Logical Equivalence

What is logical equivalence?

Two propositions are logically equivalent if both always have the same truth value. Thus two propositions $p$ and $q$, are equivalent if $p$ and $q$ both are true, or both are false at any given time or situation. We use $p \equiv q$ to express the equivalence of $p$ and $q$.

Examples

Number	Scenario	$J$	$S$
<br>	<br>	<br>	<br>
1	Height of John <br> is greater than the height of Smith.	T	T
<br>	<br>	<br>	<br>
2	Height of Smith <br> is greater than the height of John.	F	F
<br>	<br>	<br>	<br>
3	Both John and Smith <br> have the same height.	F	F

As in all the possible scenarios, the truth value of $J$ and $S$ is same, Therefore, $J \equiv S$.

No matter how complicated the given propositions are, the basic principle mentioned above remains the same to establish or observe logical equivalence.

There are many ways to express the same thing in a language. If we mean the same thing in two different statements, then those statements are equivalent. Let’s take a look at another example:

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Consider the following propositions:

• $A_1$: Adam reached the office at 8:00 a.m. on March 22, 2021.
• $A_2$: On March 22, 2021, Adam arrived in the office at 8:00 a.m.

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As we can see, both $A_1$ and $A_2$ express the same fact, and they are logically equivalent.

At times there are many things that are implicit in a sentence. To be more accurate and rigorous, we should take care of such details.

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Look at the difference between the following two propositions, $R_1$ and $R_2$.

• $R_1$: It was raining on June 15, 2021.
• $R_2$: It was raining at the Eiffel Tower in Paris at 8:30 p.m. on June 15, 2021.

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$R_1$ will serve our purpose while standing at the Eiffel Tower in Paris, as the context is understood and the meaning is clear. However, if we have any fraction of doubt that the meaning and context may confuse, $R_2$, which is equivalent to $R_1$, is a better option.

Exact meaning of equivalence

Strictly speaking, in our current context, two propositions being logically equivalent is a misnomer. According to our definition, all true propositions are equivalent, and all false propositions are equivalent. In particular, “the earth revolves around the sun” is logically equivalent to “$3 < 5$.”

The notion of equivalence is only useful if we have a proposition that depends on some variables. These variables are called free variables. In that case, we say two statements are equivalent if, for all assignments of the free variables, they are true or false simultaneously. This definition is precisely in line with our intuition. Two propositions are equivalent if they express the same thing. For example, consider the statement:

$$a > 3.$$

This statement is asserting something about a number $a$. Here, $a$ is a free variable. The set of values a free variable is allowed to take is called its domain. For integers, this is equivalent to the statement:

$$a \not< 4.$$

Logical equivalence depends on the values that the free variables are allowed to take, and the choice of domain dictates that.

The above two statements are equivalent if we are talking about integers. However, they are not equivalent if $a$ can take real values. For example, for $a = 3.5$, the first statement is true, but the second one is false.

An intuitive and correct way to think about logical equivalence is to say that two statements are equivalent if they are just different ways of expressing the same assertion.

Another place where logical equivalence is extremely useful is when we make compound statements or compound propositions.

Quiz