11.3.3: The Logic of Or

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Bradley H. Dowden
California State University Sacramento

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In any statement of the form "P or Q” the statement P is called "the left disjunct," and the Q is called "the right disjunct." The operation of “or” or “ v ” is called disjunction. Consider the statement “x = 5 or y < 19." If you were told that either the left or the right disjunct is true, but not which is true, could you be sure that the right one is true? No. The moral is the following:

However, if you knew that the left disjunct is not true, you could infer that the right one is. The general form is

Exercise \(\PageIndex{1}\)

State whether the main argument below is a deductively valid inference, then describe the logical form of the inference.

Assuming x = 4 and y < 7, as you have said, then x is not unequal to 4.

Answer

Yes, it is a deductively valid inference. Its logical form is

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Exercise \(\PageIndex{1}\)

Does this argument have a valid logical form?

Your information establishes that President Abraham Lincoln was assassinated by Ulysses Grant. Now we can be sure that John was right when he said, "Abraham Lincoln was assassinated by Grant or Booth. ”

In trying to assess whether the argument is deductively valid, you can abstract the main argument to produce the following form:

where we have defined

P = President Abraham Lincoln was assassinated by Ulysses Grant.

Q = Abraham Lincoln was assassinated either by Grant or else by Booth.

The first premise happens to be false, yet the conclusion is true. The problem, however, is not to decide which sentences are true but to decide whether the logical form is deductively valid. It is.

Answer: This is a valid form. Any argument with that form has to be valid no matter declarative sentence you replace ‘P’ with; and ditto for ‘Q’.

The truth (and falsehood) possibilities for or can be summarized in this truth table: