A Structural Approach for the Concurrent Teaching of Introductory Propositional Calculus and Set Theory

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A Structural Approach for the Concurrent Teaching of Introductory Propositional Calculus and Set Theory

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Osvaldo Skliar^*,

Sherry E. Gapper,Ricardo E. Monge

Osvaldo Skliar^*,

Sherry E. Gapper,Ricardo E. Monge

This version is not peer-reviewed

Submitted:

14 August 2024

Posted:

15 August 2024

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Abstract

A presentation is provided of a structural approach for the concurrent teaching of introductory propositional calculus and set theory. The existence of isomorphism is shown between each law (or tautology) of propositional calculus and the form of a corresponding set which is equal to the universal set U considered.

Keywords:

Subject: Computer Science and Mathematics - Data Structures, Algorithms and Complexity

1. Introduction

The objective of this article is to present a systemic-structural approach [1] – or simply, a “structural approach” – for the concurrent teaching of the basic notions of 1) propositional calculus and 2) set theory at an introductory level. The term “systemic” refers to the presentation of isomorphisms existing between the “parallel disciplines” considered (in this case, propositional calculus and set theory). For this purpose, from these disciplines the pairs of corresponding operations will be discussed in section 2, and then, the pairs of corresponding laws, or theorems, in section 3.

This instructional method is based mainly on the structure of the subject matter presented, and not on the diverse possible ways of communicating it, such as presential or virtual classes, using different amounts of audiovisual resources, or having a majority of lectures or discussion groups. The basic hypothesis of this instructional approach is the following: The structure itself is what facilitates learning. For this reason, the term “structural” is used for the approach considered.

It is supposed that those reading this article teach in high school or college and are already familiar with the notions of logic and mathematics to be considered here. For this reason, they will not be covered too formally or exhaustively; a simple review of some of their main characteristics will be provided instead. In the remainder of this section, brief consideration will be given to some of these notions which are important for this article.

A proposition – or statement – is a linguistic expression to which a truth value may be assigned: According to classical bivalent logic, each proposition is either true or false. Thus, for example, “Caballito is a neighborhood of the Argentine capital, Buenos Aires” can be considered a proposition. That does not occur with “What is your name?” or “Sit down!”.

In this article, when considering a sole proposition, it will be called “q”. When considering more than one proposition, the denominations

q_{1}

q_{2}

q_{3}

, … will be used. The negation of a proposition will be symbolized by a horizontal bar above that proposition. Thus the negation of the proposition q – that is, not q – will be symbolized as

\bar{q}

. Of course, any negated proposition, such as

\bar{q}

, is also a proposition.

In set theory, the universal set

U

is the set to which all the elements that can belong to a set – according to the topic covered – belong. If within the framework of that

U

only one other set is considered – that is, a set to which some of the elements in that

U

belong – that sole set will be denominated C. If more than one set of that type is considered, those sets will be denominated

C_{1}

C_{2}

C_{3}

, ….

The operator of complementation of a set will be symbolized by the symbol + placed above the denomination of the set whose complement is sought. Thus, the complement of C is symbolized as

\overset{+}{C}

. All of the elements belonging to the universal set

U

considered that do not to belong to C belong to the set

\overset{+}{C}

. The complement sets of

C_{1}

C_{2}

C_{3}

, … will be symbolized as

\overset{+}{C_{1}}, \overset{+}{C_{2}}, \overset{+}{C_{3}},

, … respectively.

Given that all the elements which can be considered when covering a particular topic belong to the universal set

U

symbolized as

U

, no element will belong to its complement, which will be symbolized as

\overset{+}{U}

. For this reason,

\overset{+}{U}

will be called the empty set and will be symbolized as ⌀; that is,

\overset{+}{U}

= ⌀.

In this article it will be admitted that whenever an operation involving sets is carried out, the corresponding universal set

U

considered will be known.

