 # 5. Logical conditional Por: Sergio Cohaguila

Amor a la física y matemáticas

The study of the logical conditional is not entirely easy, it is not exactly the same as the logical implication, these two have different but similar meanings that we will see progressively in this section.

Another interesting point of the conditional is that not necessarily two statements can form a conditional statement, but also two open statements can form a conditional.

There is also a little controversy and logical incongruity when we study ideal case examples, but let’s take a look at this logical connective right now.

## What is logical conditional?

The logical conditional studies the consequences of the arguments, to achieve this task it is necessary to have prior information to identify these consequences and they must also be included in the arguments.

The previous information on this type of argument we call antecedents and the consequences of these we will call consequent.

In the main section of propositional logic we discussed precisely this point, we have said that the purpose of logic is to conclude, but this conclusion would not exist if it were not for its causal premises or antecedents, a characteristic that fits very well with the logical conditional.

### Definition of the logical conditional

The logical conditional, also called material conditional or simply conditional denoted with symbol $$\rightarrow$$ is a logical connective that joins two statement $$p$$ and $$q$$ called antecedent and consequent respectively forming a new statement denoted by $$p \rightarrow q$$ such that its truth value is false if the antecedent is true and consequent is false, for other combinations of truth values of $$p$$ and $$q$$ it turns out to be true.

A statement that has the logical conditional as the dominant logical connective will be called a conditional statement.

Example

Let the following statement be:

• If she behaves well, then I will take her for a walk.

We can break it down as follows:

• She behaves well.
• I’ll take her for a ride.

By themselves it turns out to be open statements, we can also see that the sentence has two main words, this is “If …., then ….”, where:

• Ifshe behaves” indicates an affirmation of a precedent, in this case, it does not matter who she is.
• ThenI will take her for a walk” indicates a consequence, an event of a cause, here it doesn’t matter who she is to take her for a ride.

We say that

• She behaves well” is the cause
• I take her for a ride” is the consequence or conclusion.

Based on this we can find 4 possible truth combinations of the same statement:

• $$\overbrace{ \textit{If} \ \underbrace{ \textit{ she behaves well } }_{ V }, \textit{then} \ \underbrace{ \textit{I will take her for a walk} }_{V} }^{V}$$
• $$\overbrace{ \textit{If} \ \underbrace{ \textit{she behaves well} }_{ F }, \textit{then} \ \underbrace{ \textit{I will take her for a walk} }_{V} }^{V}$$
• $$\overbrace{ \textit{If} \ \underbrace{ \textit{she behaves well} }_{ F }, \textit{then} \ \underbrace{ \textit{I will take her for a walk} }_{F} }^{V}$$
• $$\overbrace{ \textit{If} \ \underbrace{ \textit{she behaves well} }_{ V }, \textit{then} \ \underbrace{ \textit{I will take her for a walk} }_{F} }^{F}$$

The first 3 statements are true and the last one is false.

Here “she behaves well” is the antecedent and “I take her for a walk” is the consequent. But if the antecedent is formed by two statements such as:

• Humans have two legs.
• Sergio is human.
• Therefore, Sergio has two legs.

Linearly it would be written like this:

• Humans have two legs, Sergio is human, Therefore Sergio has two legs.

Here, the antecedent is “Humans have two legs and Sergio is human” formed by two premises where “Humans have two legs” is the major premise and “Sergio is human” is the minor premise and the consequent is “then, Sergio has two legs”.

To take the truth values of the previous statement, we must first take the truth values formed by the antecedent where we find a logical conjunction “and” that joins the two premises of the statement, then we take the validity between the antecedent and consequent connected by the conditional.

Generally, the study of this type of proposition requires the use of parentheses to indicate which logical connector has a higher hierarchy and which connectors do not.

Keep in mind that the premises are those sentences that precede the conclusion.

## Truth table of the conditional

Based on the definition and the first example of the logical conditional, the following truth table is:

$\begin{array}{ c | c | c } p & q & p \rightarrow q \\ \hline V & V & V \\ V & F & F \\ F & V & V \\ F & F & V \end{array}$

But if we look closely at the table, 3 of the 4 combinations are true where we find a controversial point, if the antecedent is false and consequent is true, the conditional is true.

If you do not understand me, I will explain it in the following way: I can say false premises and even though the conclusion is true, the conditional is still true, or rather, I can give you a true conclusion with false premises. This little controversy we will see it later as auxiliary content.

## Types of conditional statements

The conditional statement that we will present now are derived from the statement of the type $$p \rightarrow q$$, these are classified in conversion, inverse and contraposition, let’s see each one of them.

### Conversion

The conversion results from exchanging the positions of the antecedent and the consequent in a conditional statement. Symbolically we can define it as follows.

#### Definition

If we define the conditional statement as $$p \rightarrow q$$, then it is called a conversion to the statement $$q \rightarrow p$$.

