How to understand why the premise in a logical Implication can be false when the conclusion is true

If you’ve studied some formal logic, you may have seen how a conclusion can be true when the premise of that argument is false.

How could this be? you might be wondering . . . This article will spell out the reasoning behind that and what’s known as “logical implication”.

Logical implication can take the form of:

p\rightarrow q
Sometimes different symbols are used to represent logical implication:

p\rightarrow q\: ,  p\Rightarrow q\: and\:  p\supset q\:

are all logically equivalent.

  • p is generally thought of also as the antecedent or premise of the argument.
  • q is generally thought of also as the consequent or conclusion of the argument.

The statement “p implies q” means that if p is true, then q must also be true. The statement “p implies q” is also written “if p then q” or sometimes “q if p.” Statement p is called the premise of the implication and q is called the conclusion.

(Source: http://www.math.niu.edu)

Necessity and Sufficiency

When thinking about logical implication, it is useful to be first reacquainted with necessity and sufficiency.

Necessity

To say that p is a necessary condition for q is to say that it is impossible to have q without p.

(NB: it is not necessarily the case that if you have q, then you must also have p—this is known as “affirming the consequent“)

Said another way: It is true if you do not have p you are guaranteed not to have q otherwise this logical implication does not hold true. (i.e. when not-p implies this is false)

Examples:

1. “To get a high distinction on the exam, you must study hard”.

(NB: You could study hard but not also get a high distinction on the exam.)

2. “To become the president of the United States, you must be born in the United States”.

(NB: You could be born in the United States but not become president.)

3. “A prime number must be odd if it is a whole number greater than 2”.

(NB: A whole number could be greater than two and also be odd but not necessarily be a prime number.)

Sufficiency

To say that p is a sufficient condition for q is to say that the presence of p is enough for q to be present but it is not necessary.

Said another way: It is possible to have p when q is also present, but the presence of p does not guarantee the presence of q.

Examples:

1. It is a sufficient that you can graduate by attending your graduation ceremony once you have completed the requirements for your degree.

(NB: You do not have to attend your graduation ceremony to have your degree conferred.)

2. It is a sufficient condition that if the president of the United States wins the Nobel Peace Prize he or she will be remembered by many people.

(NB: The president of the United States can still be remembered by many people even if he or she does not win the Nobel Peace Prize.)

3. It is a sufficient condition that every even integer greater than 2 can be expressed as the sum of two prime numbers (Goldbach’s conjecture).

(NB: Every even integer greater than 2 can also be expressed as the sum of non-primes, i.e. two even numbers added together will always equal an even number.)

Simultaneous necessity and sufficiency (logical equivalence)

For one statement to be a necessary and sufficient condition of another then one statement must be true when the other is also true.

The truth of one statement guarantees the truth of the another statement.

 It is true that:

p\leftrightarrow q\: ,  p\Leftrightarrow q\: and\:  p\cup q\:

are all logically equivalent and p is true if and only if q is true.

Examples:

1. “If you get a high distinction on the exam, you will score 85% or above, and will have performed well on the exam”.

(NB: There is no way that you can score 85% or above, not get a high distinction on the test, and have performed well.)

2. “Barack Obama was the president of the United States, and he was inaugurated as president”.

(NB: There is no way that Barack Obama was not the president of the United States and inaugurated as president)

3. “Prime numbers are greater than 2, only divisible by themselves and 1, and there are infinite prime numbers”.

(NB: There is no way that prime numbers are greater than 2, are not divisible by themselves and 1, and there aren’t infinitely many prime numbers)

p
Belonging to the set p is necessary for belonging to the intersection of sets p and q (p ∩ q) but not sufficient (a variable could belong to the set p but not also be in the intersecting region of sets p and q (p ∩ q). Only being in the intersection of sets p and q (p ∩ q) is necessary and sufficient for belonging to the sets p and q.  Image designed using Canva
Truth Table for Logical Implication

The truth table for logical implication is as follows:

\begin{matrix}  p &q &p\rightarrow q & \neg(p\rightarrow q)\\  T&T &T&F \\  T&F &F&T \\  F&T &T&F \\  F&F & T&F  \end{matrix}

It is also the case that:

\neg(p\rightarrow q)\Leftrightarrow p\wedge \neg q\Leftrightarrow \neg\neg p \wedge \neg q \Leftrightarrow \neg p \vee q

To illustrate with an example:

Marie Kortig

There is never an instance where somebody is the president of the United States but was also not born in the United States. This is the ONLY time that this proposition is false (the second row).

Therefore, the statement “If somebody is president of the United States then they will be born in the United States” is only violated or false when somebody becomes the president when they weren’t born in the United States.

Below are some objections to the use of logical implication, if you are still not satisfied with what has been covered above:

Further reading: Indicative Conditionals on the SEP.

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