For an introduction to topics of logic and set theory, one may consult [2]. For a more detailed treatment of topics of logic, [3] and [4] may be consulted. For a more advanced coverage of set theory, one may consult, for example [5].

2. Pairs of Operations Corresponding to 1) Propositional Calculus and 2) Set Theory

The presence of a zero – 0 – in the truth table of a proposition is equivalent to the truth value “false” of that proposition. The presence of a zero – 0 – in the membership table of a set C (or rather,

C_{1}

C_{2}

C_{3}

, …) means that an element belonging to the universal set

U

considered does not belong to that set C (that is, to

C_{1}

C_{2}

C_{3}

, …).

The presence of a one – 1 – in the truth table of a proposition is equivalent to the truth value “true” of that proposition. The presence of a one – 1 – in the membership table of a set C (or rather,

C_{1}

C_{2}

C_{3}

, …) means that an element belonging to the universal set

U

considered does belong to that set C (that is, to

C_{1}

C_{2}

C_{3}

, …).

Figure 1 presents a) the truth table of the negation

\bar{q}

(not q) of the proposition q and b) the membership table of the complement

\overset{+}{C}

of a set C.

Figure 1. a) truth table for

\bar{q}

and b) membership table for

\overset{+}{C}

Figure 1. a) truth table for

\bar{q}

and b) membership table for

\overset{+}{C}

The first row of the truth table represented in Figure 1a is

0, 1

. The first digit – 0 – in that numerical sequence in column q means that it is accepted that q is false. The second digit – 1 – in that numerical sequence means that it is accepted that

\bar{q}

is true. In other words, if the proposition q is false, then its negation (the proposition

\bar{q}

) is true.

The second row of the truth table represented in Figure 1a is

1, 0

. The first digit – 1 – in that numerical sequence in column q means that it is accepted that q is true. The second digit – 0 – in that numerical sequence means that it is accepted that

\bar{q}

is false. In other words, if the proposition q is true, then its negation (

\bar{q}

) is false.

The first row of the membership table in Figure 1b is 0, 1. The first digit – 0 – in that numerical sequence in column C means that a certain element belonging to the universal set

U

considered does not belong to the set C. The second digit – 1 – in that numerical sequence means that the element does belong to the complement set

\overset{+}{C}

of C. In other words, if any element belonging to the universal set

U

considered does not belong to a set C, then it does belong to the complement set

\overset{+}{C}

of that set.

The second row of the membership table in Figure 1b is

1, 0

. The first digit – 1 – in that numerical sequence in column C means that a certain element belonging to the universal set

U

considered does belong to the set C. The second digit – 0 – in that numerical sequence means that the element does not belong to the complement set

\overset{+}{C}

of C. In other words, if any element belonging to the universal set

U

considered belongs to the set C, then it does not belong to the complement set

\overset{+}{C}

of that set.

The symbol corresponding to the operator of the conjunction of two propositions

q_{1}

and

q_{2}

will be ∧. The symbol corresponding to the operator of the intersection of two sets

C_{1}

and

C_{2}

will be symbolized as ∩.

Figure 2 presents a) the truth table for the conjunction

q_{1} \land q_{2}

(

q_{1}

and

q_{2}

) and b) the membership table for the intersection set (

C_{1} \cap C_{2}

) of

C_{1}

and

C_{2}

Figure 2. a) truth table for q₁ ʌ q₂ and b) membership table for C₁ ∩ C₂

As seen in Figure 2a, only in the fourth row of the above truth table (where the numerical sequence 1, 1, 1 appears) is there a 1 in the column corresponding to

q_{1} \land q_{2}

. In other words, only if

q_{1}

is true as indicated by the first 1 in that numerical sequence, and

q_{2}

is also true as indicated by the second 1 in that numerical sequence, is

q_{1} \land q_{2}

also true, as indicated by the third 1 in the numerical sequence considered. In the other three possible cases, considered in the first, second and third rows of the truth table in Figure 2, there is a 0 in the column corresponding to

q_{1} \land q_{2}

, indicating that the proposition is false.