Example

Case 1: we have the following conditional statement:

• $$p$$ = If the sun comes out, then I will leave the house.

The word “the sun comes out” by itself is an affirmation, but not demonstrable, you cannot make the sun rise because simply saying “the sun comes out“, you have no will over the weather stations and the stars just by mentioning them.

But the expression “If the sun comes out” is a forecast, where it can be said to be true or false depending on the circumstances. Symbolizing “the sun comes out” as $$r$$ where this would be the antecedent of the statement $$p$$.

The expression “I will leave the house” is only an order, an action, but the expression “then I will leave the house” is a consequence, a conclusion of a cause. Symbolizing “I will leave home” as $$s$$ where this would be the consequent of the statement $$p$$.

the conditional statement would look like this:

• $$p = r \rightarrow s$$

and its conversion is represented as follows:

• $$p = s \rightarrow r$$

La nueva proposición recíproca sería:

• If I leave the house, then the sun will rise.

As we can see, this is an absurd conditional statement, we do not need to analyze so much to know that our common sense indicates how illogical this statement is. Therefore, the conditional statement $$p$$ that we define does not have a conversion.

Case 2: we have the following conditional statement:

• If $$x$$ is an even number, then it is divisible by $$2$$.

• If $$x$$ is divisible by $$2$$, then it is an even number.

Which makes sense, we see that both $$r \rightarrow s$$ and $$s \rightarrow r$$ are true, where $$r$$ = “$$x$$ is a real number” and $$s$$ = “$$x$$ is divisible by 2“.

Therefore, not all conditional statements can have a conversion, from the previous example of case 1 note also that \ (r ) and \ (s ) are not statements, they are open sentences.

Conditional statements can also have antecedents and consequents as open sentences, this peculiarity only happens with conditionals and biconditionals.

Whether absurd or not, from the point of view of logical inference, the conversion of a statement is a necessary but not sufficient condition for them to be equivalent, that is to say $$p \rightarrow q \neq q \rightarrow p$$.

It is very easy to check it from the truth table, it is known that for a conditional statement it is fulfilled that $$\mathrm{V} \rightarrow \mathrm{F} = \mathrm{F}$$, however, it does not happen the same if we invert the antecedent and consequent $$\mathrm{F} \rightarrow \mathrm{V} = \mathrm {V}$$.

### Inversion

An inversion is the negation of the antecedent and consequent of a conditional statement. As in the previous case, not always an investment can be inferred from a conditional statement.

#### Definition

From the conditional statement $$p \rightarrow q$$ we call the proposition $$\sim p \rightarrow \sim q$$ inversion.

Example

Given the proposition of the type $$p \rightarrow q$$:

• If I walk, then it advanced.

Its inverse $$\sim p \rightarrow \sim q$$, would be:

• If I don’t walk, then I don’t advance.

As we can see, both the statement $$p \rightarrow q$$ and $$p \rightarrow q$$ are true.

In the same way as the reciprocal, the inverse is not equivalent with the conditional statement. We know that $$\mathrm{V} \rightarrow \mathrm{F} = \mathrm{F}$$ and its inverse that results from negating the antecedent and consequent is $$\mathrm{F} \rightarrow \mathrm{V} = \mathrm{V}$$.

We see that the same thing happens with the reciprocal, the interesting and curious thing is that both the reciprocal and the inverse are mutually equivalent.

### Contraposition

A contrapositive of a conditional statement is like a reciprocal but with the antecedent and consequent negated, let’s see its formal definition.

#### Definition

From the conditional statement $$p \rightarrow q$$, we call contrapositive to the statement $$\sim q \rightarrow \sim p$$.

Example

Let’s look at the following statement of the type $$p \rightarrow q$$ and are:

• If the digits of $$x$$ add up to numbers that are multiples of $$3$$, then $$x$$ is a multiple of $$3$$.

The contrapositive of $$\sim q \rightarrow \sim p$$, would be:

• If $$x$$ is not a multiple of 3, then the digits of $$x$$ do not add up to numbers that are multiples of 3.

Both the original conditional statement of the type $$p \rightarrow q$$ and its contrapositive $$\sim q \rightarrow \sim p$$ are true and not contradictory.

The first two conditional statements such as conversion and inverse can be inferred from the conditional statement, they are not logically equivalent to the conditional statement.

However, the contrapositive is the only one that passes the test of sufficient and necessary condition, if we assign a truth value to both $$p$$ and $$q$$, we notice that the conditional statement $$p \rightarrow q$$ and the contrapositive $$\sim q \rightarrow \sim q$$ always have the same truth value, i.e., it is satisfied that:

$p \rightarrow q \equiv \sim q \rightarrow \sim p$

In a course of symbolic logic it is known as the law of transposition and is generally used as an indirect proof. There are different types of mathematical proofs, but we will only deal with the one that we have proposed for a future entry.

## Properties of the material conditional

We have the propositions $$p$$, $$q$$ and $$r$$, we have:

• Distributive law: $$p \rightarrow ( q \rightarrow r ) \Rightarrow ( p \rightarrow q ) \rightarrow ( p \rightarrow r )$$.
• Transitive law $$( p \rightarrow q) \rightarrow ( q \rightarrow r ) = p \rightarrow r$$.
• Commutativity of the antecedent: $$p \rightarrow ( q \rightarrow r ) = q \rightarrow ( p \rightarrow r )$$.
• Identity law: $$p \rightarrow p$$.
• $$p \rightarrow q = \sim p \vee q$$, we could say that the logical conditional is a type of disjunction.
• $$\sim ( p \rightarrow q ) = p \wedge \sim q$$, the negation of the conditional is a type of conjunction.

We finish the main content, however, you may be interested in an auxiliary content of the material conditional, its shortcomings and the difference that exists with the logical implication, with this I would close the current section.

Auxiliary content, if you wish, you can ignore this section.

## What is the problem with the material conditional?

The material conditional has a weak property, since although it is true that the antecedent is not fulfilled (from the previous example: “If she does not behave well”), it is possible that the consequent is fulfilled (“then I will take her for a walk anyway”).

We say then that the compound statement is true because despite not fulfilling what was indicated, the promise was still fulfilled.

If the antecedent, that is, the causative premise turns out to be false, and the consequent is true, then should we conclude that the compound statement must be true because the antecedent was not fulfilled, is that reason enough to be true? Let’s take a very interesting concrete example, let’s see:

A peculiar example

We have $$\phi$$ as the empty set, $$\mathrm{A}$$ any set of the Universe $$\mathrm{U}$$ and $$x$$ any element of $$\mathrm{U}$$. Prove that:

$x \in \phi \rightarrow x \in \mathrm{A}$

Como $$\phi$$ es el conjunto vacío, significa que no tiene elemento alguno y como $$x$$ es un elemento, entonces $$\phi \in \phi$$ es falso, pero para $$x \in \mathrm{A}$$ puede ser falso como también verdadera, es decir, pueda que $$x$$ pertenezca o no al conjunto $$\mathrm{A}$$. Sea cual fuese el caso, la condición de $$x \in \phi \rightarrow x \in \mathrm{A}$$ siempre será verdadera. Resumiendo:

As $$\phi$$ is the empty set, it means that it does not have any element and as $$x$$ is an element, then $$\phi \in \phi$$ is false, but for $$x \in \mathrm{A}$$ it can be false as also true, that is, it can that $$x$$ belongs or not to the set $$\mathrm{A} \in \mathrm{A}$$. Whatever the case, the condition of $$x \in \phi \rightarrow x \in \mathrm{A}$$ will always be true. To summarize:

• The statement is true because the antecedent is false and the consequent is false.
• The statement is true because the antecedent is false and the consequent is true.

Therefore, the antecedent is not legitimate proof for the consequent, based on this problem, we can say that the material conditional as we are dealing with it deserves to be applied to support solid arguments, let’s talk about this next.

The most controversial and difficult to understand logical connective is the logical conditional, it has weaknesses for the study of science, one might think that no matter how well founded a theory may be built, according to this type of connective at the argumentative level, the conclusions of many academic papers from all the libraries of the world may be correct and its theoretical development incorrect or false.

In other words, it does not matter how well explained a theory is, even if it is not true, what matters is the veracity of its actual conclusions.

According to Friedman, he says: “To be important, therefore, a hypothesis must be descriptively false in its assumptions; it must not take into account any of the many contingent circumstances because their very success reveals that they lack relevance to the phenomena it seeks to explain.”

At this point, dealing with this connective turns out to be a subject of debate and it was for a regular time and also a subject of much confusion and contradiction since to solve this point, a stronger conditional has been defined, but it is often confused with the material conditional, we refer to the implication.

But before discussing the implication, the material conditional as we put it, means that also the statements are valid for the following examples below.

Illogical examples that the material conditional takes as true

• If I eat an apple, then it will be hot today.
• I will become a frog when World War III starts.
• If elephants fly, then I am a gorilla.
• If 1+1=2, then 3+5=8.

the statements “if 1+1=2, then 3+5=8” is true, but not because the consequent “3+5=8” and the antecedent “1+1=2” are true, but because it has no relation at the semantic level, that is, the consequent cannot be deduced from the antecedent itself but by its truth value (antecedent), in other words, the conditional is only a binary truth function. it only plays with the truth values of the antecedent and the consequent.

This means that the content of the statement, that is, the argument, is excluded, this only applies to propositional logic since a conditional proposition is symbolized as $$p \rightarrow q$$, what is valid is only its validity and nothing else.

