Hypothetical syllogisms as a method to secure information

by Eugenia Bahit

This article is a sample of our technical and scientific writing work. It has been written for computer scientists, programmers, and software engineers and versioned for the Web.

What are syllogisms?

A syllogism is a logical reasoning procedure. The more classic types of syllogism are categorical and hypothetical syllogism.

Both types have three components:

1. The first statement, called the premise
2. A second statement, called the proposition
3. And a final statement, called the conclusion.

The hypothetical syllogism is a logical reasoning procedure where the premise is a conditional statement.

The difference between categorical and hypothetical syllogisms is the first statement, the premise. While the premise of a hypothetical syllogism is a conditional statement (also called the hypothetical statement), the premise of the first one is categorical. Hence, it is possible to define a hypothetical syllogism as a logical reasoning procedure where the premise (the first component) is a conditional statement.

In sciences, the hypothetical syllogism is an essential reasoning procedure because it is the base of the hypothetico-deductive method—also called the H-D method¹—used to construct the hypothesis of a scientific theory.

The hypothetical syllogism and its logic rules

A conditional statement has the form: if `p`, then `q`, where `p` is called the antecedent and `q`, the consequent. In the following example, John is tall is the antecedent and Jane is short, the consequent:

To form a hypothetical syllogism, it is necessary to have a conditional statement followed by a categorical premise, and finished by another categorical statement called the conclusion. The categorical premise could be the antecedent or the denial of the consequent. So that, in the previous example, the possibilities are: John is tall, or Jane is not short.

This is because there are two rules or ways to form a hypothetical syllogism:

1. Affirming the antecedent, also called modus ponens.
2. Denying the consequent, also called modus tollens.

When the antecedent is affirmed, the consequent must be affirmed in the conclusion:

And when the consequent is denied, the antecedent must be denied in the conclusion, such that:

Avoiding fallacies

The evident question could be why it is not possible to deny the antecedent or affirm the consequent in the proposition. The short answer is to avoid fallacies. The more complex one, is that logic and reasoning methods have rules, and in the logic sphere, rules are necessary to prevent reasoning errors.

In psychology, reasoning errors are known as thinking errors. That is because of the difference between thinking and reasoning. While thinking involves everything a person can think of (and foremost, can believe in), reasoning is limited. In that sense, thinking is infinite while reasoning is finite. Even though it is possible to use the conclusion of one argument as a premise for another one, each reasoning needs a final. A logical argument always must have a conclusion.

Thinking errors are generated by cognitive biases, which are beliefs that people have based on previous experiences, political ideologies, and social and cultural aspects, among other factors. Cognitive biases interfere with the natural reasoning process, resulting in a thinking error.

A fallacy is a psychologically persuasive argument that uses general thinking as logical reasoning.

Precisely, a fallacy is a psychologically persuasive argument that uses general thinking as logical reasoning. Formally², there are two types of formal fallacies regarding hypothetical syllogisms: denying the antecedent (or fallacy of the inverse) and affirming the consequent (or fallacy of the converse).

It should be simple to demonstrate the reasoning error with one example. When the cost of petroleum increases, it also tends to increase the price of public transport tickets, so that:

By reasoning forgetting the rules, it is possible to commit the following reasoning error:

However, what happens if the price of public transport tickets increases but not the cost of petroleum? The conclusion would be false because the premise only affirms the consequences of increasing the petroleum cost, but not the causes of the increasing public transport tickets.

So that, the hypothetical syllogism is limited. It only allows to know two things:

1. What consequences will have the fact the antecedent is true.
2. What cause can be rejected when the (supposed) consequence is not true.

A general principle of logical reasoning is that it is not possible to affirm more than has been affirmed in the premise. So that, by following two simple rules (affirming the antecedent to affirm the consequent and denying the consequent to deny the antecedent), fallacies are avoided.

The hypothetical syllogism in software security

These concepts (hypothetical syllogism and reasoning rules) are of interest to software security systems because doing categorical definitions of information makes it possible to establish a hypothesis on each of them to validate the data. For example, a categorical definition of a product price could be like that:

The previous sentence is categorical because its form (S is P) is. It has a subject (formally called the subject term), followed by a "to be" verb (linguistically called a linking verb), and finished by a predicate (formally called the predicate term). There are four forms of categorical propositions:

(A) Every S is P.
(E) No S is P.
(I) Some S is P.
(O) Some S is not P.

Forms A and E are general forms, while forms I and O are particular forms. For software security, A-form is chosen.

A categorical definition can be formally described by using mathematical logic. For example, for the sentence: A price is a real number greater or equal than 0.01 and lower or equal than 9999.99, the formal definition would be:

In this way, it is possible to establish a hypothesis on to validate a variable by saying that if p is a valid price, then p is a real number greater or equal than 0.01 and lower or equal than 9999.99.

There are only two forms to validate p:

1. Affirming that p is a valid price, but this is a paradox because it is trying to validate p precisely.
2. Denying that p is a real number greater or equal than 0.01 and lower or equal than 9999.99. The unique feasible option.

So, by following the second rule, it is possible to infer that:

In order to validate p, it is necessary to have a function `f` of p that receives p, validates p, and returns a Boolean value, which means a true or false. This function would be the following one:

The above statement can be translated to source code in a literal way. Some examples are shown then.

def is_a_price(p):
    term_1 = (type(p) == float)
    statement = (term_1 and (p >= 0.01 and p <= 9999.99))
    if not statement: return False
    return True

function is_a_price($p) {
    $term_1 = (gettype($p) == 'double');
    $statement = ($term_1 and ($p >= 0.01 and $p <= 9999.99));
    if(!$statement) return False;
    return True;
}						

The output of the above function executed in the Python shell would look like:

>>> is_a_price(1)
False
>>> is_a_price(1.1)
True
>>> is_a_price('1.1')
False
>>> is_a_price({1.1})
False
						
						
					

The output of the above function executed in the PHP shell would show:

php > print is_a_price(1);
php > print is_a_price(1.1);
1
php > print is_a_price('1.1');
php > print is_a_price([1.1]);
						
						
					

It is possible to observe that when the variable p has not a suitable data type, the AND statement is not entirely executed because, in that case, it would throw an exception due to '1.1' (and the set that contains the number 1.1) could not be managed as a number. So the language has to obtain the data type to ensure the rest of the conditional can be executed without risk.