Constructing truth tables – Filling in the wff's

About

This text is about how to construct a complete truth table for a wff from the basic part of a truth table that I introduced in an easier text. Please refer to that text for how to construct the basic part.

Adding wff’s

There are methods to construct truth tables. I know of two. Below I will explain them both.

Method #1

The first method is to represent all the wff’s from the most simple wff’s to increasingly complex wff’s until we have reached whatever wff it is we wanted to check up in a truth table. With this method all the wff’s along the way to the target wff are present individually, and so if the target wff is complex, then the table will fill a lot. Below is a truth table for the wff (P→Q)→¬P:

P Q ¬P P→Q (P→Q)→¬P
T T F T F
T F F F T
F T T T T
F F T T T

This truth table is still fairly small. But now consider the truth table for the wff ¬[(¬P↔¬Q)→R]:

P Q R ¬P ¬Q ¬P→¬Q ¬Q→¬P ¬P↔¬Q (¬P↔¬Q)→R ¬[(¬P↔¬Q)→R]
T T T F F T T T T F
T T F F F T T T F T
T F T F T T F F T F
T F F F T T F F T F
F T T T F F T F T F
F T F T F F T F T F
F F T T T T T T T F
F F F T T T T T F T

This truth table is rather large.

Method #2

Instead of representing each previous step along the way up to the targeted wff, we may simply put in the target wff after the sentence variables. We use numbers in the bottom to show how we got to the target wff. It is a bit difficult to explain how to do this, but with an example the method should be easy to understand. We will use the two same example wff’s as before. First (P→Q)→¬P:

P Q (P Q) ¬ P
T T T T T F F T
T F T F F T F T
F T F T T T T F
F F F T F T T F
1 1 1 2 1 3 2 1

And now ¬[(¬P↔¬Q)→R]:

P Q R ¬[( ¬ P ¬ Q) R]
T T T F F T T F T T T
T T F T F T T F T F F
T F T F F T F T F T T
T F F F F T F T F T F
F T T F T F F F T T T
F T F F T F F F T T F
F F T F T F T T F T T
F F F T T F T T F F F
1 1 1 5 2 1 3 2 1 4 1

It can now be seen how this method works. The numbers in the bottom row show in which order the columns have been filled out. This is done in steps and that is why some columns have the same number. Since we can’t fill out ¬P before P has been filled out, that means that negations are always a step after simple wff’s. Similarly with conditionals; We can’t fill out P→Q before both P and Q have been filled out. So conditionals are always a step after. Similarly with the other operators.

On the methods

I prefer the first method. There are a couple of reason why.

It find it easier to create the truth tables for the first method. The first method’s truth table for the first wff has 5 columns, and the second method’s truth table has 8. Similarly for the second wff; The first has 10 and the second 11. Perhaps the number of columns needed for the second method is generally larger than the number of columns needed for the first. I have not tested it properly but I guess it is so.

For the second method some of the borders/edges of the table need to be invisible, some need have a single edge etc. That takes some time to set properly.

Some platforms e.g. internet discussion boards may not support different visibility of the edges and so the second method will look terrible on such platforms. The first method works for any platform that supports tables and internet discussion boards usually do. We may note that one could take a screenshot of the truth table created using method 2 and post it as a picture. This would require some additional imagine editing (if using MSPAINT or similarly) or special screenshot software1, and of course somewhere to host the image file.2 Finally, one does not need to create a row for the step numbers in the first method.

I find the first method easier to use for evaluating wff’s. For me it is less confusing and more clear. For instance, if I want to know the evaluation for a conditional I simply need to find the conditional in the top row and then look down below it. With the second method I need to find the conditional operator between the two sentence variables that I am interesting in and look below it.

The first method takes up more space. But we may note that most of the time we don’t need to look up very complex wff’s in a truth table. So it will often not be a problem that the truth table can be large, if ever.

No matter what I prefer, it seems that the most logicians prefer the second method. Probably because it takes up less space in books.

Notes

2 comments

  1. I understand the basics of truth tables but I was completely floored with this and similar questions. Maybe you can clarify for me. I will write out the question as exactly.

    Q Suppose you had to fill in the rightmost column of the following truth table:

    p q r SHMORG(p,q) SHMORG(r, SHMORG(p, q))

    T T T
    T T F
    T F T
    T F F
    F T T
    F T F
    F F T
    F F F

    I don’t understand how to read question to answer as there has never been one example of this format.

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