Comments FUTURE LOGIC © Avi Sion, 1990 (Rev. ed. 1996) All rights reserved. CHAP HAPTER 29 29.
HYPOTHETICAL SY SYLLOGISM AN AND PR PRODUCTION. ON.
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. 1. 2. 3. 1.
Syllogism. Other Derivatives. Production. Syllogism.
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, 'whatever falls under the condition of a rule, follows the rule'. This primary mood is valid irrespective of whether the hypotheticals involved are of unspecified 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. If M, then Q if P, then M so if P, then Q This is a first figure syllogism. Its validity obviously follows from the meaning of the operator 'if-then' 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: 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 1/H2nH2nH2n, note. a.
Figure One.
(i) From the primary valid mood, we can draw up the following full list of valid, uppercase, perfect moods, in first figure, by substituting antitheses for theses in every possible combination. If M, then Q if P, then M so, if P, then Q
If nonM, then Q if P, then nonM so, if P, then Q
If M, then nonQ if P, then M so, if P, then nonQ
If nonM, then nonQ if P, then nonM so, if P, then nonQ
If M, then Q if nonP, then M so, if nonP, then Q
If nonM, then Q if nonP, then nonM so, if nonP, then Q
If M, then nonQ if nonP, then M so, if nonP, then nonQ
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, lowercase, perfect 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
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 nonM, then Q if P, not-then M so, if P, not-then nonQ
If M, then nonQ if P, not-then nonM so, if P, not-then Q
If nonM, then nonQ if P, not-then M so, if P, not-then Q
If M, then Q if nonP, not-then nonM
If nonM, then Q if nonP, not-then M
so, if nonP, not-then nonQ
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
(iii) Next, from one of the valid, uppercase, perfect moods, we derive the primary, valid, imperfect 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. If M, not-then Q if P, then nonM so, if 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. 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 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
(iv) Subaltern moods. 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 M, then Q
if P, then M so, if P, not-then nonQ. 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 P, then M so, if nonP, not-then Q. 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.
Figure Two.
(i) From one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, uppercase, perfect 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. If Q, then M if P, then nonM so, if P, then nonQ
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 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 P, then M so, if P, then Q
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
If nonQ, then nonM if nonP, then M
so, if nonP, then Q
so, if nonP, then Q
(ii) Next, from one of the valid, uppercase, perfect moods, of the first figure, we derive the primary, valid, lowercase, perfect 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 Q, then M if P, not-then M so, if 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. 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
If nonQ, then M if nonP, not-then M so, if nonP, not-then nonQ
If nonQ, then nonM if nonP, not-then nonM so, if nonP, not-then nonQ
(iii) Subaltern moods. 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.
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 so, if nonP, not-then nonQ. The following sample can be derived from moods of type (ii) by replacing the minor premise with its obvert-invertend. On this basis, 8 subaltern moods c an 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. 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 possibilities. c.
Figure Three.
(i) From one of the valid, uppercase, perfect moods, of the first figure, we derive the primary, valid, perfect mood, with lowercase major 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. If M, not-then nonQ if M, then P so, if P, not-then nonQ
deny conclusion: with same minor: get anti-major:
If P, then nonQ if M, then P if M, 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 substituting antitheses for theses in every possible combination. If M, not-then nonQ if M, then P so, if P, not-then nonQ
If nonM, not-then nonQ if nonM, then P so, if P, not-then nonQ
If M, not-then Q if M, then P so, if P, not-then Q
If nonM, not-then Q if nonM, then P so, if P, not-then Q
If M, not-then nonQ
If nonM, not-then nonQ
if M, then nonP so, if nonP, not-then nonQ
if nonM, then nonP so, if nonP, not-then nonQ
If M, not-then Q if M, then nonP so, if nonP, not-then Q
If nonM, not-then Q if nonM, then nonP so, if nonP, not-then Q
(ii) Next, from one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, perfect mood, with lowercase minor 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 so, if P, not-then nonQ
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 M, then Q if M, not-then nonP so, if P, not-then nonQ
If nonM, then Q if nonM, not-then nonP so, if P, not-then nonQ
If M, then nonQ if M, not-then nonP so, if P, not-then Q
If nonM, then nonQ if nonM, not-then nonP so, if P, not-then Q
If M, then Q if M, not-then P so, if nonP, not-then nonQ
If nonM, then Q if nonM, not-then P so, if nonP, not-then nonQ
If M, then nonQ if M, not-then P so, if nonP, not-then Q
If nonM, then nonQ if nonM, not-then P so, if nonP, not-then Q
(iii) Next, from one of the valid, lowercase, perfect moods, of the first figure, we derive the primary, valid, imperfect mood, of the third figure, by direct reduction, as follows. Note the change in polarity of the minor thesis in the conclusion, which d efines the mood as 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 Q if nonP, then M so, if nonP, then Q
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
If nonM, then Q if M, then P so, if nonP, 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
(iv) Subaltern moods. 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 the usual manner. These are perfect in nature. If M, then Q if M, then P so, if P, not-then nonQ. The following sample can be derived from moods of type (i) by replacing the major premise with its obvert-invertend, or equally well from moods of type (iii) by obvert-inverting the conclusion. On this basis, 8 subaltern mood s 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. The following sample can be derived from moods of type (ii) by replacing the minor premise with its obvert-invertend, or equally well from moods of type (iii) by
obverting 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. Note the distinct mixed polarity of the middle thesis in the two premises. If M, then Q if nonM, then P 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 course 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%. 2.
Other Derivatives.
The chaining of syllogisms into a series forming a sorites is possible with hypothetical syllogism, similarly to categorical syllogism. This is used in practise, of course, and applies irrespective of basis. The typical sorites looks as follows: If A, then B if B, then C … if G, then H therefore, if A, then H. Note that we are in the figure one, and we state the most minor premise first, and successively work up to the most major premise, an d lastly the conclusion. A sorites should be reducible to valid syllogisms to be valid. Of course, sorites is only the most regular form of continuous argument, the easiest to think without aid of paper and pencil. More broadly, any succession of premises, in any combination of figures, yielding a valid final conclusion, may be viewed as continuous, even though we have to think out the intermediate conclusions, zigzagging from figure to figure, to reach the result. We can readily reformulate all the above syllogisms using derivative forms, such as simple disjunctions. For examples, the following arguments, taken at random, are easily validated by transforming the disjunctives into standard hypotheticals:
M and/or Q P or else M P or else nonQ
Q or else M P not and/or nonM P not and/or Q.
Here again, I would not regard these as distinct valid moods. Even if they are used in practise, we are mentally required to restate them in 'If/then' form to understand them. It will however be seen, in the context of dilemma, that there are certain arguments, which mix 'If/then' forms with disjunctives, which are comprehensible on their own merit, and used in everyday discourse. Such arguments may also be regarded as 'logical compositions'. With multiple alternatives, the possible number of arguments increases and so does the mental confusion. When translating the given disjunctions into 'If-then' statements causes us as much confusion, the best course is to express each proposition in terms of the conjunctions is allows and forbids; then we can best see what conclusion, if any, may be drawn. We can also, it is noted, appeal to the above valid moods of the syllogism to clarify reasoning involving compound forms. That is, when one or both premises signifies implicance or subalternation or contradiction or contrariety or subcontrariety, we may be able to fuse the results of two or more simple syllogisms, and get a compound conclusion. Lastly, arguments may be fashioned in conditional frameworks, so that we have nested hypotheticals for premise(s) and conclusion. This may be viewed as a wider logic, concerning composite antecedents or consequents, conjunctive or even disjunctive ones. Researching the mechanics of partial or alternative theses is an area that deserves eventual attention, but presumably it can be reduced to the findings of unconditional logic. Subaltern moods are implicitly conditional; they have as hidden premises, the categorical propositions that the theses are logically contingent, rather than merely problematic or partly or wholly incontingent. The tacitly understood premises are: 'P (and nonP) is contingent, and Q (and nonQ) is contingent'. I have made no effort to develop subaltern moods with abnormal bases, because once a thesis is known to be incontingent it is rarely thereafter used in hypothetical propositions. 3.
Production.
How are hypothetical propositions produced? By their very nature they do not presuppose the reality of their theses, so how do we know that the antecedent does (or does not) engage the consequence? This question will be answered in this section. Hypothetical propositions signify a logical connection between the theses, so that any argument which is logically valid may be recast in hypothetical form. The theses involved may of course have any form, including themselves hypothetical. The term 'connection' here is to be understood in its widest sense, including any logical relationship, positive or negative, normal o r abnormal. Thus, all oppositions, eductions, deductions, are included here; overall, a valid inference of any kind produces a positive hypothetical, an invalid inference produces a negative hypothetical.
Also, the expression 'logically valid' should be taken as comprehensive of the known and the unknown; there is no presumption here that the science of logic as we know it to date is complete. It is important to stress this; while all established logical truths are capable of producing hypotheticals, it does not follow that hypotheticals cannot be produced by means not yet clarified by this science. No claim to o mniscience is required. An example of production would be recasting a categorical syllogism in hypothetical form: e.g. 'If all S are M and all M are P, then all S are P'. This is a conclusion, whose premises are the process of validation of that mood of the syllogism via the laws of logic. If we instead produced the briefer conclusion 'If all S are M, all S are P', the process to be valid must have included, after the above, a nesting (to 'If all M are P, then if all S are M, all S are P') and an apodosis (with minor premise 'All M are P'). Thus enthymeme need not be viewed as merely syllogism with a suppressed (tacit) premise, but as the end product of a series of definite arguments. However, production is not limited to relationships in terms of variables, but is especially useful for application to specific values. Using a formal relationship as major premise, we may, through the act of substitution as minor premise, produce a hypothetical with particular contents as conclusion. Continuing the above example, we might for instance produce, 'If all men were wise, they would not make war'. In short, any logical series which is incomplete, may be made to at least yield a hypothetical conclusion, and thus constitute a productive process. The missing information may simply be the exact quantity involved. Thus, if in the above example we do not know whether all or only some S are M, we can still conclude from 'All M are P' that 'If any S is M, it is P'. This produces a hypothetical proposition which seems general, but in fact only suggests that some S may be M. Incidentally, the expression 'whether' may itself be viewed as a derivative form of hypothetical, concealing a dilemma. Similarly, a negative hypothetical would express a nonsequitur. For example, 'If no S are M and all M are P, it does not follow that no S are P'. Likewise, with particular contents or indefinite quantities, as above. Clearly, the possibilities are virtually infinite. Any formal or informal sequence permitted or forbidden by the laws of logic constitutes a productive process. Ordinarily, a hypothetical would not be formed, unless information was missing or already known wrong, and only problematic elements would be included in it as theses; but there is nothing illicit in forming one even with definite theses of known truth.
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10 Hypothetical Syllogism - Presentation Transcript
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The Hypothetical Syllogism Hypothetical Syllogism is a syllogism that has a hypothetical proposition as one of its premise Kinds of Hypothetical Syllogism: 1. Conditional Syllogism (“If…, then…”) 2. Disjunctive Syllogism (“Either…, or…”) 3. Conjunctive Syllogism (“Not both…, and…”) Relationship of an Antecedent and its consequent Note: a. An antecedent is false when only one premise is false, as well as when both premises are false. b. Where the sequence is invalid, there is, strictly speaking, no sequence, antecedent, or consequent at all. (When the sequence is invalid, the apparent premises and conclusion are not related to one another.) Relationship of an Antecedent to its Consequent 1. If the antecedent is true and sequence valid, the consequent is true. (A particularized statement of the principle of contradiction.) 2. If the antecedent is true and sequence invalid, the consequent is doubtful. Every dog is an animal ; but no cat is a dog; therefore, no cat is an animal . 3. If the antecedent is false and sequence valid, the consequent is doubtful. Every dog is an animal; but every cat is a dog ; therefore, every cat is an animal. 