As seen in Figure 2b, only in the fourth row of the membership table (where the numerical sequence 1, 1, 1 appears) is there a 1 in the column corresponding to

C_{1} \cap C_{2}

. In other words, only if any element belonging to the

U

considered belongs to

C_{1}

, as indicated by the first 1 in that numerical sequence, and also belongs to

C_{2}

, as indicated by the second 1 in that numerical sequence, does that element belong to

C_{1} \cap C_{2}

(the intersection set of

C_{1}

and

C_{2}

q_{1}

is made to correspond to

C_{1}

q_{2}

C_{2}

, the operator of conjunction ∧ in propositional calculus to the operator of intersection ∩ in set theory (and as a result the correspondence between

q_{1} \land q_{2}

and

C_{1} \cap C_{2}

is established), the isomorphism existing between the truth table in Figure 2 and the membership table in 2b can be observed. In effect, for every 0 in the first table there is a corresponding 0 in the second table, and for every 1 in the first table there is a corresponding 1 in the second table.

The symbol of the inclusive disjunction (inclusive or) of the two propositions

q_{1}

and

q_{2}

will be ∨. The symbol of the exclusive disjunction (exclusive or) of two propositions

q_{1}

and

q_{2}

will be

\dot{\lor}

. The symbol corresponding to the inclusive union of two sets

C_{1}

and

C_{2}

will be ∪. The symbol corresponding to the exclusive union of two sets

C_{1}

and

C_{2}

will be

\dot{\cup}

Figure 3 presents a) the truth tables corresponding to the inclusive disjunction of

q_{1}

and

q_{2}

(

q_{1} \lor q_{2}

) and to the exclusive disjunction of

q_{1}

and

q_{2}

(

q_{1} \dot{\lor} q_{2}

), along with b) the membership tables for the inclusive union set of

C_{1}

and

C_{2}

(

C_{1} \cup C_{2}

) and of the exclusive union set of

C_{1}

and

C_{2}

(

C_{1} \dot{\cup} C_{2}

Figure 3. a) truth tables for q₁ ∨ q₂ and of q₁

\overset{\cdot}{\lor}

q₂, and b) membership tables for the sets C₁ ∪ C₂ and C₁

\overset{\cdot}{\cup}

C₂

Figure 3. a) truth tables for q₁ ∨ q₂ and of q₁

\overset{\cdot}{\lor}

q₂, and b) membership tables for the sets C₁ ∪ C₂ and C₁

\overset{\cdot}{\cup}

C₂

q_{1}

is made to correspond to

C_{1}

q_{2}

C_{2}

, the operator ∨ in propositional calculus to the operator ∪ in set theory, and the operator

\dot{\lor}

in propositional calculus to the operator

\dot{\cup}

in set theory, note may be taken of 1) the isomorphism between the truth table corresponding to

q_{1} \lor q_{2}

and the membership table for

C_{1} \cup C_{2}

(given that for every 0 in this truth table, there is a 0 in the membership table), and 2) the isomorphism between the truth table for

q_{1} \dot{\lor} q_{2}

and the membership table for

C_{1} \dot{\cup} C_{2}

, for the same reason mentioned in the above isomorphism.

Note that in the column corresponding to

q_{1} \lor q_{2}

there is a sole 0 indicating that the inclusive disjunction is false only when both

q_{1}

and

q_{2}

are false, as shown by the zeros in the first row both in the

q_{1}

column and in the

q_{2}

column. Likewise, note that in the column corresponding to

C_{1} \cup C_{2}

there is a sole 0 indicating that any element belonging to the universal set

U

considered does not belong to

C_{1} \cup C_{2}

, only when that element belongs neither to

C_{1}

nor to

C_{2}

, as shown by the zeros present in the first row both in the

C_{1}

column and in the

C_{2}

column.

