Why are the AGORA Argument Schemes Logically Valid?

The AGORA system provides currently four argument schemes that are logically valid:  modus ponens; modus tollens; disjunctive syllogism; not-all syllogism. As discussed here, an argument is logically (or deductively) valid if and only if it follows an argument scheme that is logically valid. An argument scheme is logically valid if and only if it is impossible for any argument following this scheme to have true premises and a false conclusion.

Since the AGORA system is so designed that any argument you construct is automatically presented in logically valid form, controversies are only possible with regard to the truth of the premises of your arguments, but not regarding their validity. This should simplify discussions because this way everything that might be controversial is now visible in form of a premise. But for this to happen you should understand that the argument schemes used in AGORA are indeed logically valid.

The logical validity of these schemes can be demonstrated in several ways. In the following we use the so-called “truth table” method. This method is based on the assumption that any statement can only be true or false. Let us have a look at the argument scheme modus ponens:

  • p

  • if p, then q

  • Therefore, q

According to the definition of validity mentioned above, this scheme is logically valid if—whatever statements we insert for “p” and “q”—the conclusion “q” will always be true as long as the two premises “p” and “if p, then q” are true. OK, but since “p” and “q” are variables which can represent all sorts of statements, we cannot know whether “p,” “q,” and “if p, then q” are actually true or not. However, since there are only two truth values (true and false) for each of the statements “p,” “q,” and “if p, then q,” and since there are only these three statements in a modus ponens argument, it should be possible simply to enumerate all possible combinations of these statements in which “p,” “q,” and “if p, then q” are either true or false. That means: if we simply represent all possible combinations in form of a table, then we only need to see whether we can find a case in which the premises “p” and “if p, then q” are true, but the conclusion “q” is false. If we find such a case, then the respective argument scheme cannot be logically valid because this case violates what we defined as logical validity. On the other hand, if for all cases in which the premises “p” and “if p, then q” are true also the conclusion “q” is true, then the scheme is logically valid. Since we covered all possible combinations in our table, we can be sure that there are no further cases that we did not take into consideration.

This sounds complicated. But it is quite easy because the procedure can be simplified based on the fact that the two truth values of compound statements such as “if p, then q” can be defined with reference to the truth values of their elementary statements “p” and “q.” Philo of Megara defined already the conditional statement “if p, then q” by saying that this statement, as a whole, is only false in case “p” is true but “q” false. In all other cases—that is, “p” is true and “q” is true, but also in any case in which “p” is false—the compound statement as whole is true. Since all compound statements that are used in logical argument schemes can be defined with reference to the truth values of their elements, the tables that we need for validity proofs are rather simple. This, however, means that we need to start with the definitions of these compound statements before we can do the actual proofs.

Defining logical connectors

The validity of the logical argument schemes can only be proven if we have clear definitions of the logical connectors that are used in the form of compound statements in the arguments "enabler" (the premise underneath the "therefore" which connects the reasons and the conclusion). These logical connectors need to be precisely defined before we can prove the validity of the four argument schemes.

These definitions are important because the logical connectors used in AGORA-net are formulated in ordinary language such as "if ... then" and "either ... or." Their meaning, however, is not exactly the same as in ordinary language. For logical validity, these connectors must be what is called "truth-functional," that is: they must be defined in a way that we can indeed prove, as the definition of "validity" requires, that it is impossible that a conclusion is false if all the premises of an argument are true.

Since every statement has, according to (a) above, one of two possible truth values—namely “true” and “false”—and since it is sufficient to look only at the connections between two statements p and q, any connector can be defined by a table that lists all four possible combinations of the statements p and q, in which each of them is assumed to be either true or false. (Each statement is considered to be a complete and well-formed sentence that can either be true or false, like “Paul is a rational being." Note that for defining "negation," as you will see below, just one statement is sufficient):

p q
T T
T F
F T
F F

This table is a complete representation of all possible combinations of two statements that have two possible truth values. In this table, the focus is on the truth or falsity of the elementary statements p or q. In the tables that you can find below, by contrast, this list of possibilities is enlarged by a column that represents the possible truth values of a connected statement. Here it is the connected statement that is either true or false. This difference is important. It makes a difference whether elementary statements such as “it is raining” and “the street is wet” are true or false, or connected statements like “if it is raining, then the street is wet.” The fact that a connected statement can have a truth value in itself—that is, as connected statement—becomes clear when we consider that we might probably not accept the truth of the following connected statement: “if the street is wet, then it is raining.” (The street could also be wet because it has been cleaned.) 

The fact that a connected statement can have a truth value of its own can be used to define the exact meaning of logical connectors. AGORA uses only four further logical connectors: 1. Negation; 2. if-then (and others that are logically equivalent to it); 3. either-or; and 4. it cannot be the case, at the same time, that...

