Using C-Tables to Teach Class Logic

Introduction

A highly general view of the history of formal logic is that it falls into three periods: traditional logic (founded by Aristotle in the 4th century BCE), algebraic logic (founded by Boole in the first half of the 19th century), and symbolic logic (founded by Frege and Peirce in the late 19th century). Russell, Whitehead, and others gave First-order Logic its current form in the early 20th century. Class logic has been an important part of deductive logic throughout its history [1]1. Class logic deals with arguments using properties (i.e., one- place or monadic predicates).

One notable achievement of algebraic and symbolic logicians is their development of logic graphs or diagrams. In the 18th century, mathematician Leonhard Euler (1707- 1783) developed what appears to be the earliest form of logic graphs, called accordingly “Euler diagrams.” In Conceptual Article the 19th century, several logicians developed a variety of logic diagrams. Mathematician John Venn (1834-1923) developed the most widely used graphs, “Venn Diagrams.” Author and mathematician Charles Dodgson (1832-1898) developed “Carroll diagrams” (derived from his pen-name Lewis Carroll). In the 20th century, the computer scientists Edward Veitch (1924-2013) and Maurice Karnaugh (1924- 2022) devised other logic graphs (called respectively “Veitch diagrams” and “Karnaugh maps”). This article aims to show how a tabular method— somewhat similar to Karnaugh maps—can be used instead of Venn (and Euler) Diagrams as a tool in class logic2. I will call these Class Tables (hereafter “C-tables”). In effect, c-tables combine logic graphs with Boolean operators. Class Statements and C-tables Expressing class statements as C-tables is easy. Start with the layout of a C-table. A one-class C-table is a 2-by-2

2  The diagrams are simpler than Karnaugh maps, developed and used to sim- plify circuits in electronic engineering.

table with two cells or cells (Table 1).

Here, ‘A’ is any arbitrary (simple) class, and ‘−A’ is its complement3. Cell 1 represents the things that are A, and cell 2 the things that are not A. A two-class C-table is a 3-by-3 table with 4 cells (Table 2):

A three-class C-table is a 5-by-3 table with 8 cells (Table 3):

Cell 1 represents the class of things that are A and B and C, i.e., (A ∩ B) ∩ C. Cell 2 represents the class of things that are A and not B and C, i.e., (A ∩ −B) ∩ C. The other cells are easy to identify.

Now, here we see one advantage of using C-tables. It is hard to construct Venn diagrams for more than three classes because one has to overlap ovals, making distinguishing cells difficult. However, with C-tables, we can increase the number of classes in a straightforward way. For example, for four classes or terms, we need a 5x4 table (Table 4):

3  I treat A, B, C, … as meta-linguistic variables ranging over classes.

It is again easy to identify the cells—cell 11, for example, is the class (−A ∩ B) ∩ (−C ∩ D). A five-term C-table requires a 9-by-4 table, a six-term C-table requires a 9-by-9 table, a seven-term C-table a 17-by-9, and so on.

Moving next to statements, we can represent singular statements (i.e., statements with named individuals. We can represent the statement ‘x has a property A’ (or ‘x is a member of class A’) as Table 5.

The statement that individual x is not A is represented by Table 6.

We can express general statements (i.e., universal and particular ones) by using the null symbol ‘∅’ and the existence symbol ‘√.’ We can represent the statement that everything is A by Table 7.

We can represent the statement that nothing is A as the C-table (Table 8):

We can represent the statement that something is A by the C-table (Table 9):

And the statement that something is not A by the C-table (Table 10):

The extension is easy for two-class statements with named individuals. We can represent the statement that individual x is both A and B, as in Table 11.

The representations of the statements that x is A but not B, not A but B, and neither A nor B are obvious.

The null and existence symbols readily express the four transitional categorical statements. For example, we can represent the statement that all A’s are B’s in Table 12.

We can express the statement that some A’s are B’s, as shown in Table 13.

We can express the statement that no A’s are B’s as Table 14.

Finally, we can express the statement that some A’s are not B’s as Table 15.

Assessing Categorical Syllogisms

Moving next to assessing categorical syllogisms, again, the job is straightforward, using the null and existence symbols and (as usual) symbolizing universal before particular premises. Example: 1. All A are B 2. All B are C /⸫ All A are C First, we set up the C-table (Table 16).

Second, since both the premises are universal, we can diagram them in any order (Table 17).

Does this capture the conclusion? Yes, because the conclusion requires the cells (A ∩ B) ∩ −C and (A ∩ −B) ∩ −C to be null, and they are. So the argument is valid.

Example: 1. No A are B 2. No C are A /⸫ No C are B Set up the C-table and diagram the premises (Table 18).

Does this table capture the conclusion? No, because to represent it, both the cells (A ∩ B) ∩ C and (−A ∩ B) ∩ C need to be null, but the latter cell isn’t. I have my students show what is missing by putting it in brackets (Table 19).

So, the argument is shown to be invalid.

Example: 1. Some A are B 2. All B are C /⸫ Some A are C

The argument is shown to be valid. To capture the conclusion, a checkmark must be in cell (A ∩ B ) ∩ C, and there is.

