Using the Membership Table Method (MTM) to Determine the Validity of Categorical Syllogisms

A presentation is provided of a method – the Membership Table Method (MTM) – to determine the validity of categorical syllogisms. This method makes it possible for each syllogism to be assigned a specific set. If this set is equal to the universal set U , then the categorical syllogism considered is valid, and if that set is not equal to U , then that categorical syllogism is not valid. In other words, any categorical syllogism is valid if and only if its respective set, according to the MTM, is equal to the universal set U . The conclusion of a valid categorical syllogism whose premises are true is true.


Introduction
From historical and educational standpoints, forms of reasoning known as syllogisms, especially categorical syllogisms, have been relevant in the development of logic.
The objective of this article is to present a method to determine the validity of categorical syllogisms: the membership table method (MTM). It can be applied more simply and systematically than other pre-existing methods. Up until now one of the most commonly used approaches to determine the validity of categorical syllogisms has been that of using diverse types of diagrams [1], [2], [3], [4], [5], and [6]. A previous contribution by the authors of this article, with that objective, can be classified within that approach [7].

A Correspondence between Operations of Propositional Calculus and Operations of Set Theory
Propositional variables -that is, variables that can be replaced with propositions -are usually referred to as p, q, r, etc. Given that in other articles related to this one the letter p will be used to refer to a probability and diverse probabilities will be denominated p 1 , p 2 , p 3 , etc., to prevent confusion, in this article these propositional variables will be symbolized as follows: q 1 , q 2 , q 3 , etc. (with some exceptions as specified in section 5). In addition, taking advantage of a "license" used by a number of authors, q 1 , q 2 , q 3 , etc. will be referred to as "propositions". Thus, statements such as "q 1 is true" and "q 2 is false" should be interpreted respectively in the following way: "Admit that q 1 has been replaced with a true proposition" and "Admit that q 2 has been replaced with a false proposition". If a sole proposition is considered, the subscript can be eliminated and the proposition symbolized as q.
In set theory, the universal set, or the universe of discourse, to which all the elements that may be considered when referring to a given topic belong, will be symbolized as U.
The diverse sets characterized within the framework of the universal set U will be denominated C 1 , C 2 , C 3 , etc. (with some exceptions to be specified in section 6). If only one of those sets is considered, the subscript can be eliminated and the set symbolized as C.
The (monadic) logical connective of negation in propositional calculus will be symbolized by a horizontal bar above the negated proposition. Thus, the symbol q -that is, not q -will represent the negation of that proposition q.
All the elements of the set U considered that do not belong to C belong to the complement of the set C. The complement of C is symbolized as C. In this article the symbol corresponding to the operator of complementation of a set is placed above its operand. This operand is the set C on which that operator acts, thus generating the complement of C (that is, C), which is another set.
Given that all of the elements which may be considered when dealing with a particular topic belong to the set U, no element belongs to the complement of U, known as the empty set. It is symbolized as ∅.
The relation between the universal set U and the corresponding empty set can be expressed by the following equality: Given that a) all the elements belonging to U that do not belong to C belong to the complement of any set C and that b) no element of the corresponding U belongs to the set ∅, it is obvious that Given that because of (2), ∅ and U are the same set, their complements must also be equal: If, in (3), U is replaced with ∅, admissible given the equality (1), the following equation is obtained: In addition, given that because of (1), U and ∅ are the same set, their complements will also be equal: According to (2), ∅ can be replaced in (5) with U. Thus the following equation is obtained: In general, the complement of the complement of any set C is equal to C: The truth tables (a) of q, q and q, and the membership tables (b) of C, C and C are presented in figure 1.
q q q 0 1 0 1 0 1 (a) Truth tables of q (that is, of the negation of q) and of q (that is, of the negation of the negation of q, or of the "double negation" of q).
Membership tables of C (that is, of the complement of C) and of C (that is, of the complement of the complement of C). In figure 1a, the presence of a 0 in the column corresponding to a proposition, such as q, means that the proposition is considered false. The presence of a 1 in the column corresponding to a proposition means that the proposition is considered true. Classical logic is bivalent, in the sense that there are only two possible truth values for any proposition: It is true or it is false. In that bivalent logic, the negation of any true proposition is a false proposition, and the negation of any false proposition is a true proposition.
