A valid argument has a form such that it is impossible for the
premises to be true and the conclusion false.
If a complete survey of all possible assignments of truth values
for an argument yields an assignment in which the premises are
true and the conclusion is false, the argument is invalid.
If no such line is found, the argument is valid.
1. Symbolize the argument.
2. Write out the symbolized argument with single slashes between
premises and a double slash before the conlcusion.
3. Draw a truth table as though it were a broken proposition.
4. If you can find a line on which the premises are all true and the conclusion
is false, the argument is invalid; if you can't, it's valid.
The third line in this table reveals an assignment of value in
which the premises are true and the conclusion is false. This
argument is invalid.
| A | > | B | / | ~ | A | // | ~ | B |
| T | T | T | F | T | F | T | ||
| T | F | F | F | T | T | F | ||
| F | T | T | T | F | F | T | ||
| F | T | F | T | F | T | F |
Because there is no line on which the premises are all true and
the conclusion is false, this argument is valid.
| A | > | B | / | A | // | B |
| T | T | T | T | T | ||
| T | F | F | T | F | ||
| F | T | T | F | T | ||
| F | T | F | F | F |
Sometimes it is easier not to create an entire table for an argument
but to approach it indirectly.
Setting out all the premises and the conclusion in a line, try
to find a consistent set of values that will make the premises
true and the conclusion false.
If you can, the argument is invalid; if you can't, the argument
is valid.
First set out the premises and the conclusion in a line with slashes
separating the premises and a double slash before the conclusion.
| ( A | * | C ) | V | D | / | ~ | C | // | ~ | D |
Assume you are going to assign values to make the premises true
and the conclusion false.
| ( A | * | C ) | V | D | / | ~ | C | // | ~ | D |
| T | T | F |
If ~ D is false in the conclusion, then D must be true. Repeat
that assignment wherever " D " appears.
| ( A | * | C ) | V | D | / | ~ | C | // | ~ | D |
| T | T | T | F | T |
To make ~ C true in the second premise you must make C false.
Repeat this assignment wherever " C " appears.
| ( A | * | C ) | V | D | / | ~ | C | // | ~ | D |
| F | T | T | T | F | F | T |
The first premise is a disjunction. It will be true if only one
disjunct is true. Since D is already assigned the value "true"
it doesn't matter what value you assign A.
| ( A | * | C ) | V | D | / | ~ | C | // | ~ | D |
| ? | F | F | T | T | T | F | F | T |
Thus you have found an consistent assignment of truth values that
makes the premises true and the conclusion false. The argument
is invalid.
Again we set out the argument in a line, separating the premises
and the conclusion.
| ( A | * | C) | V | D | / | ~ | C | // | D |
Set out to make the premises true and the conclusion false.
| ( A | * | C) | V | D | / | ~ | C | // | D |
| T | T | F |
But notice that C must be false to make the second premise true
and true to make the first premise true. But that would require
a contradictory assignment.
Since it is impossible to make the premises true and the conclusion
false, the argument is valid.
| ( A | * | C) | V | D | / | ~ | C | // | D |
| ? | ? | T | F | T | F | F | |||
| T | T | T | F | ? | ? | F |