1. Negation

Since negation can be defined by only one statement p, AGORA’s negation indicator “it is not the case that” is defined by the following truth table (we could, of course, also simply use “not”; but “not” can be used in natural language in a variety of positions within sentences which makes the software-generated handling of negation much more complicated to code):

p It is not the case that p
T F
F T

The meaning of "it is not the case that p" is here defined by two facts: 1. by the fact that this statement is false if p is true (first line), and 2. by the fact that it is true if p is false (second line). This is all you need to define the meaning of this logical connector because in this table every possible situation in which you might encounter the statement "it is not the case that p" is covered. (There are only two such situations here, either is true or it is false.)

2. If - then

If you look at the following table, you can say that the meaning of the connector “if …, thenis defined by how the truth values of “if p, then q” are correlated to all possible combinations of the elementary statements p and q with their possible truth values.

p q if p, then q
T T T
T F F
F T T
F F T

This truth table defines the meaning of “if …, then” by the fact that—according to this particular arrangement of truth values—the connected statement “if p, then q” is only false if p is true and q is false (as shown in the second row). In all the other possible cases, the connected statement is true, even in the strange cases that are described in lines 3 and 4 of the table.

The convention on which the definition of "if ..., then" in the table above is based can be defended when we interpret "if p, then q” as something like a universal law: any time is the case, is also the case. A universal law “if p, then q” would even be true in those situations in which p is false or both p and q are false (as in lines 3 and 4 of the table).

This can be shown with an example from mathematics (see Gamut 1991, 34): Everybody would probably agree that the statement “If a number is larger than 5, then it is larger than 3” is universally true. We would probably not even want to give up the truth of this connected statement if we replace the variable “a number” by the following instantiations in which some of the elementary statements are obviously false:

  1. If 6 > 5, then 6 > 3.
  2. If 4 > 5, then 4 > 3.
  3. If 2 > 5, then 2 > 3.

The first instantiation of the general rule corresponds to the first line in the truth table above, the second corresponds to the third line, and the third to the fourth line, if you consider the truth values of the elementary statements. The statement “If a number is larger than 5, then it is larger than 3” is only universally true if it remains true even in situations (2.) and (3.) in which p is obviously false. That means, the truth table above represents the fact that we want “if p, then q” to be universally true even if p is false. This universal statement is only false, if p is true, but q false, as in the second line of the table.

The AGORA system does not only provide “if p, then q” as a language form for modus ponens arguments, but also: “p implies q,” “p only if q,” “q provided that p,” “whenever p, q,” “p is a sufficient condition for q,” and “q is a necessary condition for p.” (Note the reversed order of the propositions p and q in the last case! The difference between the last two is the following: In the case “p is a sufficient condition for p,” you have exactly the same situation as in “if p, then q”: p is sufficient to get q. However, in the case “q is a necessary condition for p,” the only thing you can know is: if p is the case, then also q, because q is the necessary condition for p. This situation is similar to “p only if q” which means: Since q is necessary to get p, the only thing you can know is: If you have p, then also q.)

All these different language forms—in exactly the order of elementary propositions that you find in the list at the beginning of the previous paragraph—are defined exactly as “if p, then q” in the table above. That means, you can replace “if p, then q” by each of them, and you would define each of them by the same distribution of truth values as in the table. This variety of language forms for the same valid argument scheme is useful because it allows you to select the form that fits best, depending on what you want to argue for. Sometimes one formulation simply sounds better or is more convincing than another one.

3. Either - or

Either - or, that is the “disjunction” or “alternative,” is again more specifically defined than in ordinary language. In normal language “or” can be used in two different senses, as becomes clear in the following examples:

  1. Either I hang myself or I shoot myself
  2. Either Lincoln was the 16th president or Johnson was the 17th president.

The point is, the “or” in (a) is an “exclusive or” (because when I do one of these things, I don’t have a chance to do the other), whereas the “or” in (b) is an “inclusive or” (as it turns out, both statements are true). In ordinary language, the difference is often not clear or depends on the context. For example, when somebody says “I go and buy milk or orange juice,” we do not know whether the intended meaning of “or” is inclusive or exclusive.

The difference, however, is important for the validity of the disjunctive syllogism, as we will see. The disjunctive syllogism is based on an inclusive "or," and that means only the argument on the left is logically valid, but not the one on the right:

I buy milk or orange juice

I buy milk or orange juice

I do not buy milk I buy milk
Therefore, I buy orange juice Therefore, I do not buy orange juice

This means, the disjunctive syllogism always infers one of two (or more) statements from the negation of the other(s), but not the other way around. By contrast, if we would use what could be called a “XOR syllogisms,” which is based on an “exclusive or” (or short: XOR), then we could infer in both directions, from negative to positive and the other way.