The first row of the truth tables represented in figure 1a is composed of the numerical sequence 0, 1, 0. It should be interpreted in the following way: If the proposition q is false, then its negation q is true and the negation of q (that is, q) is true.
Note in figure 1a that the columns corresponding to q and q are identical: asserting q is equivalent to asserting q.
In figure 1b, the set C mentioned above is any set characterized within the framework of any universal set U; that is, any element belonging to C is an element belonging to that U.
In figure 1b, the presence of a 0 in a column corresponding to a set, such as C, means the supposition that any element belonging to U does not belong to that set. The presence of a 1 in the column corresponding to a set means the supposition that any element belonging to U does belong to that set. In classic set theory, if any element of U belongs to C then that element does not belong to C, and if that element of U does not belong to C then that element does belong to C.
The first row of the membership tables in figure 1b is composed of the numerical sequence 0, 1, 0. It should be interpreted as follows: If any element of U (that is, belonging to U) does not belong to C, then that element does belong to the complement of C (i.e., C) and does not belong to the complement of C (i.e., C).
The second row of the membership tables in figure 1b is composed of the numerical sequence 1, 0, 1. It should be interpreted as follows: If any element of U belongs to C, then that element does not belong to the complement of C (i.e., C) and does belong to the complement of C (i.e., C).
Note that the columns corresponding to C and to C are identical: C = C, as seen in (7); the double operation of complementation of a set results in the set C itself.
The first and the second rows of the truth tables in figure 1a are the same, respectively, as the first and second rows of the membership tables in figure 1b. That makes it possible to establish a) the correspondence of the proposition q with the set C, b) the correspondence of the proposition q with the set C, c) the correspondence of the proposition q with the set C, and d) the correspondence of the connective of negation in propositional calculus with the operator of complementation in set theory.
The logical connective of conjunction of two propositions is symbolized as ∧. The proposition q 1 ∧ q 2 -the conjunction of the propositions q 1 and q 2 -is read as: "q 1 and q 2 ".
The operator of the intersection of two sets is symbolized as ∩. The set which is the intersection of the sets C 1 and C 2 -two sets characterized within the framework of some universal set U -is symbolized as C 1 ∩ C 2 . Figure 2 presents a) the truth table of the proposition q 1 ∧ q 2 , and b) the membership table of the set C 1 ∩ C 2 . In the truth table of q 1 ∧ q 2 (in figure 2a), it is seen that only when both q 1 and q 2 are true, is q 1 ∧ q 2 also true. This case corresponds to the fourth row in that truth table, which is composed of the numerical sequence 1, 1, 1; if and only if q 1 and q 2 are true, is q 1 ∧ q 2 also true.
In the membership table C 1 ∩ C 2 (in figure 2b), it is seen that only when any element of U belongs both to C 1 and to C 2 , does that element belong to C 1 ∩ C 2 . This case corresponds to the fourth row of that membership table, which is composed of the numerical sequence 1, 1, 1; if and only if any element of U belongs both to C 1 and to C 2 , does that element also belong to C 1 ∩ C 2 .
Note, in figure 2, the equality of the first, second, third and fourth rows, respectively, in figure 2a and figure 2b. This makes it possible to establish a correspondence between a) the propositions q 1 and q 2 and the sets C 1 and C 2 , respectively; b) the proposition q 1 ∧ q 2 (the conjunction of q 1 and q 2 ) and the set C 1 ∩ C 2 (the set which is the intersection of C 1 and C 2 ); and c) the logical connective of conjunction in propositional calculus (∧) and the operator of intersection in set theory (∩).
The logical connective of disjunction of two propositions will be symbolized as ∨. The proposition q 1 ∨ q 2 -the disjunction of the propositions q 1 and q 2is read as: "q 1 or q 2 ".
The operator of union of two sets is symbolized as ∪. The set which is the union of the sets C 1 and C 2 -two sets characterized within the framework of some universal set U -is symbolized as C 1 ∪ C 2 . Figure 3 presents a) the truth table of the proposition q 1 ∨ q 2 , and b) the membership table of C 1 ∪ C 2 .  Note in the truth table in figure 3a that only when both q 1 and q 2 are false (in the first row of that table) is q 1 ∨ q 2 false. Note also in the membership table in figure 3b that only when any element of U belongs neither to C 1 nor to C 2 (in the first row of that table) does that element not belong to C 1 ∪ C 2 .
