Deduction Worksheet

Validity: For an argument to be deductively valid is for it to be impossible for its premises to be true and its conclusion false. Alternatively, an argument is deductively valid if and only if it’s the case that if its premises are all true, then its conclusion must be true (on pain of logical contradiction).

Here are two examples of valid arguments:

1. If Joe is late for work, then he will get fired.
2. Joe was late for work
3. Joe got fired

1. All stars are bodies that shine steadily.
2. All planets are stars.
3. All planets are bodies that shine steadily.

The second of these arguments has a false premise. But it is still valid. Again, to say that an argument is valid is only to say that if all of its premises are true, then its conclusion must be true. To see that this argument is valid, try to imagine some way the world could be such that all of the premises are true. Note that in any such possible world, the conclusion of the argument must be true as well. This thought experiment suggests an intuitive test for validity. If you can imagine some way the world could have been where the premises of an argument are true but its conclusion is false, then the argument is not valid. Again, for an argument to be valid, it must be impossible for all of the premises to be true and the conclusion false. Now, think carefully about the following arguments and determine whether or not they are valid.

Exercise 1a

1. Donna will get an A in philosophy if an only if she writes a good paper.
2. Donna got an A in philosophy
3. Therefore, she wrote a good paper.

1b

1. If Donna writes a good paper, she will get an A in philosophy.
2. Donna got an A in philosophy
3. Therefore, she wrote a good paper.

1c

1. If whales are mammals, then they are not fish.
2. Whales are fish
3. Whales are not mammals.

1d

1. If the rapture has occurred, then either some of the cars on the highway will be unoccupied or all drivers are damned.
2. Some drivers are not damned.
3. None of the cars on the highway are unoccupied.
4. Therefore, the rapture has not occurred.

1e

1. Some snarks are bandersnatches.
2. All bandersnatches are igglypoofs.
3. So, some snarks are igglypoofs

An argument’s validity or invalidity is determined by its logical form. We can represent the logical form of an argument by replacing all of the non-logical vocabulary with capital letters. Here are the logical forms of the valid sample arguments from above.

1. If Joe is late for work, then he will get fired.                 If P then Q
2. Joe was late for work.                                                 P .
3. Joe got fired.                                                               Q

1. All stars are bodies that shine steadily.                        All S are B
2. All planets are stars.                                                   All L are S
3. All planets are bodies that shine steadily.                     All L are B

The logical vocabulary in the first argument, the "If…then…", reveals a logical relationship between sentences. So the "P"s and "Q"s in the schema for its logical form stand for declarative sentences. In the second argument, the logical vocabulary "All" and "are" represent relations between predicates that express properties or categories of things. So the capital letters in the schema for that argument stand for predicates, not complete sentences. The sample arguments above are deductively valid. Since the validity of an argument is determined by its logical form, any arguments that have the same logical form as the sample arguments will also be valid.

Exercise 2: Generate valid arguments by substituting sentences for "P" and "Q" in the first form above and by substituting predicates for "S", "B" and "L" in the second form.

Exercise 3: The following arguments are invalid. Show that the arguments are invalid either by identifying their logical form and  generating an argument that has the same logical form with all true premises and a false conclusion or show that the arguments are invalid by describing a possible way the world could be that makes all of the premises true and the conclusion false.

1. If Joe fails his midterm, then he will fail calculus.
2. Joe did not fail his midterm.
3. Joe did not fail calculus

1. Some monkeys are primates.
2. Some primates are mammals.
3. Some monkeys are mammals.

Hopefully, it is now clear that for an argument to be valid, it does not have to sound convincing. A valid argument that had obviously false premises should not seem persuasive at all. For a deductive argument to provide one with a good reason for believing its conclusion, it has to be valid and have true premises. An argument that is both valid and has all true premises is called a sound argument. Valid arguments need not have true conclusions. But sound arguments must have true conclusions. In determining whether or not an argument is sound, you might have to consider further arguments in support of the premises of the original argument.

Exercise 4     Questions: