Truth Tables (conditional, biconditional)

The conditional

Another very important connective in symbolic logic is known as the material conditional. It is also referred as the material implication, the implication, the material consequence, and even simply as the conditional.

The symbol used for the conditional connective is a right arrow $\Rightarrow$⇒. 

As an example, suppose a doctor tells you that if you eat an apple you will feel better. We could think about this statement logically. We could define two propositions:

$p$p	$:$:	I will eat an apple
$q$q	$:$:	I will feel better

We could then define a compound proposition, based on the doctor's advice, that says: 

If $p$p then $q$q 

In symbolic logic we would write a compound proposition:

$p$p $\Rightarrow$⇒ $q$q

There are a number of very important things to think about before we look at the truth table.

The connective, as a right arrow, tells us that the statement can only be read one way. There is a proposition to the left of the arrow, and another to the right. We cannot switch these and expect the meaning to stay the same. For example we cannot say that if we feel better, then we must have eaten the apple, because there might be some other reason we feel better.

To keep in mind that the two propositions have different roles in the conditional, we name then differently. Proposition $p$p is called the antecedent (ante meaning before), and proposition $q$q is called the consequent (think this is what happens if $p$p happens). The compound proposition is really stating that the antecedent implies the consequent, or the antecedent is presumed to have caused the consequent.  

Truth table for the conditional 

Here is the truth table for the material conditional:

Truth Table Conditional

$p$p	$q$q	$p\Rightarrow q$p⇒q
$T$T	$T$T	$T$T
$T$T	$F$F	$F$F
$F$F	$T$T	$T$T
$F$F	$F$F	$T$T

There are two very important and immediate observations to make:

1. The only way $p\Rightarrow q$p⇒q is false is when the antecedent is true but the consequent is false.
2. If the consequent is true, then irrespective of the truth value of the antecedent, $p\Rightarrow q$p⇒q will be true.

We need to explore why the truth table is like it is, row by row. Three of the four possibilities make sense, but there is one row (row 3) that seems to be counter-intuitive.

Row 1 is straightforward. For example, if you do eat an apple, and you do feel better, then what the doctor said is entirely correct. Mind you, we can only presume to think that there is some causal link between the eating of the apple and feeling better. It might have been coincidence, or some placebo effect. In any event, we can declare that $p\Rightarrow q$p⇒q is true. 

Row 2 seems reasonable as well. Suppose we eat the apple and we don't feel better. Then the doctor was wrong, and so the conditional $p\Rightarrow q$p⇒q is false. 

Row 4 is a little trickier. If, after not eating the apple, you still don't feel better, then you can only assume that what the doctor told us was true. The doctor stated that the way to feel better is to eat the apple. You didn't take the doctor's advice and as a consequence you don't feel better. In a way, you haven't tested the doctor's theory, so on that basis, you can't say that the doctor's advice was wrong. So, in the absence of evidence to the contrary, you have to assume that what he said is true. 

Row 3 is the troublesome one. If you don't eat the apple, but still end up feeling better, then what the doctor said is deemed true. The reason is that even though you feel better, you didn't prove that the doctor was wrong in his advice. The doctor said that if you ate the apple, you would feel better. Well guess what? You felt better anyway, but just because you felt better, doesn't contradict what the doctor said.

The idea of truth functionality 

There is another more subtle reason why we need to agree on the truth values of the conditional truth table. We want a truth-functional kind of implication. That is, if we are constructing a systematic symbolic logic we want only one possible interpretation of $$.

If we allowed more than one interpretation, then the implication ceases to be functional and we lose all certainty with our conclusions. We want to be able to figure out the truth-value of a statement solely on the basis of the truth-values of its components.   

The 20th Century English philosopher Bertrand Russell coined the term "material implication" for exactly this reason. The actual content of any proposition becomes irrelevant to the truth table. This makes it possible to construct absurd conditional statements in addition to those that a far more plausible to accept.  

For example, consider the two propositions given by:

m	$:$:	The Moon is made of green cheese
c	$:$:	Cows are born with four legs

Then the statement $m\Rightarrow c$m⇒c (If the moon is made of green cheese then cows are born with four legs) is true. It's true because we know, for a fact, that the moon isn't made of green cheese (man has been there) and we know, for a fact, that cows are born with four legs, so row 3 in the truth table above informs us that the compound proposition is true. However, we know that the statement "cows having four legs" have nothing at all to do with the "Moon being made of green cheese" and so the compound statement is illogical.     

The bi-conditional

Sometimes (but certainly not always) the conditional connective works both ways. That is, sometimes when we say "If $p$p then $q$q", then we can also say "If $q$q then $p$p".

As an example, the statement:

If the triangle is equilateral then the triangle has three equal angles.

is precisely the same as the statement :

If the triangle has three equal angles, then the triangle is equilateral. 

When this happens we can replace the two statements with the if and only if clause, so that we can say:

The triangle is equilateral if and only if it has three equal angles.

Suppose we call $p$p the proposition "The triangle is equilateral" and $q$q is the proposition "The triangle has three equal angles". We happen to know that $p\Rightarrow q$p⇒q and $q\Rightarrow p$q⇒p are both true. It therefore means, from the conjunction truth table, that the compound proposition $\left(p\Rightarrow q\right)\wedge\left(q\Rightarrow p\right)$(p⇒q)∧(q⇒p) is true also.  

