Zacharias Ursinus (1534-1583) - Rules for Modal Syllogisms

 

The following is taken from Organi Aristotelei quinque priores (Neustadt Palinate: Matthias Harnisch, 1596) - Commentary on the First Five Books of Aristotle's Organon.

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First Figure

1) One Universal Premise and One Particular Premise - In the first figure, if the major is a universal contingent proposition and the minor is an affirmative necessary proposition, then it forms a perfect syllogism, with its conclusion being about the contingent, properly so-called. But if the major is a universal necessary proposition and the minor is about the contingent, whether affirmative or negative, it forms an imperfect syllogism, with its conclusion being about the contingent, not in the sense of the third mode, properly so-called and defined above, but in the sense of the second mode, commonly or not necessarily so, which is equivalent to existing either simply or in a qualified manner. However, if the minor is a negative necessary proposition or the major is particular, then no syllogism is formed, regardless of the mode of these propositions.

Zacharias Ursinus

2) Contingency signifies the negation of necessity of existence in negative propositions or the negation of non-existence in affirmative propositions of syllogisms.

3) These syllogisms are called perfect because it is evident from the premises themselves, which are taken, that they conform to the rules of dictum de omni and dictum de nullo, as is often stated.

4) Syllogisms that have an affirmative minor premise are perfected through reductio de impossibilie by substituting the contradictory of the contingent conclusion in place of the major premise and changing the contingent minor into non-existent.

5) In any valid consequence, when the antecedent is given, it necessarily follows that the consequent is. Therefore, if the antecedent is possible, the consequent must also be possible. However, the antecedent, even if false, can still be possible, as things that are only potentially possible can be brought into actuality. So, the consequent of that antecedent, even if false, will still be possible, and therefore not impossible. In other words, a false premise can lead to a possible conclusion if the antecedent, even when false, can still be possible. This demonstrates that it is not necessarily impossible.

6) A Pure Major Premise - A contingent conclusion always follows if the major is taken to be about existence simply. For example: All the afflicted are humble. It is possible for all the proud to be afflicted. Therefore, it is possible for all the proud to be humble.

7) One Necessary Premise and One Contingent Premise - When the major premise is necessary and the minor premise is contingent, particularly in affirmative modes (modes that affirm something), an imperfect syllogism of the contingent (non-necessary) conclusion can be formed. This is because the conclusion cannot be directly demonstrated from the major premise. Instead, it relies on the impossible (reduction to absurdity) to establish its truth. Here's an example illustrating this: Necessarily, every physicist is a philosopher. It is possible that some musicians are physicists (the minor premise). It is possible that some musicians are philosophers (the conclusion). The conclusion, "It is possible that some musicians are philosophers," cannot be directly derived from the major premise alone, as it involves contingent terms (musicians and philosophers) that are not necessarily related in the major premise. However, by assuming the contrary of the conclusion, "It is not possible that some musicians are philosophers," and combining it with the major premise, we arrive at a contradictory minor premise:

Given: Necessarily, every physicist is a philosopher.

And: It is not possible that some musicians are philosophers (the contrary of the conclusion).

Therefore: It is not possible that some musicians are physicists.

This contradictory minor premise demonstrates that the original conclusion, "It is possible that some musicians are philosophers," is indeed true.

So, in this case, the rule highlights the need to rely on the impossible (contradiction) to establish conclusions when dealing with necessary majors and contingent conclusions in affirmative modes.

8) Affirmative Necessary and Contingent Premises - If both the necessary and contingent premises are affirmative, the conclusion will not be necessary. Ex: A is necessarily B; B is possibly C; Ergo, A is possibly C.

9) One Necessary Premise and One Pure Premise - In the first figure, if the major premise is necessary and the minor premise is pure/simple, then the conclusion will be necessary. Ex: A is necessarily B; B is C; A is necessarily C.

10) One Universal Necessary Premise and One Particular Pure Premise - In particular syllogisms of the first figure, if the universal premise (A or E) is necessary, the conclusion will also be necessary. Ex: All A is necessarily B; Some B is C; Ergo, A necessarily belongs to some C. The conclusion will not be necessary if the particular premise is necessary in the syllogism.

11) Two Contingent Premises - In the first figure, when both premises are about the contingent, regardless of the quality of the premises, the conclusion about the contingent always follows, as long as the major premise is not particular.

Figure 2

1) Two Contingent Premises - Syllogisms in the second figure with both premises being about contingent propositions do not yield valid conclusions through direct conversion or reduction methods.

2) In EAE, AEE, and EIO, if the affirmative premise is pure, and the negative premise is contingent, nothing follows. But if the affirmative premise is contingent, and the negative is about non-existence, the conclusion is a contingent proposition in the common or equivalent sense of existence. The same applies if both premises are negative, and the contingent is converted into an affirmative. None of this, however, applies to the AOO syllogism (Baroco).

3) One Pure Premise and One Contingent Premise - A conclusion cannot follow from a second figure modal syllogism when there is an affirmative pure premise and a contingent negative premise.

But if the affirmative premise is contingent, and the pure premise is negative, then a conclusion can follow. Ex: A is not B; A is possibly C; Ergo, B is possibly not C.

4) One Necessary Premise and One Contingent Premise - In the second figure, when the negative premise is necessary, and the affirmative premise is contingent, or both premises are negative, the conclusion follows as contingent in a general sense, concerning existence. However, if the negative premise is contingent, and the affirmative is necessary, or if both are affirmative, or if both are particular or indefinite, then nothing follows.

5) One Pure Premise and One Necessary Premise - If the negative premise is necessary, the conclusion will be necessary. If the affirmative premise is necessary, the conclusion will not be necessary.

Figure 3

1) Two Contingent Premises - In the third figure, when both premises are contingent, the conclusion is also contingent, just as in the case when both premises are existent, and this is true even when one or both premises are negative. However, nothing follows in Bocardo, just as it does not follow from both premises being indefinite or particular.

2) One Contingent Premise and One Pure Premise - In the third figure, when one of the premises is pure and the other is contingent, whether the contingent is major or minor, even if the minor contingent is negative, it follows that the conclusion is not pure but only contingent, just as in the first figure. This is proven by reducing it to the first figure's particular modes, in which one premise is existent and the other is contingent, a contingent conclusion follows.

3) One Necessary Premise and One Contingent Premise - In affirmative modes, when one of the premises is necessary, and the other is contingent, even if the minor premise is a negative contingent, a contingent conclusion is drawn, just as in the first figure. In negative modes, if the affirmative is necessary, the conclusion pertains to the contingent but not necessary, and it follows that it is existent.

4) One Pure Premise and One Necessary Premise - If both of these premises (pure and necessary) are affirmative, the conclusion will be necessary. If one premise is affirmative and the other negative, whenever the negative is necessary the conclusion will also be necessary. When the affirmative premise is necessary, the conclusion will not be necessary. If one premise is universal, the other premise is particular, and both are affirmative, a universal necessary premise will result in a necessary conclusion.