An argument is valid if there is no possible way for its conclusion to be false while all its premises are true. In other words, a valid argument is one whose premises, if true, guarantee that the conclusion must be true as well. The conclusion cannot possibly be false unless at least one of the premises is false too.

An argument is invalid if there is at least one possible way for the conclusion to be false while all the premises are true. In other words, the premises of an invalid argument don’t guarantee that the conclusion is true. Even if the premises are all true, the conclusion might be false.

An argument is sound if it is valid and its premises are all true (so the conclusion must be true as well). As far as the *objective* properties of an argument are concerned, soundness is the best case scenario: the premises are all true and they guarantee that the conclusion is true as well. A sound argument is objectively flawless. Nevertheless, a sound argument might be subjectively *unconvincing*, for two reasons:

- we may mistakenly think that some of the argument’s premises are false, or
- we may fail to recognize that the argument is valid.

Some valid argument forms cannot be represented in propositional logic. In other words, some arguments have logical structures that cannot be represented with truth-functional connectives. We’ll deal with other types of logical structure (besides truth-functional structure) in later chapters.

Logic can’t help us with the first problem. Logical analysis of an argument cannot tell us whether the argument’s premises are true or false.If the argument contains a tautology or a contradiction, of course, logic can determine the truth value of that proposition. But tautologies and contradictions generally don’t (and shouldn’t) appear as premises or conclusions in an argument. But there are logical procedures we can use to determine whether an argument is valid or invalid. The validity (or invalidity) of an argument depends on its form, or logical structure, rather than on the specific claims made in its premises and conclusion. In order to examine the form of an argument, it is often useful to symbolize the argument with well-formed formulas (WFFs) of propositional logic.

To represent the form of an argument in propositional logic, we symbolize its premises and conclusion as WFFs and list them vertically, with the conclusion at the bottom of the list. The premises and conclusion are separated by a line, and the conclusion is indicated by the symbol ‘∴’ (which means “therefore”). Here’s an example:

Consider the following simple argument, which my children sometimes fail to understand:

You may have dessert only if you ate your veggies. You didn’t eat your veggies. Therefore, you may not have dessert.

Using the sentence letter ‘D’ to represent the simple proposition you may have dessert and ‘V’ to represent you ate your veggies, the argument can be symbolized as follows:

(D ⊃ V) |

~V |

∴ ~D |

In order to check all of the possibilities, we construct a truth table in which each row represents a possible combination of truth values for the sentence letters that appear in the argument. We make a column for each premise of the argument, and also a column for the conclusion, as shown in the example below. Then we simply look and see whether there is any possible combination of truth values (i.e. a row in the truth table) where the premises are all true but the conclusion isn’t.

Consider the following argument:

If Amy gets the correct answer, then Ben won’t. If Ben gets the correct answer, then Amy won’t. Therefore, either Amy or Ben will get the correct answer.

Is this argument valid or invalid? To find out, we can symbolize the argument using truth-functional connectives, then construct a truth table. Using the sentence letter ‘A’ to represent the simple proposition Amy will get the correct answer, and ‘B’ to represent Ben will get the correct answer, the argument is symbolized as follows:

(A ⊃ ~B) |

(B ⊃ ~A) |

∴ (A ∨ B) |

Now we can construct a truth table, like this:

premise 1 | premise 2 | conclusion | ||

A | B | (A ⊃ ~B) | (B ⊃ ~A) | (A ∨ B) |
---|---|---|---|---|

0 | 0 | 1 | 1 | 0 |

0 | 1 | 1 | 1 | 1 |

1 | 0 | 1 | 1 | 1 |

1 | 1 | 0 | 0 | 1 |

Look at the first row of the table, which is highlighted above. That first row represents the possibility that sentence letters A and B are false, as shown in the two columns to the left (highlighted in green). Notice that the argument has true premises and a false conclusion in that row (highlighted in yellow). This means that if the sentence letters A and B both turn out to be false, then the argument will have true premises and a false conclusion. So there *is* a possible way for the conclusion to be false while all the premises are true, and the argument is *invalid*.

A possibility where the premises are all true but the conclusion is false is called a counterexample to the argument. An argument is valid if and only if it has no counterexamples. Since the above argument does have a counterexample, it is invalid.

Here’s another example:

If aardvarks are adorable, then either baby baboons beat bongos or crocodiles consume cute capybaras. Baby baboons don’t beat bongos. Either aardvarks aren’t adorable, or crocodiles don’t consume cute capybaras. Therefore, aardvarks aren’t adorable.

This silly argument can be symbolized as follows:

(A ⊃ (B ∨ C)) |

~B |

(~A ∨ ~C) |

∴ ~A |

premise 1 | premise 2 | premise 3 | conclusion | |||

A | B | C | (A ⊃ (B ∨ C)) | ~B | (~A ∨ ~C) | ~A |
---|---|---|---|---|---|---|

0 | 0 | 0 | 1 | 1 | 1 | 1 |

0 | 0 | 1 | 1 | 1 | 1 | 1 |

0 | 1 | 0 | 1 | 0 | 1 | 1 |

0 | 1 | 1 | 1 | 0 | 1 | 1 |

1 | 0 | 0 | 0 | 1 | 1 | 0 |

1 | 0 | 1 | 1 | 1 | 0 | 0 |

1 | 1 | 0 | 1 | 0 | 1 | 0 |

1 | 1 | 1 | 1 | 0 | 0 | 0 |

Notice that there is no counterexample—no row in which all three premises are true and the conclusion is false. In other words, there’s no row that looks like this:

premise 1 | premise 2 | premise 3 | conclusion |

1 | 1 | 1 | 0 |

Since there’s no possible way for the conclusion to be false while all of the premises are true, the argument is *valid*.