In the traditional logic of the syllogism, Aristotelian logic, there are four kinds of syllogisms, Darapti, Felapton, Bramantip, and Fesapo, that are often said to be invalid in modern logic. Elementary logic students may even simply be told that they really are invalid. This is, of course, a distortion; but it is instructive to consider why this has happened and why it is that the syllogisms are considered invalid.
The problem begins, not with the syllogisms, but with other basic relationships among propositions. Aristotelian logic identifies four basic forms of propositions: the Universal Affirmative, or A, of the form, "All S is P"; the Universal Negative, or E, of the form, "No S is P"; the Particular Affirmative, or I, of the form, "Some S is P"; and the Particular Negative, or O, of the form, "Some S is not P." The relationships between these four forms of propositions are illustrated in the Square of Opposition. The diagonal lines connect Contradictories, where the truth of one implies the falsehood of the other, and vice versa. The two universal propositions are Contraries: They cannot both be true but they can both be false. The two particular propositions are Sub-Contraries: They cannot both be false, but they can both be true. Each particular proposition is the Subaltern of the universal above it: If the universal is true, then the subaltern is also true; and if the subaltern is false, then the universal is also false.
These rules mean that there are three possible assignments of truth values that can occur with the Square of Opposition: (1) where the universals are both false and the particulars both true, in "All Europeans are Italians" (false), "No Europeans are Italians" (false), "Some Europeans are Italians" (true), and "Some Europeans are not Italians" (true); (2) where the universal and particular affirmatives are true and the negatives false, in "All Canidae (dogs, wolves, etc.) are carnivores" (true), "No Canidae are carnivores" (false), "Some Canidae (e.g dogs) are carnivores" (true), and "Some Canidae are not carnivores" (false); and (3) where the universal and particular negatives are true and the affirmatives false, in "All Canidae are herbivores" (false), "No Canidae are herbivores" (true), "Some Canidae are herbivores" (false), and "Some Canidae are not herbivores" (true). The rules for the contradictories and contraries alone are sufficient to assign these values, as described in the diagrams (though the diagrams also show the subaltern inferences).
The possible assignment of truth values that cannot occur is (4) where the universals are both true and the particulars both false, though that could also occur if only the rule about contradictories were applied. In recent logic it turns out that the fourth assignment of truth values is possible. Sometimes it is simply said that the old theory was wrong. But what has happened instead is that a new interpretation of meaning has been imposed on the propositions, and this interpretation comes from Set Theory: A set is defined by its members. This is called "extensionality," and it determines the present construction of most systems of symbolic logic: Predicates are taken to refer, not to concepts, as in traditional logic, but to sets; and sets are both defined by their members and may or may not have members. A set without members is the Empty Set, and it is always possible that a set is to be identified with the Empty Set. On this interpretation, universal propositions are interpreted to be denials of existence, i.e. the identifications of certain sets that are the Empty Set; and particular propositions are taken to be affirmations of existence, i.e. identifications of certain sets as non-empty. Particular propositions, which traditionally were thought of as reports about meaning, not about existence, are therefore now typically called "existential" rather than "particular" propositions. The possibility of an Empty Set simply did not exist in Aristotelian logic, since the equivalent would be a term without meaning; but then a term without meaning would not occur or be used.
The Set Theoretical interpretation of logic is graphically illustrated through the device of Venn Diagrams, which rework the famous Circles of the great mathematician Leonhard Euler to embody the new assumptions. The Venn Diagram for the universal affirmative (A) proposition thus simply blanks out that side of the subject (S) circle that lies outside the predicate (P) circle, so that there are not any S's that are not P's. The Diagram for the universal negative (E) proposition blanks out that side of the subject (S) circle that lies inside the predicate (P) circle, so that there are not any S's that are P's.
The Diagram for the "existential" affirmative (I) proposition indicates, by a mark, that a member exists where the subject (S) circle and the predicate (P) circle overlap, so that there is at least one S that is a P. This is the contradictory of the E proposition, so that the subset which is identified as empty by the E is identified as non-empty by the I. The Diagram for the "existential" negative (O) proposition indicates, by a mark, that a member exists where the subject (S) circle lies outside the predicate (P) circle, so that there is at least one S that is not a P.