Note that the first, second, third and fourth rows in figure 3a are the same, respectively, as the first, second, third and fourth rows in 3b. Therefore, a correspondence can be established between a) the propositions q 1 and q 2 and the sets C 1 and C 2 , respectively; b) the proposition q 1 ∨ q 2 and the set C 1 ∪ C 2 ; and c) the logical connective of disjunction in propositional calculus and the operator of union in set theory.
The logical connective of material implication in propositional calculus will be represented as →. The proposition q 1 → q 2 is read as: "If q 1 , then q 2 ". The proposition q 1 is known as the antecedent of q 1 → q 2 and the proposition q 2 is known as the consequent of that proposition.
The operator of the material implication in set theory will be represented as −→ | . The result of that operator acting on the ordered pair of sets The proposition q 2 → q 1 is read as: "If q 2 , then q 1 ". The proposition q 2 is known as the antecedent of q 2 → q 1 and the proposition q 1 is known as the consequent of that proposition.
The result of the operator of material implication acting on the ordered pair of sets {C 2 , C 1 } is equal to the set C 2 −→ | C 1 . Figure 4 presents a) the truth tables of the propositions q 1 → q 2 and q 2 → q 1 ; and b) the membership tables of the sets C 1 −→ | C 2 and C 2 −→ | C 1 .
(a) Truth tables of q 1 → q 2 and q 2 → q 1 . Figure 4: a) Truth tables of the propositions q 1 → q 2 and q 2 → q 1 , and membership tables of the sets C 1 −→ | C 2 and C 2 −→ | C 1 .
Note in figure 4a that the only case in which the proposition q 1 → q 2 is false (in the third row in the truth table of that proposition) is that in which the antecedent q 1 is true and the consequent q 2 is false. In the other three cases, q 1 → q 2 is true. Note also in figure 4b, that the only case in which any element whatsoever of the universal set U does not belong to the set C 1 −→ | C 2 (in the third row of the membership table of that set) is that in which the element belongs to C 1 and does not belong to C 2 . In the other three cases, that element does belong to C 1 −→ | C 2 .
Note in figure 4a that the only case in which the proposition q 2 → q 1 is false (in the second row in the truth table of that proposition) is that in which the antecedent q 2 is true and the consequent q 1 is false. In the other three cases, q 2 → q 1 is true. Note also in figure 4b that the only case in which any element whatsoever of the universal set U does not belong to the set C 2 −→ | C 1 (in the second row of the membership table of that set) is that in which that element belongs to C 2 and does not belong to C 1 . In the other three cases, that element does belong to C 2 −→ | C 1 .
Observe that the first, second, third and fourth rows of the truth table of q 1 → q 2 are the same respectively, as the first, second, third and fourth rows of the membership table of C 1 −→ | C 2 . Observe also that the first, second, third and fourth rows of the truth table of q 2 → q 1 are the same, respectively, as the first, second, third and fourth rows of the membership table of C 2 −→ | C 1 . Therefore, a correspondence may be established between a) the propositions q 1 and q 2 and the sets C 1 and C 2 , respectively; b) the propositions q 1 → q 2 and q 2 → q 1 and the sets C 1 −→ | C 2 and C 2 −→ | C 1 , respectively; and c) the logical connective of material implication in propositional calculus and the operator of material implication in set theory.
The logical connective of material bi-implication, or of logical equivalence in propositional calculus, will be represented as ←→. The proposition q 1 ←→ q 2 is read as: "The proposition q 1 is logically equivalent to the proposition q 2 ". The operator of material bi-implication in set theory will be represented as ←→ | . Figure 5 presents a) the truth table of q 1 ←→ q 2 , and b) the membership table of the set C 1 ←→ | C 2 In figure 5a it is seen that the two cases in which the proposition q 1 ←→ q 2 is true are those (in the first and fourth rows of the truth table of that proposition) in which q 1 and q 2 have the same value of truth. In the case represented in the first row both q 1 and q 2 are false, and in the case represented in the fourth row both q 1 and q 2 are true.