Note that in the truth table for the proposition

q_{1} \dot{\lor} q_{2}

this proposition is true if only one of the two propositions

q_{1}

and

q_{2}

is true. Likewise, in the membership table corresponding to

C_{1} \dot{\cup} C_{2}

it can be seen that only the elements belonging to the universal set

U

considered that belong only to one of the two sets

C_{1}

and

C_{2}

belong to that set (

C_{1} \dot{\cup} C_{2}

The operator of material implication in propositional calculus is symbolized as →. The proposition

q_{1} \to q_{2}

(that is,

q_{1}

materially implies

q_{2}

) can be read as “If

q_{1}

, then

q_{2}

”. The proposition

q_{2} \to q_{1}

(that is,

q_{2}

materially implies

q_{1}

) can be read as “If

q_{2}

, then

q_{1}

”. In the proposition

q_{1} \to q_{2}

q_{1}

is known as the antecedent of that proposition and

q_{2}

is its consequent. In the proposition

q_{2} \to q_{1}

q_{2}

is the antecedent of that proposition and

q_{1}

is its consequent.

Given two sets

C_{1}

and

C_{2}

, the use of the operator of membership implication —|→ makes it possible to generate the sets C₁ —|→ C₂ and C₂ —|→ C₁.

Figure 4 presents 1) the truth table for the proposition

q_{1} \to q_{2}

and the truth table for the proposition

q_{2} \to q_{1}

, and 2) the membership table for the set

C_{1}

—|→ C₂

and the membership table for the set

C₂ —|→ C₁.

Figure 4. a) Truth tables for the propositions q₁ → q₂ and q₂ → q₁, and membership tables for the sets C₁ −|→ C₂ and C₂ −|→ C₁.

Note that the proposition

q_{1} \to q_{2}

is false only when

q_{1}

is true and

q_{2}

is false. Likewise, any element belonging to the universal set

U

considered does not belong to the set

C_{1}

—|→ C₂ only if that element belongs to the set C₁ —|→ C₂.Note also that the proposition

q_{2} \to q_{1}

is false only when

q_{2}

is true and

q_{1}

is false. Likewise, any element belonging to the universal set

U

considered does not belong to the set

C_{2}

—|→ C₁

only if that element belongs to the set

C₂

but not to the set

C₁.

The truth table for

q_{1} \to q_{2}

is isomorphic to the membership table for

C_{1}

—|→ C₂ Both tables are the same from a purely numerical view point.) This also occurs with the truth table for q₂ —|→ q₁ and the membership table for C₂ —|→ C₁.

The operator of logical equivalence (or of material bi-implication) in propositional calculus will be symbolized as ⟷. The operator of membership bi-implication in set theory will be symbolized as

\leftarrow \to

. The operator of membership bi-implication in set theory will be symbolized as

\leftarrow | \to

. Figure 5 presents a) the truth table for the proposition q₁ ←→ q₂ (that is, “q₁ is logically equivalent to q₂” and b) the membership table for the set C₁ —|→ C₂.

Figure 5. a) truth table for the proposition q₁ ↔ q₂ and b) membership table for the set C₁ ←|→ C₂

In the truth table for

q_{1} ⟷ q_{2}

it is seen that the proposition is true in only two of the four possible cases: the cases in which

q_{1}

and

q_{2}

have the same truth value (that is, if both propositions are false or if both propositions are true). These cases are considered in rows 1 and 4 of the truth table. In the membership table for

C_{1}

—|→ C₂ it is seen that any element belonging to the universal set

U

considered belongs to the set C₁ —|→ C₂ only in two of the four possible cases: if that element belongs neither to the set C₁ nor to the setC₂, or if that element belongs both to the set C₁ and to the set C₂. These cases are consideres in rows 1 and 4 of the membership table.

