The Logical `If`

Posted on February 14, 2014 by Vincent Engelmann
      p | q | (p -> q)
     ________________   
1)    T | T |    T
2)    T | F |    F
3)    F | T |    T
4)    F | F |    T

Lines 3 and 4 just made no sense to me.

If p is FALSE and q is TRUE, how does that make the statement true?

If p is FALSE and q is FALSE, how does that make the statement true?

Luckily, as is usually the case, there are a wealth of explanations about this on the StackExchange site (thank you, Mr Atwood).

Two concepts have proved most helpful:

  1. P and Q are linked to each other. You can’t affect one without affecting another (it’s a sort of interdependence)
  2. You need context for your Ps and Qs to really see why this is the case

How do you provide context? Well, you can make those lone Ps and Qs actual mathematical functions instead of just symbols. It’ll help give more “meaning” to the concept of True and False.

So let’s make a couple of function examples (it appears we actually need two in order to complete our context). We’ll start by giving an example of the fourth line of the truth table.

Example 1:

p =  x > 0

q = 2x > 0

Example 2:

p =     x > 0

q = x + 1 > 0

Remember, the results of these functions is a simple True or False (we are “assigning” this boolean to the p or q at the beginning of these example functions).

Now in human-speak, example 1 becomes:

2x is positive when x is positive.

Example 2 becomes:

If x is positive, then so is x+1.

Let’s run example 1 into reality. Meaning, we’ll choose a number and plug it into x. We’ll make x = -1.

In doing so, our first example becomes:

p = (-1) > 0

p here will return False!

and

q = 2(-1) > 0

q here will also return False!

So let’s put that situation into plain English.

If you make p false, you’re making q false, given what p and q stand for.

Here’s the important part. What you are checking here is the truth of that very statement!

Rephrased as a question:

Is it true that if you make p false, that q is false… given what p and q stand for?

…whereupon you should find it in your mind to say “Yes!” (e.g., True)

It is true that if you make p false, you will make q false.

This is a slight re-write of Jyrki’s response here.

Anyway, that was line 4 of the truth table. Now for line 3.

For line 3 we go to example 2. Let’s put example 2 through a reality check. We’ll use x = -1/2.

p = (-1/2) > 0

p returns False!

and

q = (-1/2) + 1 > 0

q returns True!

Again we put this into plain English.

*Under the conditions given, if you make p false, you’ll make q true.

And again - you are verifying the truth of this above statement, not the individual ps and qs.

I find the question-form helpful, so here it goes:

Under the conditions given, if you make p false, will this make q true?

To which you’d answer, “Yup, if you use those functions, that’s what happens.” (e.g., the result is True.).

There are two plain English phrases I have used that should show context:

Both of these mean, we have provided an environment for our if-then – it cannot stand alone!