On the Set Theoretical interpretation, the fourth assignment of truth values in the Square of Opposition becomes possible: Both universals can be true and both "existentials" false because the S set may be the Empty Set. Thus every possible place for members is blanked out. This produces a different Venn Diagram than the standard ones for the A, E, I, and O propositions. One might call it the U proposition: There are no S's. Since the fourth assignment of truth values is possible, this means that the rules for Contraries, Sub-Contraries, and Subalternation are no longer valid. Only the rule of Contradictories works in Venn Diagrams.
The question now arises whether this proves anything, whether something new has been "discovered" in modern logic that was overlooked, and whether Aristotelian logic affirmed certain things that simply turn out to be false. But it is clear that what has happened is that the premises of the issue have been changed, and it doesn't prove anything just to say, "This is the way we are going to do things now." If logicians want to adopt the principles of Set Theory and build a system of logic around them, there is nothing wrong with that; but it doesn't discredit the older approach. On the other hand, if it is to be contended that Extensionality, one of the axioms of Set Theory, is the True construction of the nature of meaning, this is clearly false: If a set is defined by its members, this cannot mean "defined" in any cognitively familiar and meaningful sense; for words in ordinary language, like "dog," cannot be "defined" by members most of whom we cannot be acquainted with.
Aristotelian logic is often said to be a "class" logic, as opposed to propositional logic (where the elements are true or false). This not true. Calling it "class" logic is itself an artifact of Set Theory. "Classes" are collections of individuals that do not have the restrictions that have been placed on true Sets (to avoid the paradoxes discovered by Bertrand Russell and others). A "class" logic would thus be a rather primitive and undeveloped version of Set Theory. However, as noted, Aristotelian logic is not about any kind of collection of individuals, but about concepts. Concepts refer to individuals, but this is just one of their properties. Mischaracterizing Aristotelian logic simply preserves Extentionality and so perpetuates the misconceived critique of traditional logic.
Extensionality works in Set Theory because it works for mathematical objects, which are abstract entities and so do not "exist" in the same way that dogs, tables, and cities exist. One might even say that Extensionality works in the World of Forms, where no objects have contingent existence. This means that the category of "members" as treated in Set Theory contains an important ambiguity. It is effective and unobjectionable for mathematical or abstract objects, but not for concrete objects in space and time. This ambiguity may have been overlooked or dismissed by logicians who accepted Leibniz's views that space doesn't exist and that individuality is conferred by logical "discernability." Since Einstein's replacement of Newtonian physics was thought to imply that Leibniz must have been correct in his debate with the Newtonians over the nature of space, Leibniz came into vogue; and it was an easy step to the mistaken idea that abstract objects are no different from concrete objects.
Extensionality as a theory of meaning seems to have been finished off by Wittgenstein. But the tendency behind Extensionality, to explain meaning with natural objects of some sort, continues, as Wittgenstein himself saw meaning as "usage" and so behavior. This issue is so unsettled that there has been no strong motivation for logicians to reconstruct the extensional assumptions behind symbolic logic; but the beginning of the way out of naturalistic theories of meaning may be found in Jerrold Katz.
The possibility of the Empty Set is what creates problems for the syllogisms Darapti, Felapton, Bramantip, and Fesapo as translated into symbolic logic. This is most easily examined through Venn Diagrams again.
First of all, as a syllogism uses three terms, the subject (S) of the conclusion, the predicate (P) of the conclusion, and the middle term (M), which only occurs in the premises, a Venn Diagram for a syllogism uses three circles. After the diagrams for the two premises of the syllogisms are constructed, the result is examined to see if the conclusion may simply be read off. Thus in the "ideal" syllogism in Aristotelian logic, Barbara, the two premises between them (All M is P, and All S is M) end up blanking out all of the set S that is outside the set P (All S is P). In the syllogism Darii, the "major" premise (the one with the predicate of the conclusion) blanks out all of M outside of P (All M is P); and since the "minor" premise affirms that S has a member which is an M (Some S is M), this member can only be placed where S, M, and P all overlap, which enables us to read off the conclusion that there is an S which is a P (Some S is P).
