However, once you have decided whether the deductive arguments are valid or invalid, it is meaningful to ask whether the premises are true. If a deductive argument is valid and has true premises, then we say it is a sound argument. Deductive arguments that are invalid or possess one or more false premises are unsound. In other words, if a deductive argument is invalid it is unsound; if it has one or more false premises, it is unsound. A rational person will accept the conclusions of sound arguments.

Indicate (1) deductive or inductive (D or I), (2) valid or invalid or strong or weak (V or IV or S or W) Hint on (1): find the inductive ones; the others are deductive.

Explanation on inductive and deductive.

Only the arguments in III are inductive (clearly indicated by "probably"). The rest are deductive. Immediate inferences based on the patterns of categorical statements, i.e., on the Square of Opposition, are typically deductive, even if the word "necessarily" is not used. Arguments that make predictions are often inductive even if they do not include a word indicating it like "probably." Historians, journalists, and detectives also make extended use of inductive reasoning.

I. Immediate Inferences

1. All Greeks are philosophers, so some Greeks are philosophers.
This is valid, because an A-type (universal affirmative) entails its corresponding I-type (particular affirmative). Although it's valid, it's unsound, because the premise is false.

2. Some Christians are philosophers; therefore it is false that no Christians are philosophers.
This is valid because an I-type (particular affirmative) is the contradictory of its corresponding E-type (universal negative). Contradictories have opposite truth values. So if the I-type is true, the corresponding E-type must be false. Since the premise is true and the argument is valid, the argument is sound.

3. Since it's false that all Greeks are philosophers, no Greeks are philosophers.
This is invalid because the A-type (universal affirmative) and the E-type (universal negative) are contraries. We can reason between contraries from the truth of one to the falsity of the other, but we cannot go from the falsity of one to the truth of the other. Since the argument is invalid, it is unsound, even though the premise "it is false that all Greeks are philosophers" is true.

4. Because no Christians are philosophers, some Christians are not philosophers.
This is valid because the E-type (universal negative) entails the corresponding O-type (particular negative). If the E-type is true, then the corresponding O-type would have to be true. This argument is unsound because the premise is false.

5. Since some Christians are absolutists, all Christians are absolutists.
This is invalid. You cannot infer a universal affirmative from its corresponding particular affirmative. The argument is unsound even though the premise is true, because the argument is invalid.

6. Given that all students are relativists, it's false that no students are relativists.
This is valid. Since the universal affirmative and the universal negative are contraries, if one is true, the other must be false. Although it is valid, it is unsound, because the premise is false.

7. Since some philosophers are not skeptics, it is false that all philosophers are skeptics.
This is valid. The O-type (particular negative) and its corresponding A-type (universal affirmative) are contradictories. Therefore, if the O-type is true, the corresponding A-type must be false.

II. Syllogisms

1. No agnostics are believers. No atheists are believers. So all atheists are agnostics.
Even if the premises are true, that's no guarantee the conclusion is. The three groups could be entirely separate from one another.

2. All Moslems are believers and all Christians are believers. So some Moslems are Christians.
The premises can both be true while the conclusion is false. Suppose, for example, that "believer" is broad enough to include two distinct groups. (This is similar to: all cows are mammals, all horses are mammals, so some horses are cows.)

3. Because all prime numbers are odd numbers and all odd numbers are integers, all prime numbers are integers.

This is valid. If the premises were true, the conclusion would have to be true. However, the first premise is false (2 is a prime number). So the argument, though valid, is unsound.

4. Plato is a founder of the Academy (in ancient Athens). All founders of the Academy (in ancient Athens) are Greek philosophers. So Plato is a Greek philosopher.
This is valid, because if its premises are true, it would not be possible for the conclusion to be false. Now, its premises are true. So it is sound.

5. Because Martha is an absolutist and (she is) a Moslem, some (i.e., at least one) Moslems are absolutists.
This is valid. If we can assume that its premises are true, it is also sound.

III. Other

The claim in both these cases is that the premises make the conclusion probable.

But is that claim justified? Not in the first case. The first argument is very weak. Being a "swell guy" and an excellent opthamologist do not provide a strong basis for accurate predictions about the stock market. The second argument is much stronger. Its premises are each sufficient to create a suspicion that X is the culprit. Together, they make it highly probable that X is.

1. My golfing partner is a swell guy. He is an excellent eye doctor. He thinks the stock market will crash on Tuesday. So it probably will.

2. X was known to be angry with Y and had threatened to kill Y. X was observed entering Y's apartment shortly before a shot was fired. Y was discovered dying of a bullet wound just minutes later. X was observed leaving Y's apartment within a minute after the shot was fired. X was apprehended with Y's blood on his clothes five minutes after the shot was fired. Probably X shot Y.