In view of this small detail, two types of conditionals were differentiated, one of them we have already studied and we call it material conditional, the other, the one we are going to refer to next is the implication.

Therefore, the material conditional concludes its truth value from the truth values of the statements that comprise it and not from its argument.

Note that propositional logic only focuses on the truth values of statements in an exclusive way, the only limitation of propositional logic.

## Difference between implication and material conditional

Of course, there is a difference between the material conditional and the logical implication in spite of its subtlety; to begin with, we must understand that the implication is a non-probabilistic statement, the conditional is the opposite, it talks about what could happen or happen, but does not affirm if something will happen with total certainty, that is why different truth values are assigned to it.

There is also another point to consider, the material conditional is expressed at the syntactic level with respect to arguments (only in propositional logic) and its only semantic values would be true or false.

On the other hand, implication focuses more on the semantic aspect (it takes into account literally the arguments, this is studied as a broader scope in predicate logic), that is, it takes into account the meaning of both premises and conclusions, for example: “If I eat an apple, then today it will be hot”, it would be meaningless for the implication.

Finally, for the implication there must be a cause for an effect, this does not happen with the material conditional, although it is true that the cause can be false and the effect true, it means that the conclusion did not occur according to the premises, then, ¿ What has generated it ?, This means that there must be some causal premise for the conclusion to be taken for granted, which means that data is missing.

Based on the previous paragraph we say then that the material conditional is incomplete, apparently, this type of logical connectives lacks data to affirm the rationale of the conclusion as it is. Let us look at these differences.

Example

We have the following propositions:

1. If tomorrow is New Year’s Day, then I will go for a walk.
2. Tomorrow is New Year’s Day, so I will go for a walk.

The first statement predicts an occurrence, an event, the second statement confirms something, it tells us that it is obvious, that the conclusion occurs because of the premise.

Let us write proposition 1 as follows:

• $$\overbrace{ \text{If} \ \underbrace{ \text{tomorrow is New Year’s Day} }_{F}, \text{then} \ \underbrace{ \text{I will go for a walk} }_{V} }^{V}$$

For implication, it always turns out to be true because it affirms or denies a statement in the following way:

• $$\text{ Tomorrow is New Year’s Day, so I will go for a walk }$$

The implication simply states, it cannot be blindly given truth values such as true or false, simply if I say that if “the sun comes out”, therefore and obligatorily “I will have to go for a walk” it is an inevitable event.

## Fallacy of the material conditional: negation of the consequent

In the previous example, the conclusion depends only on the premise, because if we deny the premise, we also deny the conclusion and therefore, the proposition will always be true, but the same cannot be done with the material conditional, denying the premise does not imply that the conclusion is always true, but it is often believed that it is.

This fallacy is called negation of the consequent and it turns out that the conclusion can be given even when the antecedent is false, but it is only possible if we treat the material conditional from the semantic perspective of its arguments and not only from its truth values (but that is another topic to deal with).

Therefore, the conclusion will always depend on the causal premises and that is what the logical implication intends.

It is also said of the implication that the antecedent carries the truth of the consequent so it will always be a definitive truth without contradictions.

Therefore, implication is the basis of mathematical theorems, because there is always a hypothesis and a thesis, where hypothesis carries the truth of the thesis.

This is contrary to the material conditional, since any false or true proof of an antecedent can result in an always true conclusion, therefore, we choose implication as the main basis for any mathematical proof.

In the section where I deal with logical inference I describe in more detail these differences, thus ending the confusion that exists between these two. (a topic that is in Spanish and will soon have its translation).

But we must be careful when we use the material conditional colloquially (where the semantics of the arguments is always unavoidable), I already explained in the example the difference between the conditional and the implication where I mention that this type of arguments can be a fallacy that many times can be overlooked.

## Symbolic representation of the implication

For two propositions $$p$$ and $$q$$, the symbolic representation of the implication is $$p \Rightarrow q$$ where it turns out to be only true. therefore, its truth table would be the following:

$\begin{array}{ c | c | c } p & q & p \Rightarrow q \\ \hline V & V & V \\ F & F & V \end{array}$

If we find two molecular statements with a dominant material conditional where we see that all truth values are true, we say that it is a tautology, then the material conditional could fulfill the truth values of the logical implication.

Although considering it as a tautology may also present a problem since we would have to accept the combination ” False $$\Rightarrow$$True = True “.

This relation also does not say anything whether the meaning of the arguments is considered or not. As a practical matter, the logical implication would be transformed into the material conditional only by operational questioning but maintaining the semantics of the arguments of the original statement.

The only way for two propositions to form a material conditional and be a logical implication (taking into account their arguments), is that the conditional must always be true as long as we do not find the combination “False $$\Rightarrow$$ True = True“. That would be all for today.