For the "invalid" syllogisms, the problem is that the premises are all universals but that the conclusions are particulars. Since universal propositions are just denials of existence and "existential" propositions are affirmations of existence, these conclusions will not follow from the premises.
Darapti and Felapton are in the Third Figure, which means that the subject of the conclusion is the predicate in the minor premise. In Darapti the premises (All M is P, and All M is S) blank out all of M except where S, P, and M overlap. If there are any M's, then an M can only be an S and a P also, which makes the conclusion follow (Some S is P). So only if M is the Empty Set, which is meaningless in Aristotelian logic, would Darapti be invalid. In Felapton the premises (No M is P, and All M is S) blank out all of M except where S and M lie outside of P. If there are any M's, then an M can only be an S which is not a P, which makes the conclusion follow (Some S is not P). So only if M is the Empty Set, which is meaningless in Aristotelian logic, would Felapton be invalid.
Bramantip and Fesapo are in the Fourth Figure, which means that the subject of the conclusion is the predicate in the minor premise and the predicate of the conclusion is the subject in the major premise. This is exactly the reverse of the First Figure, which led Kant and others to argue, somewhat pointlessly it now seems, that there are really only three Figures of the syllogism. In Bramantip the premises (All P is M, and All M is S) blank out all of P except where S, P, and M overlap. If there are any P's, then a P can only be an S and a P also, which makes the conclusion follow (Some S is P). So only if P is the Empty Set, which is meaningless in Aristotelian logic, would Bramantip be invalid. In Fesapo the premises (No P is M, and All M is S) blank out all of M except where S and M lie outside of P. If there are any M's, then an M can only be an S which is not a P, which makes the conclusion follow (Some S is not P). So only if M is the Empty Set, which is meaningless in Aristotelian logic, would Fesapo be invalid.
So there is nothing wrong with the four "invalid" syllogisms except when a new interpretation of meaning introduces the possibility of the Empty Set. If it is postulated that the circle representing an entire term cannot be blanked out, because it cannot be the Empty Set, then Darapti, Felapton, Bramantip, and Fesapo are all quite valid inferences. If a restructured symbolic logic, which eliminates the presupposition of Extensionality, ever becomes common, then the four syllogisms will return to such dignity as they ever had in the theory of the syllogism. Of course, the theory of the syllogism is no longer of much importance in philosophy, or even in logic; but this episode illustrates two important features of philosophy in the 20th century: (1) Philosophers are often willing to believe they have discovered something new, or have disproved something old, just because they decide to do something in a different way. If "knowledge" is merely the conventional agreement of some community of scholars, then this may be true; but it will certainly not be true if discovery and proof are supposed to be about the independent nature of things. (2) It should now be obvious that logic is not just the neutral and unbiased description of forms of rational inference.
Problematic philosophical assumptions, whether from Set Theory, from Leibniz, etc., can be built into a system of logic, which is then used to "discover" the consequences of these assumptions in some area. Such question begging "discoveries" can then be used to trumpet the power and usefulness of the new system. Such circularity was especially conspicuous in Logical Positivism, where there seems to have been an expectation that symbolic logic would mechanically solve all the problems of philosophy. Since the Positivist logicians could simply build their assumptions, semantic, epistemological, and metaphysical, into their logic, it is not surprising that they felt vindicated in their conclusions. Symbolic logic as it exists today still reflects many of those assumptions, probably, as I have said, because there is no obvious or consensus alternative to Positivist principles, however unsatisfactory those clearly are.
On the other hand, while Logical Positivism is gone, and most of its trendy successors are not very interested in logic, the trick of defining and assuming one's way to favorable conclusions nevertheless continues. One can see this in the tactic of politicized deconstruction or "post-modernism" in condemning contrary opinion as "biased," when the theory itself actually contends that everything is biased and that "knowledge" is just the construction of "power." If that is true, of course, it is easy to prove that contrary opinion is biased -- it cannot be otherwise -- but to condemn it for such "bias" is then tautologous. The issue simply becomes power, which is an interesting parallel to the Logical Positivist position that there are no ad rem arguments in ethics, only expressions of emotion. Now they become expressions of power instead. The lesson of Bramantip, and the other "invalid" syllogisms defined into invalidity by a theory about meaning, is thus a symptom and a caution for all the workings of 20th century philosophy.