In logic, the inclusive either - or is defined by the following truth table:

p q either p or q
T T T
T F T
F T T
F F F

AGORA does not offer a XOR syllogism because if you want to formulate an argument that infers a negative statement from a positive one, you can use what is called in AGORA a “not-all syllogism.” 

4. It cannot be the case, at the same time, that ...

This last logical connector in AGORA is used in the so-called "not-all syllogism" (see below). Whereas in a disjunctive syllogism an affirmed statement can be inferred from one or more negated statements, the not-all syllogism works the other way: you can infer a negated statement from one or more affirmed statements. Since we need only two statements to define the meaning of "it cannot be the case, at the same time, that ...," we define as follows:

p q It cannot be the case, at the same time, that p and that q
T T F
T F T
F T T
F F T

Proving the validity of the AGORA argument schemes

Based on these definitions of the logical connectors used in the AGORA system, we can now prove the validity of the four argument schemes currently used in AGORA.

A. Modus ponens 

Modus ponens, also known as "affirming the antecedent," is structured as follows:

  • p
  • If p, then q
  • Therefore, q

We can arrange this list of two premises and a conclusion in the form of three columns that follow two columns in which all possible combinations of truth values for the two elementary statements and are listed:

p q Premise: p Premise:
if p, then q
Concl.:
q
T T T T T
T F T F F
F T F T T
F F F T F

The first two columns, as in the definitions of connectors above, list simply all possible truth values for all possible combinations of the two statements p and q. The third column presents the possible truth values for the first premise, simply copied from the first column which defines the truth values for p. The truth values of the fourth column are based on the definition of “if …, then” that we developed above. And the values of the last column are again simply copied from the second one which defines the truth values of q.

Since logical validity is defined by the fact that the conclusion is true if all the premises of an argument are true, we only have to check those rows in our truth table in which all the premises are true. In the truth table for modus ponens above, this is the case only in the first row (in all the other rows at least one of the premises is false so that these rows are not relevant for checking the validity). Since in this row also the conclusion is true (last column), the argument scheme is logically valid. That’s it; that is the entire proof. Since all possible combinations are covered in the table, and since there is none among them in which all the premises are true but the conclusion false, the scheme presented in the table is logically valid.

In order to get a better understanding of the method, let us, by contrast, prove that the so-called “affirming the consequent” is an invalid argument scheme. Affirming the consequent is structured as follows:

  • If p, then q
  • q
  • Therefore, p

To prove whether this argument is logically valid or not, we construct again a truth table along the lines described above for modus ponens. Again, simply following the definitions, we get the following truth-table:

p q Premise:
if p, then q
Premise:
q
Conclusion:
p
T T T T T
T F F F T
F T T T F
F F T F F

Again, guided by the definition of logical validity we check whether there is a line in the table in which all premises are true but the conclusion false. In this table, however, we find such a case in the third row. This proves that the scheme is invalid (because it can happen that the conclusion is false, even though all the premises are true, violating thus the definition of logical validity).

B. Modus tollens

According to the same procedure, the validity of modus tollens can be proven by the following truth table. Modus tollens is structured as follows:

  • If p, then q
  • not q
  • Therefore, not p

(Note that the truth values of the fourth column are the opposite of those in the second, based on our definition of negation above; the same happens in fifth column compared to the first one):

p q Premise:
if p, then q
Premise:
not q
Concl.:
not p
T T T F F
T F F T F
F T T F T
F F T T T

There is only one row in which all the premises are true (the fourth), and in this row also the conclusion is true, satisfying thus the requirement of logical validity.

C. Disjunctive syllogism

Disjunctive Syllogism is structured as follows:

  • either p or q (in disjunctive syllogism, the meaning of "or" is that of an inclusive "or." See above)
  • not q
  • Therefore, p

(Note that it does not matter whether you infer p from not q or q from not p):

p q Premise:
either p or q
Premise:
not q
Concl.:
p
T T T F T
T F T T T
F T T F F
F F F T F

There is only one row in which all the premises are true (the second), and in this row also the conclusion is true, satisfying thus the requirement of logical validity.

D. Not-all syllogism

Not-All Syllogism is structured as follows:

  • It cannot be the case, at the same time, that p and that q
  • p
  • Therefore, not q

(Note that it does not matter whether you infer not p from q or not q from p):

p q Premise: It cannot be the case, at the same time, that p and that q Premise: p Concl.: not q
T T F T F
T F T T T
F T T F F
F F T F T

There is only one row in which all the premises are true (the second), and in this row also the conclusion is true, satisfying thus the requirement of logical validity.
 

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Gamut, L. T. F. (1991). Logic, language, and meaning. Introduction to Logic. (Back to text)