4. If the antecedent is false and sequence invalid, the consequent is doubtful. Every cat is a monkey ; but no cat is a dog; therefore, no dog is an monkey . Relationship of a Consequent to its Antecedent 1. If the consequent is false and the sequence valid, the antecedent is false. ( Only truth can flow from truth, every antecedent from which a false statement can flow must itself be false. ) 2. If the consequent is false and the sequence invalid, the antecedent is doubtful. ( When the sequence is invalid, anything can come after anything, since the consequent and the antecedent are not related to one another at all .) Every cat is a dog ; but no cat is a terrier; therefore, no terrier is a dog . 3. If the consequent is true and sequence valid, the antecedent is doubtful. Squares have three sides ; but triangles are squares ; therefore, triangles have three sides. 4. If the consequent is true and the sequence invalid, the antecedent is doubtful. ( If the antecedent of a true consequent is doubtful even when the sequence is valid, it is also doubtful when the sequence is invalid .) The Basic Laws which serve as basis of Valid Inference 1. If the antecedent is true and the sequence valid, the consequent is true. 2. If the consequent is false and the sequence valid, the antecedent is false. CONDITIONAL SYLLOGISM A Conditional Syllogism is one whose major premise is a conditional proposition. 2 Types of Conditional Syllogism: 1. Mixed Conditional (the minor premise is a categorical proposition) 2. Purely Conditional (both of whose premises are conditional propositions) Conditional Propositions is a compound proposition of which one member (the “then” clause) asserts something as true on the c ondition that the other member (the “if” clause) is true. “ If it is raining, the roof is wet.” - The “if” clause or its equivalent is called the antecedent . - The “then” clause or its equivalent is called the c onsequent . Rules of the Mixed Conditional Syllogism: 1. If the antecedent is true and the sequence valid, the consequent is true. Procedure: 1. Posit the antecedent in the minor premise and posit the consequent in the conclusion. 2. If the consequent is false and the sequence valid, the antecedent is false. 2. Sublate the consequent in the minor premise and sublate the antecedent in the conclusion. Example of a Valid Form Conditional Syllogism: Major Premise “ If your have acute appendicitis, you are very sick.” Conclusion Posit the Consequent “ Therefore you are v ery sick.” Sublate the Antecedent “ Therefore you do not have acute appendicitis” Minor Premise Posit the Antecedent “ But you have acute appendicitis.” Sublate the Consequent “ But you are not sick.” Example of a Invalid Form Conditional Syllogism: Major Premise “ If your have acute appendicitis, you are very sick.” Conclusion Posit the Antecedent “ Therefore you have acute appendicitis.” Sublate the Consequent “ Therefore you are not very sick.” Minor Premise P osit the Consequent “ But you are very sick.” Sublate the Antecedent “ But you do not have acute appendicitis.” Purely Conditional Syllogism The Purely Conditional Syllogism, which has conditional propositions for both its premises, has exactly the same forms and the same rules as the mixed conditional syllogism except that the condition expressed in the m inor premise must be retained in the conclusion. If A is a B, then C is a D; but if X is a Y, then A is a B; therefore, if X is a Y, then C is a D. Exercise: Indicate the form, or procedure, illustrated by each of the following, and state whether the example is valid or invalid. If the dentist is not skillful, he will cause his patient much pain; but the dentist is skillful;
o
therefore he will not cause his patient much pain.
2. If this book possesses literary merit, it will be widely read; but it will surely be a best seller; therefore it must possess literary merit. 15. 3. “If you have bad eyes, you will never make the team.” “ But my eyes are all right; therefore you must admit that I will make the team.” 4. If materialism is true, you would expect an intimate connection between the condition of a man’s brain and his powers of thinking; but there is such connection; therefore materialism must be true. 5. If that bill passes, rents will rise; but the bill will not pass; therefore rents will not rise. 16. DISJUNCTIVE SYLLOGISM A Disjunctive Syllogism is one whose major premise is a disjunctive proposition, whose minor premise sublates (or posits) one or more members of the major premise, and whose conclusion posits (or sublates) the other member or members. A Disjunctive Syllogism is one that presents various alternatives and asserts that an indeterminate one of them is true. It consists of two or more members joined by the conjunctions “either … or…”. It is s ometimes called an alternative proposition . 17. 2 Kinds of Disjunctive Syllogism: 1. Strict Disjunctive (only one member is true and the others are false. If all the members except one are false, the remaining member must be true; and if one is true, the remaining members must be false). 2. Broad Disjunctive (at least one member is true but more than one may be true). 