Truth Tables
From Logic
Contents |
Using Truth Tables to Test Arguments for Validity
Truth tables provide a useful method of assessing the validity or invalidity of the form any argument. We can use the table to determine whether the entire form of the argument is true or false, based on one very simple rule: Any argument that allows for a set of all true premises with a false conclusion must be invalid. This elegant process provides us with a means of providing a logical, deductive proof that an argument form is valid. In addition, this process allows us to identify which forms are invalid, allowing for refutations by logical analogy: a process wherein one may use a ridiculous version of an opponents argument, using the precise form of his argument, to show where it the argument form reaches illogical conclusions.
To use a truth table to test an argument:
1. Make a column for each of the components used in the argument. There must be a line for each possible combination of truth values for these components. Each additional component will double the number of lines needed. A single component will need two lines. Thus, if there are n components there must be 2^{n} lines. By using a set pattern we can be sure to have included all the possible combinations and that no combination occurs more than once. The set pattern is to alternate true with false in the first column, in the second column (if needed) alternate 2 trues with 2 falses, in the third column (if needed) alternate 4 trues with four falses, and so on (doubling the number kept together for alternation with each column one adds) until the final component's column consists of the top half true and the bottom half false.
2. Add a column for each premise and for the conclusion. This may require additional columns to enable the calculation of complex premises or conclusions. Calculate the truth values for each line in the premises' and conclusion's columns.
3. Identify the lines where the conclusion is false and check those lines to see whether there is at least one false premise on that line.
4. If there is at least one false premise on every line where the conclusion is false, the argument is valid. Otherwise, you have demonstrated the possibility of all the premises being true at the same time as the conclusion is false, which is the mark of an invalid argument.
Modus Ponens
1st Premise | 2nd Premise | Conclusion | ||
p | q | p ⊃ q | p | q |
---|---|---|---|---|
T | T | T | T | T |
T | F | F | T | F |
F | T | T | F | T |
F | F | T | F | F |
p ⊃ q
p
_______
q
This argument reads as follows: "If p is true, then q is true. P is true, ergo, q is true". The truth-table shows that there are no cases with all true premises and a false conclusion. We can determine that this form is therefore valid.
Modus ponens, therefore, is a valid argument form.
Affirming the Consequent
On the other hand, consider the superficially similar argument form known as The fallacy of Affirming the Consequent is not valid:
1st Premise | 2nd Premise | Conclusion | ||
p | q | p ⊃ q | q | p |
---|---|---|---|---|
T | T | T | T | T |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | F | F |
p ⊃ q
q
_______
p
This argument reads "If p is true, then q is true. q is true, therefore, p is true." As you recall, we've already seen that this form is invalid. Let's prove this claim. Looking to the table, we can see in the table that there are two instances where there both premises are true. Pay strict attention to the third line, where we have two true premises and a false conclusion. This argument form permits a case where all the premises are true while the conclusion is false - the inference is therefore invalid as a deductive argument. The form therefore, does not guarantee the truth of its conclusion. It can only be possibly true. Thererore, if your opponent uses such a form, you can offer a refutation through logical analogy by presenting a similiar, and more obviously false conclusion, to demonstrate that his argument does not necessarily support his conclusion.
Modus Tollens
1st Premise | 2nd Premise | Conclusion | ||
p | q | p ⊃ q | ~ q | ~ p |
---|---|---|---|---|
T | T | T | F | F |
T | F | F | T | F |
F | T | T | F | T |
F | F | T | T | T |
p ⊃ q
~ q
_______
~ p
This argument reads: "If P is true, then q is true. q is not true, ergo p is not true." As the truth-table shows, there is no case where we have all true premises and a false conclusion. So we have proven that this form is valid in all cases.
Modus Tollens is a valid argument form.
Denying the Antecedant
As with Modus Ponens, there is an argument form superficially similar to modus tollens, that is actually invalid:
1st Premise | 2nd Premise | Conclusion | ||
p | q | p ⊃ q | ~ p | ~ q |
---|---|---|---|---|
T | T | T | F | F |
T | F | F | F | T |
F | T | T | T | F |
F | F | T | T | T |
p ⊃ q
~ p
_______
~ q
This argument reads: "If p is true, then q is true. p is not true, ergo q is not true." This is the fallacy of Denying the Antecedent. As the truth-table shows, this form allows for the case of an unreliable inference: line three contains all true premises with a false conclusion. Therefore, arguments that rely on this form are not valid! They will not work as deductive arguments.
Disjunctive Syllogism
1st Premise | 2nd Premise | Conclusion | ||
p | q | p Ú q | ~ p | q |
---|---|---|---|---|
T | T | T | F | T |
T | F | T | F | F |
F | T | T | T | T |
F | F | F | T | F |
p v q
~ p
_____
q
This argument reads: "Either p is true, or q is true. p is not true, ergo q must be true." Again, as with the other forms, there is no case of a set of all true premises with a false conclusion. This argument form is valid, valid, valid!
Affirming the Alternative
Once again, there is a similar form that embodies an invalid inference, the fallacy of affirming the alternative:
1st Premise | 2nd Premise | Conclusion | ||
p | q | p v q | p | ~ q |
---|---|---|---|---|
T | T | T | T | F |
T | F | T | T | T |
F | T | T | F | F |
F | F | F | F | T |
p v q
p
_____
~q
This argument reads: "Either p is true or q is true. p is true, ergo q is not true." However, if you recall our discussion of disjunctives in the previous section, you will remember that it IS possible for both the p and q atomic statements to be true in a disjuctive. So, using our inclusive sense of the ( v ) conjunction, it is possible for both p and q to be true, meaning that there is a case here where the premises aree true while the conclusion false.
Hypothetical Syllogism
Hypothetical Syllogisms involve three statement variables instead of two, meaning we must consider eight possible permuations of truth-values. It should already becoming apparent that Truth Tables become unweildy as we add more elements to an argument.
1st Premise | 2nd Premise | Conclusion | |||
p | q | r | p ⊃ q | q ⊃ r | p ⊃ r |
---|---|---|---|---|---|
T | T | T | T | T | T |
T | T | F | T | F | F |
T | F | T | F | T | T |
T | F | F | F | T | F |
F | T | T | T | T | T |
F | T | F | T | F | T |
F | F | T | T | T | T |
F | F | F | T | T | T |
p ⊃ q
q ⊃ r
_______
p ⊃ r
This statement reads: "If p is true, q is true. If q is true, r is true. Ergo, if p is true, r is true." As with any valid form, there are no cases here where there are all true premises, and a false conclusion.
We have affirmed the validity of this form.
Hypothetical Syllogism, therefore, is a valid argument form.
Review of Truth Tables
Truth Tables are a means of providing a rigorous proof of the validity of an argument form. By taking an argument through every possible permutation, it can be demonstrated that there are no cases where all true premises lead to a false conclusion.
However, there is a limit to the utility of Truth Tables: it most likely became apparent to you that the size of a Truth Table grows exponentially, with the addition of each new element to the equation. For this reason, Abbreviated Truth Tables will be introduced in the next lesson (They are the 7th option on the next page.
Those following the Course in Logic 101 ought to now proceed to the Formal Fallacies section.
References
- Copi, I. M, Cohen, C., (2001), "Introduction to Logic", 11th Edition.
- Hurely, P. J. (2000) A Concise Introduction to Logic - 7th Edition