The Sheffer stroke operator – or NAND, the negation of the conjunction of two propositions

q_{1}

and

q_{2}

– will be symbolized as ↑. The Sheffer stroke operator in set theory will be symbolized as

-_{|}^{↑}

. Figure 6 represents a) the truth table for the proposition

q_{1} ↑ q_{2}

, and b) the membership table for the set

C_{1} -_{|}^{↑} C_{2}

Figure 6. a) truth table for the proposition q₁ ↑ q₂; and b) membership table for the set C₁7

-_{|}^{↑}

C₂

Figure 6. a) truth table for the proposition q₁ ↑ q₂; and b) membership table for the set C₁7

-_{|}^{↑}

C₂

In the truth table for the proposition

q_{1} ↑ q_{2}

it is seen that in only one of four possible cases is the proposition false: that in which both

q_{1}

and

q_{2}

are true. In the membership table for

C_{1} -_{|}^{↑} C_{2}

it is seen that in only one of four possible cases, does any element whatsoever belonging to the universal set

U

considered not belong to the set

C_{1} -_{|}^{↑} C_{2}

: that in which that element belongs both to

C_{1}

and to

C_{2}

If the proposition

q_{1}

is made to correspond to the set

C_{1}

, the proposition

q_{2}

is made to correspond to the set

C_{2}

, the operator ↑ in propositional calculus to the operator

[] \to [] \to

in set theory – and therefore,

q_{1} ↑ q_{2}

C_{1} -_{|}^{↑} C_{2}

– it can be seen that the truth table for

q_{1} ↑ q_{2}

is isomorphic to the membership table for

C_{1} -_{|}^{↑} C_{2}

. (Note that both tables are the same from a purely numerical viewpoint.)

The operator Peirce’s arrow – or NOR, the negation of the inclusive disjunction of two propositions

q_{1}

and

q_{2}

– will be symbolized as

↓

. The operator Peirce’s arrow in set theory will be symbolized as

-_{↓}^{|}

. Figure 7 presents a) the truth table for the proposition

q_{1} ↓ q_{2}

and b) the membership table for the set

C_{1} -_{↓}^{|} C_{2}

Figure 7. a) truth table for the proposition q₁ ↓ q₂ and b) membership table for the set C₁

-_{↓}^{|}

C₂

Figure 7. a) truth table for the proposition q₁ ↓ q₂ and b) membership table for the set C₁

-_{↓}^{|}

C₂

In the truth table for the proposition

q_{1} ↓ q_{2}

it is seen that in only one of four possible cases is the proposition true: that in which both

q_{1}

and

q_{2}

are false. In the membership table for

C_{1} -_{↓}^{|} C_{2}

it is seen that in only one of four possible cases does any element whatsoever belonging to the universal set

U

belong to the set

C_{1} -_{↓}^{|} C_{2}

: that in which that element belongs neither to the set

C_{1}

nor to the set

C_{2}

If the proposition

q_{1}

is made to correspond to the set

C_{1}

, the proposition

q_{2}

to correspond to the set

C_{2}

, the operator ↓ in propositional calculus to the operator

-_{↓}^{|}

in set theory – and therefore,

q_{1} ↓ q_{2}

C_{1} -_{↓}^{|} C_{2}

– it can be seen that the truth table for

q_{1} ↓ q_{2}

is isomorphic to the membership table

C_{1} -_{↓}^{|} C_{2}

. (Note that both tables are the same from a purely numerical viewpoint.)

3. Each Law – or Tautology – of Propositional Calculus Is Isomorphic to an Expression Corresponding to a Set Equal to the Universal Set Considered

Any proposition resulting from operations between n propositions, for

n = 1, 2, 3, \dots

, which is true because of its logical form, regardless of the truth value of each of those n propositions, is known as a law (or tautology) in propositional calculus.

In this section reference will be made to several laws (or tautologies) of propositional calculus. For each a specification will be given, according to section 2 above, of the corresponding set which is isomorphic to that law. A representation will be provided of 1) the truth table for that law and 2) the membership table for that set.