18. Rules for Disjunctive Syllogism: 1. If the minor premise posits one or more members of the major premise, the conclusion must sublate each of the other members. It is either raining or not raining ; but it is raining ; therefore it is not not raining . It is either raining or not raining ; but it is not raining ; therefore it is not raining . 19. 2. If the minor premise sublates one or more of the members of the major premise, the conclusion posits the remaining members, one of which must be true. If more than one member remains, the conclusion must be a disjunctive in the strict sense. It is either raining or not raining ; but it is not raining ; therefore it is not raining . 20. It is either raining or not raining ; but it is not not raining ; therefore it is raining . Broad Disjunctive In a Broad Disjunctive Syllogism, the major p remise is a disjunctive proposition in a broad or improper sense. There is only one valid procedure: to sublate one (or more – but not all) of the members in the minor and posit the remaining member (or members) in the conclusion. It is either A, or B, or C, or D – at least one of them; but it is either A nor B; therefore it is either C or D – at least one of them. 21. Exercise: If possible, complete the following syllogism. Are the major premises disjunctive propositions in the strict sense or in the broad sense? o He is either not speaking or lying; o but he is not speaking; o therefore he is ….. 2. He is either not speaking or lying; but he is lying; therefore he is ….. 22. 3. John failed to pass such and such an exam, and is therefore either lazy or lacking in talent; but John is lacking in talent; therefore John is ….. 4. John is either lazy or lacking in talent; but John is not lacking in talent; therefore John is ….. 5. Either the m an who drafted the Constitution of the United States were animated by the desire to protect their property and privileges, or they were trying to create a just government based on the ethical standards of right. Historical research has shown that the members did indeed wish to protect their property and privileges. And so it is certain that they ….. 23. Criticize the following. Some are valid, others are not. Examine the disjunctive propositions to see if they include all possible alternatives. Are they disjunctive propositions in the strict or in the broad sense? o The order in the world owes its origin to mere chance or to an intelligent designer; o but the order of the world cannot be due to mere chance; o therefore it must be due to an intelligent designer.
2. He either violated the law, or else he was arrested unjustly; but he did violate the law; therefore he was not arrested unjustly. 24. 3. Jesus Christ is either God or the world’s greatest deceiver; but it is impossible to admit that He is the world’s greatest deceiver; therefore we are compelled to admit that He is God. 25. CONJUNCTIVE SYLLOGISM A Conjunctive Syllogism is one whose major premise is a conjunctive proposition, whose minor premise posits one or more members of the major premise, and whose conclusion sublates the other member of the major premise. A Conjunctive Syllogism is one that denies the simultaneous possibility of two alternatives. “ A thing cannot both be and not be in the same respect” 26. Rules for Conjunctive Syllogism: 1. Posit one member in the major premise and sublate the other in the conclusion. He cannot be in Manila and Cebu at the same time; but he is now in Manila; therefore he cannot now be in Cebu. 27. Exercise: o You cannot be married and be single too; o but he is married; o therefore he cannot be single. 2. A diplomat, it is sometimes s aid, is either not honest or not successful; but John is a diplomat who is not successful; therefore it looks as though John is at least honest. 3. It is impossible to study properly and at the same time to listen to the radio; but he is listening to the radio; therefore he cannot be studying properly.
http://www.slideshare.net/ulrick04/10-hypothetical-syllogism
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 cond itional 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 wh ich are invalid. The VALID forms are:
(AA) Affirming the Antecedent
If p, then q. p. q
or “Modus Ponens”
(DC) Denying the Consequent
If p, then q. Not q. Not p.
or “Modus Tollens” 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)
2. Denying the Antecedent (DA)
If Tweety is a bird, then Tweety flies. Tweety is a bird.
If Tweety is a bird, then Tweety flies. Tweety is not a bird.
Tweety flies
Tweety doesn’t fly.
3. Affirming the Consequent (AC)
4. Denying the Consequent (DC)
If Tweety is a bird, then Tweety flies. Tweety flies. Tweety is a bird
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: ii. If not q, then p. q. Not p.
P only if Q SOLUTION: Whenever Q, not R If p, then q. Not r, given p. If q, then not r. If p, then not r.
Invalid (DA)
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.
10. P else q.
Q only if r. P only if r.
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. Answers to odd exercises: 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)
http://faculty.unlv.edu/beiseckd/Courses/Phi-102/HypotheticalSyllogisms.htm
