Truth Tables to Determine Validity
According to Johnson (2007), "The truth table method for evaluating arguments consists of three steps:
Step 1 Translate the statements of the argument into symbolic form.
Step 2 Construct a truth table for the whole argument.
Step 3 Look for a row in which the truth-values of the premises are T and the truth-value of the conclusion is F. If there is such a row, the argument is invalid. If there is not such a row, the argument is valid" (p. 150).
In essence, this system has the reader look for incongruities within an argument. An invalid argument would have a sequence of true premises leading to a false conclusion.
So if we take the example containing both true premises and a true conclusion:
A dog is a canine.
Canines are mammals
Therefore dogs are mammals.
The reader can determine that the two premises are both true and that the conclusion is true. That then means the argument is valid.
If we take the example with a false conclusion:
If my dog is hungry, then she barks.
My dog is barking
Therefore my dog is hungry.
Here both of the premises can be true. My dog can bark when she's hungry, and my dog could be barking right now. However, the conclusion could be false. My dog could be barking because a cat went by the door. So the argument is invalid.
There are a variety of flavors of evaluating a truth table. For example, Hurley (2008) indicates "[A] truth table shows that a conjunction is true when its two components are true and false in all other cases. This definition reflects ordinary language usage ..." (p. 303). The rules of grammar we learned in English carry over nicely into logic, to help us evaluate how pieces of an argument are pulled apart and evaluated.
Hurley, Patrick J. (2008). A Concise Introduction to Logic. Belmont, CA: Thomson Wadsworth.
Johnson, Robert M. (2007). A Logic Book. Belmont, CA: Wadsworth Cenage Learning.
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