In each truth table it will be possible to note in the corresponding column that, regardless of the truth values of n propositions considered to construct it, the proposition that qualifies as a law is true. In each membership table it will be possible to note how, regardless of whether any element of the universal set

U

considered belongs or not to one of the n sets considered to construct the set corresponding to that law – which is isomorphic to it – that element belongs to this latter set. Therefore, that set is equal to the universal set

U

considered.

First of all consider in Figure 8 the “law of the excluded middle”:

q \lor \bar{q}

. In this case

n = 1

Figure 8. a) truth table for the law q ∨

\overline{q}

and b) membership table for the set C ∪

\overset{+}{C}

Figure 8. a) truth table for the law q ∨

\overline{q}

and b) membership table for the set C ∪

\overset{+}{C}

The set corresponding to the law

q \lor \bar{q}

– or which is isomorphic to it – is the following:

\overset{+}{C}

Consider in Figure 9 one of the De Morgan’s laws in propositional calculus:

\bar{(q_{1} \land q_{2})} ⟷ ({\bar{q}}_{1} \land {\bar{q}}_{2})

. In this case

n = 2

Figure 9. a) truth table for the law (q₁ ∧ q₂) ↔ (q₁ ∨ q₂) and b) membership table for the set (C₁ ∩ C₂) ←|→ (C₁ ∪ C₂)

The set corresponding to the law

\bar{(q_{1} \land q_{2})} ⟷ ({\bar{q}}_{1} \land {\bar{q}}_{2})

– or which is isomorphic to it – is the following:

\overset{+}{(C_{1} \cap C_{2})} \leftarrow | \to (\overset{+}{C_{1}} \cap \overset{+}{C_{2}}

Consider in Figure 10 another De Morgan’s law in propositional calculus:

\bar{(q_{1} \lor q_{2})} ⟷ ({\bar{q}}_{1} \land {\bar{q}}_{2})

. In this case

n = 2

Figure 10. a) truth table for the law (q₁ ∨ q₂) ←→ (q₁ ∧ q₂) and b) membership table for the set (

\overset{+}{(C_{1} \cup C_{2})} \leftarrow | \to (\overset{+}{C_{1}} \cap \overset{+}{C_{2}})

Figure 10. a) truth table for the law (q₁ ∨ q₂) ←→ (q₁ ∧ q₂) and b) membership table for the set (

\overset{+}{(C_{1} \cup C_{2})} \leftarrow | \to (\overset{+}{C_{1}} \cap \overset{+}{C_{2}})

The set corresponding to the law

\bar{(q_{1} \lor q_{2})} ⟷ ({\bar{q}}_{1} \land {\bar{q}}_{2})

– or which is isomorphic to it – is the following:

\overset{+}{(C_{1} \cup C_{2})} \leftarrow | \to (\overset{+}{C_{1}} \cap \overset{+}{C_{2}}

Consider in Figure 11 the law of propositional calculus “modus ponendo ponens”:

((q_{1} \to q_{2}) \land q_{1}) \to q_{2}

. In this case

n = 2

Figure 11. a) truth table for the law ((q₁ → q₂) ∧ q₁) → q₂ and b) membership table for the set ((C₁ −|→ C₂) ∩ C₁) −|→ C₂

The set corresponding to the law

((q_{1} \to q_{2}) \land q_{1}) \to q_{2}

– or which is isomorphic to it – is the following:

((C_{1} ⟶ ∣

C₂) C₁) ⟶∣ C₂.