Now, one of the trickier skills the student needs to learn in doing any logic graph for testing syllogisms is representing a particular claim when diagramming the universal premises (if any) does not force it into a specific cell. In Venn diagrams, the student does this by placing the check mark on a boundary line—which can be awkward and hard to read. With C-tables, it is fairly easy. Example: 1. Some A are C 2. Some not B are not C /⸫ Some A are not B Since both premises are particular, we can do them in any order. Let’s start with premise 1, shown in Table 21.

The student should read the numbered checkmarks: “There is at least one individual in the cells marked by √1, but we don’t know in which one or ones.” That is, we know that there is at least one thing in A and C, but we don’t know whether it is in B or not. Similarly, diagramming the second premise yields Table 22.

Does this capture the conclusion? No, because that would require check marks with the same number in the cells (A ∩ −B) ∩ C and (A ∩ −B) ∩ −C, and we don’t have that. For all the premises tell us, both of those cells might be empty. We signal this with check marks marked with an index ‘i’ indicating that they have to have the same number and again bracketed to show what is needed to capture the conclusion, shown in Table 23.

Example: 1. No A is C 2. Some not B are not C /⸫ Some A are not B

This is invalid, as Table 24 shows.

The premises require a checkmark in the cell of (A ∩ −B) ∩ −C (since the cell (A ∩ −B) ∩ C is null). And all we have is ‘√1’—which signals that it could, in fact, be empty. Again, we represent this with brackets.

One last example: 1. Some A are B. 2. Some A are not C. /⸫ Some B are not C.

Table 25 shows that this argument is invalid.

Extended Syllogisms

Another advantage of using C-tables is that we can use them to check extended syllogisms mechanically. Example:

1. All A are B.
2. All B are C.
3. Some A are D. /⸫ Some C are D.

Table 31 shows that representing the premises also captures the conclusion, showing that the argument is valid.

A more interesting example:

1. All A are B.
2. No B are C.
3. Some C are D. /⸫ Some A are D.

Table 32 shows that this argument is invalid.

We can extend this further. Example:

1. All A are B.
2. All B are C.
3. No C are D.
4. All D are E. /⸫ No A are E.

This argument is invalid, as shown by Table 33.

A more complicated example:

1. All are B.
2. Some B are C.
3. No C are D.
4. Some D are not E. /⸫ Some A are not E.

Table 34 shows that this argument is invalid.

Finally, an example with six terms:

1. No A are B.
2. No B are C.
3. No C are D.
4. No D are E.
5. Some E are F. /⸫ Some A are not F.

Table 35 shows that this argument is invalid.

We can also examine extended syllogisms with named individuals. Example:

1. x is A.
2. Some A are B.
3. All B are C.
4. No C are D. /⸫ x is not D.

Table 36 shows that this argument is invalid.

Syllogisms with Compound Classes

Some texts extend the power of syllogistic by using “compound classes.” A compound class is a class defined by two properties conjoined. We can represent this by concatenating the property constants. For example, we might express “All friendly dogs are happy” as “All FD are A.” We can use expanded C-tables to express statements with compound classes. For example, “All AB are C” can be represented by Table 37.

However, the table becomes simpler if we treat ‘AB’ as if it were an atom (Table 38).

It becomes simple because we are interested only in two classes: A ∩ B and –(A ∩ B), that is, individuals with both A and B and those without both A and B.

Syllogisms using categorical statements with compound classes are easily assessed with C-tables. Example: 1. All AB are C. 2. No C are D. /⸫ No AB are D.

Table 39 shows that this argument is valid.

Example: 1. All AB are C. 2. Some C are D. /⸫ Some AB are D.

Table 40 shows that this argument is invalid.

The fact that the cell [(AB) ∩ −C] ∩ D is null means that to capture the conclusion, we need a checkmark squarely in the cell [(AB) ∩ C] ∩ D. But all we have from premise 2 is a qualified checkmark indicating that something is either in that cell or another (or both).

1. No AB are CD. 2. No CD are EF. /⸫ No EF are AB.

Table 41 shows that the argument is invalid.

And, of course, we can have extended syllogisms with compound classes. Example:

1. All AB are CD.
2. No CD are EF.
3. Some EF are GH.
4. No GH are IJ. /⸫ Some IJ are not AB.

Conclusion

To summarize, using C-tables instead of Venn (or Euler) diagrams has several advantages. First, they are easy to construct. They require no talent for drawing circles on the board or special templates for word-processing the diagrams.

More importantly, C-tables more easily extend logical assessment to syllogisms of more than three terms. You can represent an extended syllogism with six terms in a C-table more easily than in a Venn diagram. Of course, C-tables (like truth tables, Venn diagrams, Euler diagrams, Carroll diagrams, and Karnaugh maps) are mechanical (i.e., effective) methods of deciding validity. So, they grow exponentially in complexity with adding each new atom. A C-table for an argument with

10 terms would require 1,024 cells. One had better be able to write in a very tiny print. However, syllogisms with (say) six terms are easy to construct, especially when compared with Venn diagrams.

Finally, C-tables easily handle arguments with named individuals and compound terms.