Any truth table can be cumbersome, and truth tables with four variables can be very difficult to manage. Fortunately, there is a short-cut method, one that does not require the construction of a complete truth table.
Recall the counter-example method that we used with categorical syllogisms. It was based on the idea that no valid syllogism can have true premises and a false conclusion. This is true of any deductive argument. If a valid deductive argument has true premises, the conclusion will be true. Accordingly, if we can assign consistently truth-values to an argument that results in all true premises and a false conclusion, then we know the form is invalid. (By "consistent" I mean that if the truth-value of a proposition is true in one occurrence in a particular argument, then it must be true in its other occurrences in that same argument. Similarly with a false proposition.)
Here is how the method works. First display the argument in a linear form:
A Ú B B É C A C
Then assign T to all the premises and F to the conclusion. Put the truth-values above the premises and conclusion:
T T T F A Ú B B É C A C
That is the easy part. Now comes the test. Figure out what are the truth values of the simple propositions. This may take some effort. But given the assumption that the premises are true and the conclusion is false, there are some givens: The A proposition will be true and the C proposition will be false.
T T T F A Ú B B É C A C t f
If the C is false then, to be consistent, we must enter an f underneath the C of the second premise. But this means the B must be false as well, for if the B were true and the C were false then the conditional statement would be false. So we put in two fs underneath the second premise:
T T T F A Ú B B É C A C f f t f
Now we can put in the remaining truth values, for we know the A is true and the B is false:
T T T F A Ú B B É C A C t f f f t f
We have now shown that there is a consistent set of truth-values that produces a line in which the premises are true and the conclusion is false. Therefore the argument form is invalid.
Note that the line we discovered would be one of the lines in a full truth table. But instead of going to the trouble to display the whole table, we, with a little imagination and reasoning, were able to produce only what we needed: a line with true premises and a false conclusion. This is all it takes to show that a deductive argument form is invalid.
But, just as some truth tables have multiple lines with true premises and false conclusions, some short-cut truth table will have more than one possible consistent assignment of truth-values that show invalidity. Your short-cut truth table may have more than one assignment and still be invalid.
NOTE: A valid argument form will not have any lines that have true premises and a false conclusion. Look at the the argument from Exercise 8:
If determinism holds, people do not have free will.
People have free will.
Therefore determinism does not hold.
Here it is in symbols: D É ~F, F / \ ~D
Here it is in a short-cut truth table:
Assume: T T F D É ~F F ~D t f t f But: F T F
No actual assignment of true premises and false conclusion is possible. Therefore the argument form is valid!
Exercise 9 provides practice opportunities.
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Copyright © 1999, Michael Eldridge