Your statement made a category mistake (so it is a false question), so you use the terminology of propositional logic in the field of lexical logic.

explain

Only for hypothetical propositions (that is, propositions that can be written in the form of conditional sentences and propositions written in the form of "if P, then Q") can we talk about their inverse propositions, negative propositions and negative propositions.

This is discussed in propositional logic. Propositional logic only considers propositional connectives (or, and, not, if …… then), and propositional connectives connect simple propositions (outspoken propositions) into compound propositions (including negative propositions, conjoined propositions, alternative propositions, hypothetical propositions, etc.). ); In the view of propositional logic, "some S are P", "all S are P", "some S are not P" and "all S are not P" are simple propositions (or "atomic propositions"), and their internal structures will not be analyzed, but will only be written as P, Q, R and S. For a simple proposition (or an outspoken proposition or an atomic proposition), there is no inverse proposition, negative proposition or inverse negative proposition (unless you can equivalently convert it into the form of "if P, then Q").

Lexical logic (syllogism is its proof theory) only focuses on the internal structure of simple propositions. In other words, propositional logic is regarded as a simple proposition. As far as lexical logic is concerned, it is not simple, but has internal structures, which can be classified (divided into A, E, I, O) and related (summarized as the corresponding square matrix). There is no such thing as a negative proposition in lexical logic.

expand

With the advent of predicate logic technology, the situation has become different again. Predicate logic can also analyze simple propositions that cannot be analyzed by propositional logic, but it is different from lexical logic in that it takes all generic names (referring to the names of a class of things, such as "people" and "college students") as predicates, even though it is the subject in daily language (occupying the position of the subject), only singular lexical items (referring to a single thing, mainly proper names) are the real subjects. At the same time, the full-name quantifier ("for any") and the existential quantifier ("for some") are introduced, so that four propositions of lexical items can be written as follows:

SAP (full name affirmation): for any x, if x is s, then x is p.

SEP (full name negation): for any x, if x is s, then x is not p.

SIP (especially positive): there is x, x is s, and x is p.

SOP (special negation): there is x, x is s and no x is p.

It can be seen that in the full name proposition, the predicate logic is rewritten with "if … then ……", and in the proper name proposition, the predicate logic is rewritten with "and". Therefore, for the full-name proposition, it is possible to talk about its negative proposition (only in the sense of derivation) after rewriting the predicate logic.

For example, after "all s are p" is written in the form of predicate logic, its negative proposition is "for any x, if it is not x or p, then it is not x or s", and then it is written in the form of word logic in turn, that is, "all non-p are non-s"; Namely:

SAP is equivalent to [non-P]A[ non-S]

However, "A certain S is P" is a special proposition, even in the sense of the above deduction, it is not a negative proposition.

However, looking back at the logic of the entry, there is actually the so-called transposition reasoning. From SAP to [non-P]A[ non-S], we can get: sap-> se [non-p]-> [non-p] es-> [non-p] e [non-s].

But it is impossible to get [non ]PO[ non ]S from SIP, because SOP cannot be transposed directly.