nonP so, if P, then nonQ
If nonM, then nonQ if M, then nonP so, if P, then nonQ
The following sample can be derived from moods of type (ii) by replacing the minor premise with its obvertinvertend, or equally well from moods of type (iii) by obverting the conclusion. On this basis, 8 s ubaltern moods can be derived in the usual manner. These are imperfect, since the minor thesis changes polarity in the conclusion. Note the distinct mixed polarity of the middle thesis in the two premises.
If M, then Q if nonM, then P
(iv) . These are valid only with normal hypotheticals, unlike the preceding, because they are derived from the latter by subalternating a lowercase premise or being subalternated by an uppercase conclusion. Their premises are always both uppercase, and their conclusion lowercase. The following sample can be derived from moods of type (i) by replacing the major premise with its obvertend, or equally well from moods of type (ii) by replacing the minor premise with its obvertend. On this basis, 8 subaltern moods can be derived in t he usual manner. These are perfect in nature.
so, if nonP, not-then nonQ.
In summary, we thus have a total of 3X8 = 24 primary valid moods in the third figure, plus 3X8 = 24 subaltern valid moods. Or a total of 48 valid moods, out of 8X8X8 = 512 possibilities.
d. With regard to the fourth figure , it can be ignored in hypothetical syllogism. Since the first figure here (unlike with categorical syllogism) includes imperfect moods, the fourth figure here would introduce no new valid moods for us. Its valid moods can of c ourse all be reduced directly to the first figure, by transposing or
contraposing the premises, but they do not represent a movement of thought of practical value. We therefore have, in the three significant figures taken together, a total of 24+16+24 = 64 primary valid moods, plus 16+24+24 = 64 subaltern valid moods. Or a total of 128 valid moods, out of 3X512 = 1536 possibilities; meaning a validity rate of 8.33%.
