16: On Reduplicative Propositions
Reduplicative Propositions universally of the same antecedent. For example, by means of 'Every man who is white is running', taken in one sense, it is asserted that the predicate 'running' is said of everything that the whole phrase 'man who is white' is said of. And in that case for the truth of this proposition two propositions are required, namely 'Some man is white' and "Each such man is running'.
However, in the other sense it is asserted that these two propositions are true: 'Every man is white' and 'Every man is running'.
Nevertheless, it should be noted, as will be said below,' that sometimes the relation is not personal, and in such a case it is not necessary that in the corresponding conjunctive the same term has the same supposition in both parts. But sometimes the relation is personal, and in that case it should have the same supposition.
16: On Reduplicative Propositions With respect to reduplicative propositions it should first be noted that a proposition is called reduplicative when it contains the expression 'insofar as' or some equivalent, taken reduplicatively. For, according to some, this expression can sometimes be understood reduplicatively, in which case it renders the proposition reduplicative; and it can sometimes be understood specificatively, in which case it does not render the proposition reduplicative.
Second, it should be noted that in a reduplicative proposition the reduplication itself, namely, the expression 'insofar as' or its equivalent, is sometimes affirmed, since a negation does not precede it, and sometimes denied, since a negation does precede it— as in 'It is not the case that Socrates, insofar as he is a man, is munning'. Now if the reduplication is not denied, then either it occurs in an affirmative proposition such as 'Socrates, insofar as he is a man, is visible'; or it occurs in a negative proposition such as 'Socrates, insofar as he is a man, is not Tunning'.
Reduplicative Propositions Thus, we must consider, first, what is required for the truth of an affirmative reduplicative proposition; second, what is required for the truth of a negative proposition in which, nevertheless, the reduplication is not negated; third, what is required for the truth of a proposition in which the reduplication is negated.
As far as the first point is concerned, it should be noted that such a proposition can be distinguished, since the reduplication can be made to express either concomitance or a cause. If the teduplication is made to express concomitance, then for the truth of the proposition four propositions, which, as it were, expound it, are required. One is a proposition in which the principal predicate is truly predicated of the principal subject. The second is a proposition in which what falls under the reduplication is predicated of the principal subject. The third is a proposition in which the principal predicate is predicated of that which falls under the reduplication, taken universally. The fourth will be a true conditional proposition in which one argues from that which falls under the reduplication to the principal predicate—in the way in which there is said to be a valid consequence from an inferior to a superior, and in the way in which it is said that one of two interchangeable propositions follows from the other.
For example, for the truth of 'Socrates, insofar as he is a man, has a color' the truth of these propositions is required: 'Socrates has a color', 'Socrates is a man', 'Every man hasa color', and 'If a man exists, then something which hasa color exists' (or: 'If A is a man, then A has a color'). And since such a conditional is false,' it follows that the reduplicative proposition 'Socrates, insofar as he is a man, has a color' is likewise false—since, namely, it has a false exponent.
On the other hand, if the reduplication is made to express a cause, then for the truth of such a reduplicative, besides the four exponents just mentioned, it is required that what falls under the reduplication express the cause of what is introduced by the predicate, or that it be that in which the principal predicate inheres primarily, or that the principal predicate inhere in it prior to its inhering in a pronoun referring to that which the principal subject supposits for. For example, by means of 'An isosceles, insofar as it is a triangle, has three angles' it is asserted that an Reduplicative Propositions isosceles has three angles, and that an isosceles is a triangle, and that every triangle has three angles, and that if something is a triangle, then it has three angles—and, besides this, it is asserted that the predicate 'has three angles' is truly predicated of 'triangle' prior to its being truly predicated of 'isosceles', in the sense in whicha logician uses 'prior' and 'posterior', namely, for relations among propositions. Similarly, by means of 'Fire, insofar as it is hot, is warmth-producing' what was just mentioned is asserted.
And therefore the proposition is true. Likewise, 'A man, insofar as he has an intellective soul, is susceptible to learning' is true.
For, in addition to the four exponents, an intellective soul is the cause of learning—extending the name 'cause' to the subject of some [quality]. And this is sufficient for the truth of such a reduplicative. But 'A man, insofar as he is risible, is susceptible to learning' is false, although it is true if the reduplication is made to express concomitance. The reason for this is that the four exponents alluded to above are true.
From what has been said the following rule can be inferred: there is always a formal consequence from a reduplicative proposition to its non-reduplicative counterpart. This is because its nonreduplicative counterpart is always one exponent of a reduplicative. Hence, this is formally valid: 'A man, insofar as he is an animal, is sentient; therefore a man is sentient'. Similarly, this follows: 'Fire, insofar as it is hot, is warmth-producing; therefore fire is warmth-producing'.
From these considerations it follows that many propositions which are conceded by many both in philosophy and theology are, strictly speaking, simply false. They are propositions like the following: 'A creature, insofar as it is in God, is really the divine essence', 'The Father and the Son generate [the Spirit] insofar as they are one', 'A man, insofar as he is a creature, is not a being', 'An object, insofar as it is intelligible, has diminished being', and so on. For all such propositions have some false exponent. And the same thing, mutatis mutandis, should be said about propositions which are just like these. And if some such propositions or ones just like them are found in the authorities—whether saints or philosophers—they should be glossed.
For authorities frequently do not speak literally, as is obvious to anyone who examines their books carefully.
Reduplicative Propositions What has been said enables us to solve such sophisms as 'Some things, insofar as they agree, differ' and 'Some things, insofar as they differ, agree'. For if we take 'to agree' in the sense in which it is truly predicated of all things which exist in reality, and if we take 'to differ' for all things which differ in any way whatsoever, i.e. which exist and are not the same, then all such propositions and ones just like them are true, if the reduplication is made to express concomitance. For, as is manifestly clear, the four previously mentioned exponents of such propositions are true. On the other hand, if the reduplication is made to express a cause, then in that case they are false. For 'to agree' neither agrees with nor is said of differing things prior to its being said of things which agree. Nor does it designate the cause of their differing. Therefore, the propositions are false if taken in this sense.
However, the same does not hold for these propositions: 'Some things, insofar as they are dissimilar, are similar'. For neither the following consequence nor its converse is valid: 'These things are similar; therefore they are dissimilar'. Nor does this follow: 'They agree in the predicable "dissimilar"; therefore they are similar'.
For in order for them to be similar it is required that they have qualities of the same lowest level species.
Another rule is that it is a valid consequence when one argues from an inferior to a superior, without distribution, on the part of the principal subject. Hence, this is a good consequence:
'Socrates, insofar as he is a man, is risible; therefore an animal, insofar as he is a man, is risible'. For it is impossible for the exponents of the antecedent to be true unless the exponents of the consequent are true. And this holds whether the reduplication is made to express concomitance or to express a cause.
Similarly, it is a valid consequence when one argues in the same way with respect to the principal predicate. However, when one argues from that which falls under the reduplication to its superior, there is a fallacy of consequent, as in the following:
'Socrates, insofar as he is a man; is risible; therefore Socrates, insofar as he is an animal, is risible'. For if that consequence were valid, then it would follow that this consequence is valid: 'Every man is risible; therefore every animal is risible'. It would follow in accord with this rule: 'When a reduplicative entails a redupliReduplicative Propositions cative, the exponents of the antecedent entail the exponents of the consequent'.
Nevertheless, it should be noted that someone could use these propositions in some other sense and not in their literal sense. In that case such consequences could be denied. And this is the way in which one would have to respond to authorities if they were found to say the opposite of some of the things said here.
The second principal task is to discuss a negative reduplicative, in which, nevertheless, the reduplication is not denied. A proposition of this sort is 'A man, insofar as he is rational, is not a donkey'.
And it should be noted that such a reduplicative proposition, if the reduplication is made to express concomitance, has four. exponents: two negative, one affirmative, and one conditional.
The affirmative exponent is a proposition in which what falls under the reduplication is truly affirmed of the principal subject. One of the negative exponents is the non-reduplicative counterpart of the reduplicative proposition in question, namely, it is a proposition in which the principal predicate is denied of the principal subject. The other negative exponent is a proposition in which the principal predicate is denied of that which falls under the reduplication, taken universally. The conditional is a proposition in which the negation of the principal prediate follows from the positing of that which falls under the redpulication. For example, 'A man, insofar as he is-risible, is not a donkey' has these exponents: 'A man is risible', 'A man is not a donkey', 'No risible thing is a donkey', 'If something is risible, then it is not a donkey'.