In figure 5b it is seen that the two cases in which any element whatsoever of the universal set U considered belongs to the set C 1 ←→ | C 2 are the following: a) in the case in which the element belongs neither to C 1 nor to C 2 (represented in the first row of the membership table of that set) and b) the case in which that element belongs both to C 1 and to C 2 (represented in the fourth row of the membership table of that set).
Observe that the first, second, third and fourth rows of the truth table of q 1 ←→ q 2 are the same, respectively, as the first, second, third and fourth rows of the membership table of C 1 ←→ | C 2 . Thus a correspondence can be established between a) the propositions q 1 and q 2 and the sets C 1 and C 2 , respectively; b) the proposition q 1 ←→ q 2 and the set C 1 ←→ | C 2 ; and c) the logical connective of material bi-implication in propositional calculus and the operator of material bi-implication in set theory.
For propositions resulting from the use of connectives, such as q 1 → q 2 or q 3 ∨ q 4 , it can be suitable to express them in parentheses as (q 1 → q 2 ) and (q 3 ∨ q 4 ), respectively. Thus, if a connective is used with those propositions to obtain another proposition, it is clear how that has operated. For example, (q 1 → q 2 ) → (q 3 ∨ q 4 ) is the proposition of a conditional nature: "If q 1 → q 2 , then q 3 ∨ q 4 ", in which q 1 → q 2 is the antecedent, and q 3 ∨ q 4 is the consequent. The proposition (q 1 → q 2 ) → (q 3 ∨ q 4 ) has been obtained by the action of the connective of material implication on the following ordered pair of propositions can be expressed in parentheses if an operation is carried out on it and on some other proposition, by using some logical connective. Hence, for example, ((q 1 → q 2 ) → (q 3 ∨ q 4 )) ∧ (q 1 → q 5 ) is the proposition obtained through the action of the logical connective of conjunction on the propositions (q 1 → q 2 ) → (q 3 ∨ q 4 ) and (q 1 → q 5 ).
Given the correspondences mentioned, a) between propositions and sets, and b) between logical connectives and set theory operators, considerations of this same type concerning the use of parentheses are valid in that theory. Therefore, the set ( , the set C 1 −→ | C 5 corresponds to the proposition q 1 → q 5 , and the set ( If in the logical operations carried out there are n propositions -q 1 , q 2 , . . . , q n -in the corresponding truth tables, there will be 2 n rows because each of those propositions can have two truth values: true or false. Each row of those tables will correspond to each possible case of different assignments for the truth values of each of those n propositions. Since for each of these cases there are two possible assignments of truth value for the logical function to be specified, there are 2 ( For each of the 2 (2 n ) logical functions of n propositions, for n = 1, 2, 3 . . ., there is, according to the approach used, a function of n sets, which also is a set.

Isomorphism between Each Law -Theorem
or Tautology -of Propositional Calculus and the Corresponding Expression of the Universal Set This section will provide only a) the characterizations of the main notions concerning the topic discussed; and b) three examples of isomorphism existing between the tautologies of propositional calculus and the corresponding sets (according to section 2). If a function of n propositions, for n = 1, 2, 3, . . ., is true regardless of the truth values of each of those n propositions, then that function, which is also a proposition, is considered a law -or tautology -of propositional calculus. A tautology is true given its logical form, or structure.
The negation of a tautology which is a function of n propositions is known as a contradiction and is false, regardless of the values of truth of each of those n propositions.
The law of propositional calculus known in Latin as modus tollendo ponens -that is, "the mode that, by denying, affirms"-is stated below in (8); and the corresponding set, which is isomorphic to it according to section 2, in (9). Figure 6 presents a) the truth table of the proposition specified in (8), and b) the membership table of the corresponding set specified in (9).   In the truth table in figure 6a it can be seen that the proposition ((q 1 ∨ q 2 ) ∧ q 2 ) → q 1 is a tautology because it is true given its logical form; that is, it is true regardless of the truth values of q 1 and q 2 .
In the membership table in 6b it can be seen that for any element of the universal set U there are four possibilities: 1) that it belongs neither to C 1 nor to C 2 (as in the first row of the membership table); 2) that it does not belong to C 1 and does belong to C 2 (as in the second row of the membership table); 3) that it belongs to C 1 and not to C 2 (as in the third row of the membership table); and 4) that it belongs both to C 1 and to C 2 (as in the fourth row of the membership table). In each of these cases, the element of U considered belongs to the set ((C 1 ∪ C 2 ) ∩ C 2 ) −→ | C 1 . Therefore, given that any element of U belongs to that set, it is inferred that the set is equal to U: The law of propositional calculus known in Latin as modus tollendo tollens -that is, the mode that by denying, denies -is stated in (10) below; and the corresponding set, which is isomorphic to it, is stated in (11). Figure 7 presents a) the truth table of the proposition specified in (10) and b) the membership table of the corresponding set, isomorphic to the proposition.   In figure 7 it is seen a) that ((q 1 → q 2 ) ∧ q 2 ) → q 1 is a tautology of propositional calculus, and b) that the corresponding set ((C 1 −→ | C 2 ) ∩ C 2 ) −→ | C 1 , which is isomorphic to it, is equal to the universal set U: A law of propositional calculus -one of the laws of De Morgan -is stated in (12), and the corresponding set, which is isomorphic to it, is stated in (13) below. Figure 8 presents a) the truth table of the proposition specified in (12) and b) the membership table of the set specified in (13).    (13) In figure 8 it is seen a) that (q 1 ∧ q 2 ) ←→ (q 1 ∨ q 2 ) is a tautology of propositional calculus and b) that the corresponding set (C 1 ∩C 2 ) ←→ | (C 1 ∪C 2 ), which is isomorphic to it, is equal to the universal set U:

Categorical Propositions and Their Corresponding Sets
Categorical propositions are assertions about two sets that affirm or deny that one of those sets is totally or partially included in the other set. In this section those sets are called C 4 and C 5 respectively. Those names have been chosen for the following reason: When referring to categorical syllogisms, reasoning of a particular type, the names C 1 , C 2 and C 3 will be used to refer systematically to certain sets which play an important role in that reasoning. To avoid confusion, those three names are not used in this section. Of course, instead of using C 4 and C 5 , names such as F and G respectively could have been used. Four examples of categorical propositions are given below: In each of the four categorical propositions, reference is made to the following two sets: The sets referred to in each of these other four categorical propositions are these: C 4 : philosophers; C 5 : honest persons If both sets of categorical propositions are compared, it is noted that each proposition of the first set has the same logical form as the proposition to which the same number was assigned in the second set. Thus, four types of categorical propositions can be distinguished. Any categorical proposition, given its logical form, may be ascribed to one of these four types.
The propositions assigned number 1, in both sets of four propositions, are examples of those known as affirmative universal propositions. The symbolic expression of either of them is the following: According to the preceding proposition, if any element whatsoever of the universal set U considered belongs to the set C 4 , then it belongs also to the set C 5 . If that proposition is true, then the possibility that any element whatsoever of U could belong to C 4 and not also to C 5 is eliminated. According to section 2, any element of U belonging to the set C 4 −→ | C 5 satisfies that condition; according to the membership table of C 4 −→ | C 5 , the possibility that the element could belong to C 4 and not to C 5 is eliminated.
The symbolic expression of any affirmative universal categorical proposition is stated once more below, with the corresponding set at the right.
The propositions assigned number 2, in both sets of four propositions, are examples of those called negative universal propositions. The symbolic expression of either of them is the following: According to the preceding proposition, if any element of the universal set U considered belongs to the set C 4 , then it does not belong to the set C 5 ; that is, it also does belong to the set C 5 . If that proposition is true, then the possibility of any element whatsoever of U belonging to C 4 , but not belonging also to C 5 , is eliminated. Any element of U belonging to the set (C 4 −→ | C 5 ) satisfies this condition: According to the membership table of (C 4 −→ | C 5 ), the possibility of that element belonging to C 4 , but not belonging also to C 5 , is eliminated.
Below the symbolic expression of any negative universal categorical proposition is given once more, with the corresponding set on the right.
