Chapter 14 Truth table shortcuts
With practice, you will quickly become adept at filling out truth tables. In this chapter, we consider (and justify) some shortcuts which will help you along the way.
14.1 Working through truth tables
You will quickly find that you do not need to copy the truth value of each sentence letter, but can simply refer back to them. So you can speed things up by writing:
$P$  $Q$  $(P$  $\vee $  $Q)$  $\leftrightarrow $  $\mathrm{\neg}$  $P$ 
T  T  T  F  F  
T  F  T  F  F  
F  T  T  T  T  
F  F  F  F  T 
You also know for sure that a disjunction is true whenever one of the disjuncts is true. So if you find a true disjunct, there is no need to work out the truth value of the other disjunct. Thus you might offer:
$P$  $Q$  $(\mathrm{\neg}$  $P$  $\vee $  $\mathrm{\neg}$  $Q)$  $\vee $  $\mathrm{\neg}$  $P$ 
T  T  F  F  F  F  F  
T  F  F  T  T  T  F  
F  T  T  T  
F  F  T  T 
Equally, you know for sure that a conjunction is false whenever one of the conjuncts is false. So if you find a false conjunct, there is no need to work out the truth value of the other conjunct. Thus you might offer:
$P$  $Q$  $\mathrm{\neg}$  $(P$  $\wedge $  $\mathrm{\neg}$  $Q)$  $\wedge $  $\mathrm{\neg}$  $P$ 
T  T  F  F  
T  F  F  F  
F  T  T  F  T  T  
F  F  T  F  T  T 
A similar short cut is available for conditionals. You immediately know that a conditional is true if either its consequent is true, or its antecedent is false. Thus you might present:
$P$  $Q$  $((P$  $\to $  $Q$)  $\to $  $P)$  $\to $  $P$ 
T  T  T  
T  F  T  
F  T  T  F  T  
F  F  T  F  T 
So ‘$((P\to Q)\to P)\to P$’ is a tautology. In fact, it is an instance of Peirce’s Law, named after Charles Sanders Peirce.
14.2 Testing for validity and entailment
In chapter 12, we saw how to use truth tables to test for validity. In that test, we look for bad lines: lines where the premises are all true and the conclusion is false. Now:

•
If the conclusion is true on a line, then that line is not bad. (And we don’t need to evaluate anything else on that line to confirm this.)

•
If any premise is false on a line, then that line is not bad. (And we don’t need to evaluate anything else on that line to confirm this.)
With this in mind, we can speed up our tests for validity quite considerably.
Let’s consider how we might test the following:
The first thing we should do is evaluate the conclusion. If we find that the conclusion is true on some line, then that is not a bad line. So we can simply ignore the rest of the line. So, after our first stage, we are left with something like this:
$J$  $L$  $\mathrm{\neg}$  $L$  $\to $  $(J$  $\vee $  $L)$  $\mathrm{\neg}$  $L$  $J$ 
T  T  T  
T  F  T  
F  T  ?  ?  F  
F  F  ?  ?  F 
where the blanks indicate that we won’t bother with any more investigation (since the line is not bad), and the question marks indicate that we need to keep digging.
The easiest premise to evaluate is the second, so we do that next, and get:
$J$  $L$  $\mathrm{\neg}$  $L$  $\to $  $(J$  $\vee $  $L)$  $\mathrm{\neg}$  $L$  $J$ 
T  T  T  
T  F  T  
F  T  F  F  
F  F  ?  T  F 
Note that we no longer need to consider the third line on the table: it is certainly not bad, because some premise is false on that line. And finally, we complete the truth table:
$J$  $L$  $\mathrm{\neg}$  $L$  $\to $  $(J$  $\vee $  $L)$  $\mathrm{\neg}$  $L$  $J$ 
T  T  T  
T  F  T  
F  T  F  F  
F  F  T  F  F  T  F 
The truth table has no bad lines, so the argument is valid. Any valuation which makes every premise true makes the conclusion true.
It’s probably worth illustrating the tactic again. Consider this argument:
Again, we start by evaluating the conclusion. Since this is a disjunction, it is true whenever either disjunct is true, so we can speed things along a bit.
$A$  $B$  $C$  $A\vee B$  $\mathrm{\neg}(B\wedge C)$  $(A$  $\vee $  $\mathrm{\neg}$  $C)$ 
T  T  T  T  
T  T  F  T  
T  F  T  T  
T  F  F  T  
F  T  T  ?  ?  F  F  
F  T  F  T  T  
F  F  T  ?  ?  F  F  
F  F  F  T  T 
We can now ignore all but the two lines where the sentence after the turnstile is false. Evaluating the two sentences on the left of the turnstile, we get:
$A$  $B$  $C$  $A\vee B$  $\mathrm{\neg}($  $B$  $\wedge $  $C)$  $(A$  $\vee $  $\mathrm{\neg}$  $C)$ 
T  T  T  T  
T  T  F  T  
T  F  T  T  
T  F  F  T  
F  T  T  T  F  T  F  F  
F  T  F  T  T  
F  F  T  F  F  F  
F  F  F  T  T 
So the entailment holds! And our shortcuts saved us a lot of work.
We have been discussing shortcuts in testing for validity. But exactly the same shortcuts can be used in testing for entailment. By employing a similar notion of bad lines, you can save yourself a huge amount of work.
Practice exercises
A. Using shortcuts, check whether each sentence is a tautology, a contradiction, or neither.

1.
$\mathrm{\neg}B\wedge B$

2.
$\mathrm{\neg}D\vee D$

3.
$(A\wedge B)\vee (B\wedge A)$

4.
$\mathrm{\neg}[A\to (B\to A)]$

5.
$A\leftrightarrow [A\to (B\wedge \mathrm{\neg}B)]$

6.
$\mathrm{\neg}(A\wedge B)\leftrightarrow A$

7.
$A\to (B\vee C)$

8.
$(A\wedge \mathrm{\neg}A)\to (B\vee C)$

9.
$(B\wedge D)\leftrightarrow [A\leftrightarrow (A\vee C)]$