Because of this, propositions such as the following are literally false: 'A logician, insofar as he is a logician, differs from a grammarian', since the exponent 'Every logician differs from a grammarian' is false; 'A white thing, insofar as it is white, differs from a sweet thing'; and others just like them. Similarly, propositions such as the following are false: 'An intellect, insofar as it is an intellect, does not will', 'A soul, insofar as it is an intellect, does not will', 'A soul, insofar as it is active, is not passive', 'A soul, insofar as it is passive, does not act', 'Fire, insofar as it is hot, is not dry', 'A body, insofar as it is altered, does not undergo local motion', and so on for many others. For all such propositions have some false exponent.
Reduplicative Propositions On the other hand, if the reduplication is made to express a cause, then it is required that the principal predicate be denied of what falls under the reduplication primarily or prior to its being denied of a pronoun referring to what the principal subject supposits for—always assuming that the proposition has the four exponents mentioned above. For this reason, if the reduplication is taken in the sense in question, then 'A man, insofar as he is risible, is not a donkey' is false.
Similarly, it should be understood, just as before, that there is always a formal consequence from a reduplicative proposition to its non-reduplicative counterpart. Therefore, this consequence is formal: 'A logician, insofar as he is a logician, differs from a grammarian; therefore a logician differs from a grammarian'-—from which it follows that a logician is not a grammarian.
Third, we must deal with a proposition in which the reduplication is negated. And it should be noted that such a proposition is the contradictory of a reduplicative proposition in which the reduplication is affirmed. And, therefore, the opposite of any exponent of its contradictory reduplicative is sufficient for the truth of such a proposition. For it is a general rule that when a proposition is the contradictory of some proposition which has several exponents, then the truth of the opposite of any exponent is sufficient for the truth of the proposition in question.
Hence, for the truth of 'It is not the case that Socrates is a man insofar as he is white' it is sufficient that any of the following be true: 'Socrates is not a man', 'Socrates is not white', 'Some white thing is not a man', 'It is not the case that if it is white, then it is a man'.
It should also be noted that the expressions 'since it is.
. .', 'qua', 'by reason of its being .
. .', etc., are equivalent to the expression 'insofar as', and so they render propositions reduplicative. And the propositions in which they occur should be treated in the same way that propositions in which 'insofar as' occurs have already been treated.
On the other hand, if such an expression is understood specificatively rather than reduplicatively, then it is not required that that to which such an expression is added be the subject with respect to the principal predicate in a universal proposition.
Rather, it is required that what falls under the reduplication Reduplicative Propositions designate that in virtue of which the principal predicate belongs to the primary subject. For example, if in the proposition 'Fire, insofar as it is hot, is warmth-producing' the expression 'insofar as' is taken specificatively rather than reduplicatively, then it is not required that 'Every hot thing is warmth-producing' be true.
Rather, it is required that the name 'hot' convey the heat in virtue of which fire produces warmth. For to produce warmth belongs to heat prior to and more per se than its belonging to fire. Or at least 'hot' must convey that heat is a principle of producing warmth. Thus, for the truth of such a proposition it is required that the principal predicate be predicated of the principal subject and of that to which the reduplicative expression is added, and that the latter be predicated of the principal subject.
But it is not required that the principal predicate be predicated universally of that to which the reduplicative expression is added.
Rather, it is required that the latter designate that by reason of which the principal predicate truly agrees, through predication, with the principal subject. And in this way propositions like the following are verified: 'Socrates, insofar as he is white, dilates vision',? 'Socrates, insofar as he has a free will, sins', 'A being, insofar as it is a being, is the subject of metaphysics', and so on.
But one should not be concerned with the examples.
It is also necessary to point out that the expression 'insofar as' and, likewise, the expressions 'since it is . .', 'by reason of its being .. .', etc., are in some propositions equivalent to a temporal adverb. For example, one sense of the proposition
' "Dog", insofar as it signifies an animal which can bark, makes this proposition true: "Every dog is an animal" ' is this: 'The proposition
"Every dog is an animal" is true only when "dog" stands for an animal which can bark'. And, similarly, we say that the following proposition is true: 'This image is a man, since "'man" is being used in an improper sense'. That is, the proposition is true when
'man' is being used in an improper sense. Thus, the expression 'insofar as' can also be used in various senses and equivocally.