The propositions assigned number 3, in both sets of propositions, are examples of those called affirmative particular categorical propositions. The symbolic expression of either of them is the following: According to the preceding proposition there is at least one element of the universal set U considered which belongs both to C 4 and to C 5 . If there is not at least one element of U that belongs both to C 4 and to C 5 , then that proposition is false. According to section 2, any element of U that belongs both to C 4 and to C 5 belongs to the set which is the intersection of C 4 and C 5 -C 4 ∩ C 5 . Thus, C 4 ∩ C 5 is the set corresponding to that proposition.
The symbolic expression of any affirmative particular categorical proposition is stated again below, with the corresponding set at the right.
The propositions assigned the number 4, in both sets of four propositions, are examples of those known as negative particular categorical propositions. The symbolic expression of either of them is the following: Some C 4 are not C 5 .
According to the preceding proposition, there is at least one element of the universal set U that belongs to C 4 and does not belong to C 5 ; that is, it belongs to C 5 . If there is not at least one element of U that belongs both to C 4 and to C 5 , then that proposition is false. Any element of U that belongs both to C 4 and to C 5 belongs to the set that is the intersection of C 4 and C 5 -C 4 ∩ C 5 . Thus, C 4 ∩ C 5 is the set corresponding to that proposition.
The symbolic expression of any particular negative categorical proposition is given below once more, with the corresponding set at the right.

Characterization of Categorical Syllogisms
A syllogism is a type of deductive reasoning in which a categorical proposition known as a conclusion is inferred, or deduced, from two other categorical propositions known as premises.
In this article the first premise of each categorical syllogism will be named s 1 , given that "statement" or "sentence" are considered synonyms of "proposition". The second premise and the conclusion will be called s 2 and s 3 , respectively. These denominations have been preferred to q 1 , q 2 and q 3 , for example, to emphasize that these are not just any three propositions, but rather three categorical propositions constituting a categorical syllogism, linked together due to their internal structures.
The predicate term of the conclusion -s 3 -is known as the major term of the syllogism and the subject term of s 3 is called the minor term of the syllogism. The major premise of the syllogism, which in this article will correspond systematically to s 1 , is that which contains the major term. The minor premise of the syllogism, which in this article will correspond to s 2 , is that which contains the minor term.
The third term of the syllogism does not appear in the conclusion s 3 , but it does appear in each of the premises. This third term is called the middle term.
Each term of a syllogism may be assigned to a particular set. In this article the set C 3 will correspond to the major term, C 1 will correspond to the minor term, and C 2 will correspond to the middle term.
The seventh example of a categorical syllogism considered in section 7 is the following: s 1 : All engineers are pragmatic. s 2 : Some engineers are wealthy. ∴ s 3 : Some wealthy persons are pragmatic.
The symbol ∴ preceding s 3 , means "therefore". According to the above explanations, the sets corresponding to the different terms of that syllogism are the following: Given these conventions, the syllogism under consideration can be presented as follows: 6 Specification of the Membership Table Method (MTM) With the terminology of propositional calculus, the most important characteristic of each categorical syllogism -that s 3 may be inferred, or deduced, from s 1 and s 2 -can be expressed as follows: (s 1 ∧ s 2 ) → s 3 . The internal structure of the propositions s 1 , s 2 and s 3 is not made evident by their names. It is thus not feasible to use a resource directly from propositional calculus, such as a truth table to prove in the case of the syllogism considered that it is valid that (s 1 ∧ s 2 ) → s 3 is a tautology, or law, of that calculus. However, the proposition (s 1 ∧ s 2 ) → s 3 does make it possible to show one of the most relevant characteristics of valid categorical syllogisms: If (s 1 ∧ s 2 )that is, the conjunction of s 1 and s 2 which is the antecedent of that proposition -is true, then s 3 (the consequent of that proposition) is true.
According to section 2, a certain set corresponds to each proposition. The sets corresponding respectively to s 1 , s 2 and s 3 will be denominated S 1 , S 2 and S 3 . Each of those sets does reveal the relation between the terms of the corresponding proposition.
According to the MTM, the corresponding set S should be obtained for each categorical syllogism. If S is equal to the universal set U -that is, if S = U -then that categorical syllogism is valid. If S ̸ = U, then that syllogism is not valid. The different types of invalid categorical syllogisms will not be considered in this article. The MTM will be used only to determine whether the categorical syllogism considered is valid or not valid.
