Abbreviated Truth Tables

D. Yeakel

The ConceptWhen using abbreviated tables to check an argument

for validity we are looking for the same thing we were looking for with regular tables: a row with all true premises and a false conclusion. However the approach is reversed.

Instead of starting with assignments of true and falseto component letters, with abbreviated tables we begin by assuming that the premises are true and the conclusion is false. Then we work backwards to an assignment of true and false to the component letters (if there is one).

Consider the simple argument form on the next slide:

Suppose we wanted to check this argument form (commonly called ‘Modus Ponens’) for validity. A regular table wouldn’t be too much effort for this one, but there’s an even shorter method.

Modus Ponens

We know that if the argument is invalid then there is a way to make the premises all true and the conclusion false.

Let’s suppose that the premises are both true and the conclusion false. We’ll temporarily denote that supposition this way:

The ‘T’s and ‘F’ show that we are supposing that the premises are true and the conclusion false. If that’s possible then there must be a row of the truth table where A ⊃ B is true, A is true and B is false.

But if A is true and B is false then premise one (A⊃B) cannot be true! It’s a conditional with a true antecedent and false consequent. Its truth-table definition requires that it be false in that circumstance.

So Modus Ponens is Valid

Now we see that no argument with this form can be invalid. Any assignment of true and false to the components (A and B) that makes the second premise true and the conclusion false will also make the first premise false. So there cannot be an assignment that gives it all true premises and a false conclusion.

The next slide shows the same thing as the last three pictures, except that now the argument is presented (almost) in the format we will use for abbreviated tables.

As an abbreviated table:

The argument arranged horizontally

Assuming the premises are true and the conclusion false

But if A is true and B is false then premise one cannot be true! Since we ran into a contradiction, the argument is valid.

Now, a slightly tougher argumentThe argument to the left has four letters and a regular truth table would be sixteen rows long. The abbreviated table will take much less effort. First, write the argument as you would if you were going to do a full table. Include the sentence letters on the left side as you would in a regular table. The result is below.

1-Start by making the premises true and the conclusion false.

4-So D is true everywhere on the row, and ~D is false.

2-Since we are looking for a row where the conclusion is false, B must be false everywhere on that row.

3-Since it is a conjunction, premise 2 could only be true if both parts are true. So the row we are looking for must make ~C true and D true.

So A is false everywhere it appears.

Since ~ C is true, C is false everywhere.

In premise 1: if B is false and ~C is true then the conjunction B∙ ~ C is false. In premise 3: If the premise (A≡~D) is true and the right side (~ D) is false then the left side (A) must also be false.

There’s no contradiction on the table since a false ⊃ false conditional is true.

We’ve learned that if we had done a full 16 row table then the FFFT row would have all true premises and a false conclusion and show the argument to be invalid.

Choices• The idea of abbreviated tables is that if trying to give the argument

true premises and a false conclusion leads to contradiction then the argument is valid, but success at finding such a row is failure (invalidity) for the argument.

• The examples so far have been nice at least in that every step of the abbreviated table was ‘forced.’ For example, there’s only one way to make A≡D true if D is true (A must be false).

• But what if all we knew was that A≡D was true and we had no information about A or D? We wouldn’t be forced. We’d have to make a choice.

• Running into a contradiction after making an unforced choice doesn’t imply validity. Some other choice might have led to a row that shows the argument invalid. Finding a row along a choice path does imply invalidity though, because you only need one row with true premises and a false conclusion to show an argument invalid.

• Next is an example of an abbreviated table with a choice.

Notice that we aren’t forced anywhere. There are three ways to make a disjunction true, three ways to make a conditional true and two ways to make a biconditional false. Below, I made a choice in the conclusion since there are just two possibilities (instead of three). The circled ‘1’ to the right of the table below shows that A true and B false is the first option for making the conclusion false.

To the left is the argument in standard form.

Above, the argument is set up for an abbreviated table with premises set to true and conclusion set to false.

3- We don’t have to think about C because we already have a contradiction. Premise 2 can’t be true if it has a true antecedent and a false consequent. However, because we made a choice earlier, we aren’t finished. We’ve only showed that the option we chose for the conclusion leads to contradiction, not that every possible way of making the premises true and the conclusion false leads to contradiction!

1- Since A is true in the conclusion it must be true everywhere it appears.

2- Since B is false in the conclusion it must be false everywhere it appears.

4-So, we go back to the point of the choice and try the other way to make the conclusion false.

3- Since ~A is true, premise one is true no matter what value C has. So we have to make another choice, but this one obviously doesn’t matter. We have a row that makes the premises true and the conclusion false. In fact, we know that a complete table would have two rows that show the argument invalid: the FTT row and the FTF row.

1- A is false everywhere on the row.

2- B is true everywhere on the row.