Logicians created a shorthand symbol for the proposition $\left(p\Rightarrow q\right)\wedge\left(q\Rightarrow p\right)$(p⇒q)∧(q⇒p). They created the double arrow symbol $\Leftrightarrow$⇔ and called it the bi-conditional implication, often abbreviated as the bi-conditional.  

Of course, just like other connectives, the bi-conditional has truth values for each of the four combinations of truth values for $p$p and $q$q. 

				$\Leftrightarrow$⇔
$p$p	$q$q	$p\Rightarrow q$p⇒q	$q\Rightarrow p$q⇒p	$\left(p\Rightarrow q\right)\wedge\left(q\Rightarrow p\right)$(p⇒q)∧(q⇒p)
$T$T	$T$T	$T$T	$T$T	$T$T
$T$T	$F$F	$F$F	$T$T	$F$F
$F$F	$T$T	$T$T	$F$F	$F$F
$F$F	$F$F	$T$T	$T$T	$T$T

Look carefully at the table. Note that when finding truth values for $p\Rightarrow q$p⇒q or for $q\Rightarrow p$q⇒p the only false occurs when the consequent is false.

The only times that the bi-conditional is true is when $p$p and $q$q have the same truth value - either both true or both false.

Think about the conditional statement:

The Moon is made of green cheese if and only if the the Earth is flat.   

In a strange way we could accept that, even though both separate propositions are plainly false, the bi-conditional seems true. In colloquial terms, the only way we can accept a false proposition is to accept some other false proposition.

If $p$p and $q$q have different truth values, then the bi-conditional is false. As an example, consider the statement:

The Moon is made of green cheese if and only if 11 is a prime number.

We immediately sense that this is false, because we know that, even though 11 is in fact prime, the moon is not made of green cheese. 

A summary of connectives on p and q  

There are other connectives that we haven't as yet discussed, but it might be beneficial to summarise our results so far. We have encountered the negation ($\sim$~ ), the conjunction ($\wedge$∧ as in "and"), the disjunction ($\vee$∨ as in "inclusive or"), the conditional ($\Rightarrow$⇒ ) and the bi-conditional ($\Leftrightarrow$⇔ ).

Summary Table

		Negation	And	Or	Conditional	Bi-conditional
$p$p	$q$q	$\sim p$~p	$p\wedge q$p∧q	$p\vee q$p∨q	$p\Rightarrow q$p⇒q	$p\Leftrightarrow q$p⇔q
$T$T	$T$T	$F$F	$T$T	$T$T	$T$T	$T$T
$T$T	$F$F	$F$F	$F$F	$T$T	$F$F	$F$F
$F$F	$T$T	$T$T	$F$F	$T$T	$T$T	$F$F
$F$F	$F$F	$T$T	$F$F	$F$F	$T$T	$T$T

 

The general procedure for truth tables

The truth table shown under the sub-heading The Bi-conditional above was constructed by an alternate method to that  actually used by most logicians. The preferred method, often called the general procedure, is far simpler to deal with. The method hinges on knowing what the main connective is. 

We will illustrate the method using the proposition $\sim p\Rightarrow\left(p\vee\sim q\right)$~p⇒(p∨~q).

To begin with we note that there are two propositions $p$p and $q$q to consider.

We also see that the main connective is the conditional $\Rightarrow$⇒ with the antecedent as $\sim p$~p and the consequent as $$. Whenever the general method is used, the main connective becomes the target, and is the last column filled out.

We lay out the truth table with $p$p and $q$q and the proposition we seek broken up into parts as shown here:  

$p$p	$q$q	$\sim p$~p	$\Rightarrow$⇒	( $p$p	$\vee$∨	$\sim q$~q )
$T$T	$T$T		?
$T$T	$F$F		?
$F$F	$T$T		?
$F$F	$F$F		?

Our aim is to find the truth values in the column with the question marks. 

1. We first determine the values for the third and seventh columns ($\sim p$~p and $\sim q$~q).
2. This allows us to determine the values for the sixth column ($\vee$∨ ) which gives us the truth values for the antecedent ( $$ ).
3. Finally we can then determine the values for the main connective using the third and sixth columns. 

You may need to read the procedure a couple of times to really understand it. You are jumping around a little on the table, but the general method is far easier to use in practice. 

Here is the completed table:

$p$p	$q$q	$\sim p$~p	$\Rightarrow$⇒	( $p$p	$\vee$∨	$\sim q$~q )
$T$T	$T$T	$F$F	$T$T	$T$T	$T$T	$F$F
$T$T	$F$F	$F$F	$T$T	$T$T	$T$T	$T$T
$F$F	$T$T	$T$T	$F$F	$F$F	$F$F	$F$F
$F$F	$F$F	$T$T	$T$T	$F$F	$T$T	$T$T

 

Tautologies and self-contradictions

If, after constructing a truth table for a particular proposition, we find the column under the main connective contains all true truth values, then the proposition is called a tautology.

If on the other hand we find the column under the main connective contains all false truth values, then the proposition is known as a self-contradiction.

Worked Examples

Question 1

Question 2

Question 3

Question 4