Recall that in the column for S -the set corresponding to the syllogism considered -in the membership table of that set, the following should be examined: a) If each element of that column is equal to one (1) then S = U, and the syllogism considered is, therefore, valid; b) If at least one element of that column is equal to zero (0), then S ̸ = U, and the syllogism considered is, therefore, not valid.
7 Determining the Validity, or the Non-Validity, of Diverse Categorical Syllogisms Using the MTM: Examples The first six examples of the categorical syllogisms considered in this section were taken from [17].
For each example considered, the following aspects are specified: 1) the categorical syllogism expressed in natural language; 2a) that syllogism expressed in terms of the sets C 1 , C 2 and C 3 , and 2b) the corresponding sets S 1 , S 2 and S 3 , respectively; 3) the set S corresponding to the syllogism; and 4) the membership table which makes it possible to determine whether S = U, in which case the syllogism is valid, or whether S ̸ = U , in which case the syllogism is not valid. Example 1 s 1 : All men are mortal. s 2 : All Greeks are men. ∴ s 3 : All Greeks are mortal. C 1 : Greeks C 2 : men C 3 : mortal beings C 1 : Greeks C 2 : men C 3 : perfect beings     The categorical syllogism considered is valid.
s 1 : All Greeks are men. s 2 : Some mortals are not men. ∴ s 3 : Some mortals are not Greeks. C 1 : mortal beings C 2 : men C 3 : Greeks If it is taken into account that (C 1 ∩ C 2 ) = (C 2 ∩ C 1 ), it is noted that the set corresponding to the categorical syllogism in example 7 has the same form as the set corresponding to the categorical syllogism in example 3. The latter categorical syllogism is valid. Therefore, the categorical syllogism in example 7 is valid.
It is noted that the set corresponding to the categorical syllogism in example 8 has the same form as the set corresponding to the categorical syllogism in example 4. The latter categorical syllogism is valid. Therefore, the categorical syllogism in example 8 is valid. Example 9 s 1 : All men are rational. s 2 : All Spaniards are men. ∴ s 3 : All Spaniards are rational. C 1 : Spaniards C 2 : men C 3 : rational men It is noted that the set corresponding to the categorical syllogism in example 9 has the same form as the set corresponding to the categorical syllogism in example 1. The latter categorical syllogism is valid. Therefore, the categorical syllogism in example 9 is valid.  The categorical syllogism considered is valid.  The categorical syllogism considered is valid.        Given that in the column of S there is a membership value equal to zero (0), S ̸ = U. Therefore, the categorical syllogism considered is not valid. Example 16 s 1 : All cosmologists are scientists. s 2 : Some scientists are polyglots. ∴ s 3 : Some polyglots are cosmologists. C 1 : polyglots C 2 : scientists C 3 : cosmologists Given that in the column of S there is a membership value equal to zero (0), S ̸ = U. Therefore, the categorical syllogism considered is not valid.

Discussion and Perspectives
Reference was made to the relation existing between the truth tables of propositional calculus and the membership tables of set theory, used in the MTM, for example. Within the framework of canonical fuzzy logic (CFL) [18], the latter tables can be considered truth tables, in the strict sense. One of the main objectives of this series of articles on logic presented by the authors is to shed light on the relations existing -both in classical logic and in diverse variants of some non-classical logics -between different calculi of these logics, such as propositional calculus and predicate calculus, the last of which is expressed in the terminology of set theory. (Another of the objectives of this research program is to show how classical logic can be considered a "limit case" of those variants of non-classical logics.) Categorical syllogisms, simple types of reasoning which have been studied a great deal, are important from historical and educational perspectives. In this research program they are a "testbed" for different methods which will be evaluated according to their possibilities of 1) being applied to determine the validity of more complex types of reasoning, and 2) being automatized for their use in disciplines such as control engineering and artificial intelligence. A method different from the MTM to determine the validity of categorical syllogisms will be presented in another article.
Note: A computer program based on the Membership Table Method (MTM) for the determination of the validity of categorical sylllogisms has been published on the follolwing sites: