1. Introduction to Logic Lecture 14 The truth functional argument
By David Kelsey

2. Truth functional arguments vs. Categorical syllogisms
If you will remember, a categorical syllogism is an argument composed of 3 standard form categorical claims.
A truth functional argument, on the other hand, isn’t limited to 3 claims and neither must the claims be standard form categorical claims.
Truth functional arguments aren’t limited in such ways. They need only be composed of sentences and so can come in an infinite number of valid argument patterns.
There are three different methods for testing the validity of a truth functional argument.

3. The truth table test
We can use the Truth Table Test to test a truth functional argument for validity
To do so construct a truth table of the argument.
The truth table for an argument is a pictorial representation of the argument, in terms of all of the possible circumstances of the argument.
Thus, a truth table displays the premises and the conclusion of the argument.
And it displays all the possible truth values for the premises and conclusion of the argument.
Once we have constructed a truth table we look at the various rows of it.
If there is at least one row in which the premises of the argument are true and the conclusion is false, then the argument is invalid.
Otherwise it is valid!

4. Testing an argument: UCLA and Arizona
We might test the following argument:
1) If UCLA become national champs then Arizona aren’t.
2) But UCLA aren’t national champs.
3) Thus, Arizona are national champs.
Lets assign ‘U’ to ‘UCLA become national champs’
Let’s assign ‘A’ to ‘Arizona are national champs’

5. Constructing the reference columns
Now let us formalize the argument by substituting in claim variables:
1. U  ~A
2. ~U
3. A
Now that we have formalized the argument let’s construct a truth table for it:
First we need reference columns:
A U
T T
T F
F T
F F

6. Adding the premises
Our next task in constructing our truth table is to add to it the premises of the argument.
But notice that the first premise is a compound claim and one of its parts, ~A, is something we have yet to add to our table.
A U ~A U~A ~U
T T F F F
T F F T T
F T T T F
F F T T T

7. Testing the argument
Now we can add the conclusion of the argument to the table.
(the conclusion of our argument is seen in the first reference column)
A U ~A U~A ~U A
T T F F F T
T F F T T T
F T T T F F
F F T T T F
Now that we have completed our truth table let us look to see if the argument is valid:
We are looking for any row of the truth table in which the premises are true and the conclusion is false.
And we do in the fourth row as is shown in bold.
Thus, the argument is invalid!

8. Scarlet example The passage:
If Scarlet is guilty of the crime, then Ms. White must have left the back door unlocked and the colonel must have retired before ten o’clock. But either Ms. White didn’t leave the back door unlocked or the colonel didn’t retire before ten. Thus, Scarlet is not guilty of the crime.
The first thing we need to do is symbolize the argument:
S = Scarlet is guilty of the crime.
W = Ms. White left the back door unlocked.
C = The colonel retired before ten.
Let us now formalize it:
S(W&C)
~W v ~C ~S

9. Constructing our truth table
So here is our argument:
S(W&C)
~W v ~C ~S
Now we can construct our truth table. First we need reference columns, to which we can add the first premise:
C S W W&C S(W&C)
T T T T T
T T F F F
T F T T T
T F F F T
F T T F F
F T F T T
F F T F T
F F F T T

10. Testing the argument
Now we can add the second premise and the conclusion and all of their simpler parts:
C S W W&C S(W&C) ~W ~C ~Wv~C ~S
T T T T T F F F F
T T F F F T F T F
T F T T T F F F T
T F F F T T F T T
F T T F F F T T F
F T F F F T T T F
F F T F T F T T T
F F F F T T T T T
Let’s try to determine if the argument is valid now:
We are looking for any row in which the premises are all true and the conclusion is false.
As you can see though the only rows in which the premises are all true, the 4th and the last two, are also rows in which the conclusion is true.
Thus, the argument is valid!

11. The short truth table method
Constructing a full truth table can be a long and tedious process.
Lucky for us we have the short truth table method.
To use the short truth table method one constructs a truth table.
But because an argument is invalid if there is just one row in its truth table in which the premises are all true and the conclusion is false, we look only to construct just such a row.
Thus, we simply try to construct a row of a truth table in which the premises are true and the conclusion is false.
If we can do this then the argument is invalid!

12. Using the short truth table An example with 3 conditionals
Here is an example:
1. PQ
2. ~QR
3. ~PR
Let’s first try to make the conclusion false.
Because the conclusion is a conditional claim it is false just when its antecedent is true and its consequent false.
Thus, our row of the truth table starts like this:
P Q R ~P ~PR
F F T F

13. Making both premises true
Let us now try to make both premises true.
Of course, if we make Q true both premises turn out true.
Do you see why?
Hint: both premises are conditional claims.
Thus, we get this row:
P Q R ~P PQ ~Q ~QR ~PR
F T F T T F T F
And so the argument is invalid!

14. Testing with the short truth table A second example
Here is another example to test:
1. (PvQ)R
2. SvQ
3. SR
Notice that we can make the conclusion false by assigning S a ‘T’ and R an ‘F’.
Let us now try to make both premises true.
Given we have assigned R an ‘F’ we must make both P and Q False to make our first premise true.
Do you see why?
Thus, so far we have this
P Q R S (PvQ)R SvQ SR
F F F FFF F F F F F

15. Making the second premise true
Now let’s try to make the second premise true.
The second premise is a disjunction so to make it true we must simply make one of its disjuncts true.
But since we have already made Q false and yet we haven’t assigned a truth value to S, we can make S true.
So here is the final truth table:
P Q R S (PvQ)  R SvQ SR
F F F T FFF F F TTF T F F
Thus, the argument is invalid!

16. Using the short truth table on another example
Another example to test:
1. P&(QvR)
2. RS
3. PT
4. S&T
Let’s first try to make the conclusion false.
But because the conclusion is a conjunction it is false if both S and T are false or if either one is false.
So instead of making the conclusion false, let’s first try to make one or more of the premises true.
Consider the first premise.
To make it true we must make both of its conjuncts true. Thus we must make P true.

17. Making the premises true and the conclusion false
But now consider the third premise.
Given P has been assigned ‘true’, to make the third premise true we must make T true as well.
So far we have made P and T true.
But now look at the conclusion.
To make it false we must make S false.
Thus far we get this:
P Q R S T P & (QvR) RS PT S&T
T F T T T T FFT

18. Making all the premises true
Continuing with our argument, let’s look at the second premise.
Since we have already made S false we must make R false as well.
But in looking at our first premise and given we have already made P true and R false we must make Q true in order to make the premise true.
Thus, we get this finished truth table:
P Q R S T P&(QvR) RS PT S&T
T T F F T TT TTF F T F T T T FFT
And so the argument is invalid!

19. Trial and error
Although in our prior three examples we were always forced to make some truth assignment to make the conclusion false this won’t always be the case.
Sometimes there will just be more than one way to make the conclusion false.
In such cases we use trial and error.
This just means that we try out all of the possible ways of making the conclusion false until we either have exhausted them all or have found a way of making the premises true and the conclusion false.
If we cannot succeed in making the premises true and the conclusion false then the argument is valid.
Otherwise it is invalid.