A Proposition is said to have existential import if and only it is used to assert the existence if objects of any sort. For example, when we say, ‘Some politicians are scholars’ we mean that mere exists at least one politician who is a scholar.
Similarly, the assertion of ‘Some politicians are not scholars’ asserts that there are politicians who are not scholars. Thus, by looking at the meaning of “Some” or “Some… not” in our ordinary language, we notice that I and O-propositions clearly have existential import.
We have already seen that I proposition follows from the corresponding A-proposition and an O-proposition follows from the corresponding E-proposition. Since I-proposition has existential import and also follows from an A-proposition, A-proposition must have existential import. By the same reasoning, an E-proposition must have existential import. Hence, every universal proposition will have existential import.
In general, we may say that all categorical propositions of syllogistic logic have existential import. Note that accepting the existential import of particular propositions poses no problem but to assert the existential import of every universal proposition is problematic. Consider the following case. We know from the square of opposition of propositions that A and O-propositions are contradictories.
Hence, both cannot be true together nor can be false together i.e. if one of them is false, the other must be true and vice-versa. But the proportions “All inhabitants of Mars are intelligent” and “Some inhabitants of Mars are not intelligent” could be false together if Mars has no inhabitants. Same is the case with the other contradictory pairs such as ‘I’ and E-propositions. Hence, E and I-propositions would not be contradictories which are absurd in syllogistic logic.
Further, the square of opposition of propositions also teaches that I and O propositions are sub-contraries. Being sub contraries the falsity of one would imply the truth of the other. Thus they cannot be false together. Let us show that I and O-propositions having the same subject and same predicate can be false together.
For example, “Some inhabitants of Mars are intelligent” and “Some inhabitants of Mars are not intelligent” would be both false if Mars has no inhabitants. This shows that I and O propositions are not sub-countries. This contradicts the traditional interpretation concerning the square of opposition of propositions.
Therefore, traditional square of opposition of propositions seems to be wrong. Of course, it is correct when it tells us that A and E validly imply their corresponding I and O propositions. It is wrong with respect to contradictories (such as A and 0 and E and I) and sub-contraries (such as I and O). Now the problem is how to save our square of opposition of propositions.
As we know, the square of opposition of propositions expresses the following relations holding between categorical proposition (i) A and O and E and I are contradictories, (ii) A and E are contraries. (iii) I and O are sub-contraries and (iv) Subalterns I and O follow validity from their super-alterns A and E respectively.
To preserves all these relations, we have to assume or presuppose that all classes designated by our terms as well as the complements of these classes do have members. In other words, all terms of Aristotelian logic must refer only to non-empty classes. If we presuppose this or accept the existential presupposition as explained above then we may continue with the square of the opposition of propositions.
This existential presupposition accepted to protect the square of opposition of propositions really problematic because of the following reasons. Firstly, accepting the existential presupposition leads to curtail the power of expressing the propositions that denies that it has -.embers.
Secondly, the ordinary use of our language is often inconsistent with the existential presupposition. Consider this example. “All trespassers will be prosecuted”. Here we do not intend to say that there are trespassers or that the class of trespassers is non-empty. We do not mean that there are actual trespassers who will be punished. We simply mean to say that if any person will trespass he or she will be punished. Thus, we do not assume anything about the existence of members of the class of trespassers.
The objection of the above sort has led the modern logicians to reconstruct syllogistic logic of Aristotle without existential presupposition in case of universal propositions. This is done of empty set or class and allowing the term variables of syllogistic logic to designate either empty or non-empty class. This move was due to George Boole. The rejection of existential presupposition entails the following:
(i) I and O propositions continue to have existential import. Hence, “Some S is P” and “Some S is not P” would both be false if ‘S’ is empty.
ii) Since I and O-propositions have existential import and both can be false (if the subject class is empty), I and O-proposition with the same subject and the same predicate would fail to be sub-contraries.
iii) A and E propositions do not now have existential import. Thus, an A-proposition of the form “All S is P” would be true even if ‘S’ is empty. The same is the case for an E- proposition (i.e. “No S is P” would be true even if ‘S’ is empty). Since in Boolean interpretation A and E-propositions could be true together, they fail to be contraries.
(iv) In Boolean interpretation the sub-alternation (i.e. inferring an I-proposition from an A- proposition and an O- proposition from an E-proposition) is not generally valid.
Thus, under Boolean interpretation, the traditional square of opposition of proposition – is been transformed in the following way. All relations between propositions along the sides of are square had been rejected and only the contradictory relations represented along the diagonal retained. Thus, the following is the square of opposition of propositions under Boolean interpretation.