Consider in Figure 12 the law of propositional calculus “modus tollendo tollens”:

((q_{1} \to q_{2}) \land {\bar{q}}_{2}) \to {\bar{q}}_{1}

. In this case

n = 2

Figure 12. a) truthtable for the law ((q₁ → q₂) ∧ q₂) → q₁, and b) membership table for the set ((C₁ −|→ C₂) ∩ C₂) −|→ C₁

The set corresponding to the law

((q_{1} \to q_{2}) \land {\bar{q}}_{2}) \to {\bar{q}}_{1}

– or which is isomorphic to it – is the following:

((C_{1}

—|→ C₂) ∩

\overset{+}{C_{2}}

—|→

\overset{+}{C_{2}}

Consider in Figure 13 the law

(q_{1} \dot{\lor} q_{2}) ⟷ \bar{(q_{1} ⟷ q_{2})}

in propositional calculus. In this case

n = 2

Figure 13. a) truth table for the law (

(q_{1} \dot{\lor} q_{2}) ⟷ \bar{(q_{1} ⟷ q_{2})}

) and b) membership table for the set (

(C_{1} \dot{\cup} C_{2})

) ←|→ (

\overset{+}{(C_{1} \leftarrow | \to C_{2})}

)

Figure 13. a) truth table for the law (

(q_{1} \dot{\lor} q_{2}) ⟷ \bar{(q_{1} ⟷ q_{2})}

) and b) membership table for the set (

(C_{1} \dot{\cup} C_{2})

) ←|→ (

\overset{+}{(C_{1} \leftarrow | \to C_{2})}

)

The set corresponding to the law

(q_{1} \dot{\lor} q_{2}) ⟷ \bar{(q_{1} ⟷ q_{2})}

– or which is isomorphic to it – is the following: (

(C_{1} \dot{\cup} C_{2})

) ←|→ (

\overset{+}{(C_{1} \leftarrow | \to C_{2})}

Consider in Figure 14 the law of transitivity of material implication in propositional calculus:

((q_{1} \to q_{2}) \land (q_{2} \to q_{3})) \to (q_{1} \to q_{3})

. In this case

n = 3

Figure 14. a) truth table for the law ((q₁ → q₂) ∧ (q₂ → q3)) → (q₁ → q3) and b) membership table for the set ((C₁ −|→ C₂) ∩ (C₂ −|→ C3)) −|→ (C₁ −|→ C3)

The set corresponding to the law

((q_{1} \to q_{2}) \land (q_{2} \to q_{3})) \to (q_{1} \to q_{3})

– or which is isomorphic to it – is the following:

((C_{1}

—|→ C₂) C₂ —|→ C3)) —|→ (C₁ —|→ C3).

4. Discussion and Perspectives

The importance given in the instructional approach proposed here to the structure of the subject matter presented, explains and justifies why reference is made to a contribution of the general systems theory of Ludwig von Bertalanffy [1], regarding the detection and use of isomorphisms between the laws and regularities of diverse areas of knowledge.

In a future article on this topic, a broader and more detailed characterization of this approach will be provided. In addition, in this and other papers clear examples of its usage will be given. This has been done here by pointing out certain correspondences between propositional calculus – the most basic level of logic – and set theory. In these articles, emphasis will be given to correspondences existing between propositional calculus, predicate calculus, set theory, Boole’s algebra, and syllogistics, a historically and instructionally important discipline of logic.

Author Contributions

Conflicts of Interest

References

Bertalanffy, L. (1968). In General System Theory: Foundations, Development, Applications; George Braziller.
Grassmann, W. K. & J.-P. Tremblay (1996). In Logic and Discrete Mathematics; Prentice Hall.
Quine, W. V. (1952; 1982). In Methods of Logic, 4th ed.; Harvard University Press.
Swaart, H. (2018). In Philosophical and Mathematical Logic; Springer.
Jech, T. J. (2013). In Set Theory (The Third Millenium Edition); Springer.

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A Structural Approach for the Concurrent Teaching of Introductory Propositional Calculus and Set Theory

Abstract

1. Introduction

2. Pairs of Operations Corresponding to 1) Propositional Calculus and 2) Set Theory

3. Each Law – or Tautology – of Propositional Calculus Is Isomorphic to an Expression Corresponding to a Set Equal to the Universal Set Considered

4. Discussion and Perspectives

Author Contributions

Conflicts of Interest

References

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