Syllogisms

valid categorical syllogism
Ex: All dogs are animals
All collies are dogs
Therefore, all collies are animals. They are all universal affirmatives, or “A” propositions. There are a total of three different classes mentioned: dogs, animals and collies.
Major term: Prediciate term (P) of the conclusion- animals.
Minor term: Subject term (S) of the conclusion- collies.
Middle term: (M) appears in both premisses , but not in the conclusion- dogs.
syllogisms
Have two premisses and a conclusion.
Major and minor premisses
the premise of a syllogism that contains the major term (which is the predicate of the conclusion). So “all M are P” is the major premiss. So “All S are M” is the minor premiss.
Syllogism standard form
We list the statements in our syllogism in the following order: Major premiss, minor premiss, conclusion. So our syllogism in standard form is All M are P
All S are M
Therefore, all S are P. When the S, P and M are “collies,” “animals,” and “dogs,” both premisses are true and the conclusion is proved to be true. Therefore, this syllogism is valid and sound (valid and all true premisses)
Mood of a categorical syllogism
the mood is simply the listing of letters. Mood= the letter (A, E, I, or O) of the major premiss, minor premiss and the conclusion in that order. Since all of our propostions are A propositions, our syllogisms mood is AAA.
Syllogism validity
A syllogism is valid if it meets all the requirements or “rules” for validity. It is invalid if it breaks one or more of the rules. There are six rules or requirements
Rule One
All of the terms in a valid categorical syllogism must have the same meaning each time they occur. Remember each term will appear twice in each categorical syllogism. Ex: All dogs are animals. All collies are dogs. Therefore, all collies are animals.
Rule Two
The middle term of a valid categorical syllogism must be distributed in at least one of the premisses. The middle term is the term that appears in both premisses. If the middle term is not distributed in either premiss, then the rule is broken. The middle term in our syllogism is dogs. Since both premisses are A propositions and since any A proposition distributes its subject term but not its predicate term, we see that the major premiss “All dogs are animals,” does indeed distribute the middle term.
Rule Three
Any term which is distributed in the conclusion of a valid categorical syllogism must also be distributed in the premiss where that term appears. Rule 3 simpley says that if the minor term is distributed in the conclusion, then it must be distributed in the minor premiss where it appears and if the major term is distributed in the conclusion, then it must be distributed in the major premiss where it appears. If this doesn’t happen, then Rule Three has been broken. The A proposition distributes the subject term, but not the prediciate term. IN the conclusion then, only “collies” is distributed.Collies is also distributed in the premiss where it appears (the minor premiss). So our syllogism passes the test.
Rule Four
The premisses of a valid categorical syllogism cannot both be negative. This means that any syllogism that is valid must have at least one affirmative premiss, at least one A or I proposition. Any syllogisms whose premisses are both negative, any combination of E or O propositions is invalid because it breaks Rule Four. Both premisses are A propositions and are affirmative.
Rule Five
Says any valid categorical syllogism with a negative premiss (E or O) must have a negative conclusion (E or O) and vice versa. So if there is an E or O proposition as one of the premisses, then the conclusion must be an E or O. At the same time, if the conclusion is an E or O, then one premiss must be an E or O.
Rule Six
Rule Six has to do with quantity. The quantity of a categorical proposition refers to whether it is universal or particular. A and E propostions are universal in quantity and the I and O propostions are particular in quantity. Says any valid categorical syllogism with two universal premisses (A or E) must have a universal conclusion (A or E). Ex: A syllogism with two A propositions for premisses and an I proposition for a conclusion would be invalid because it would break Rule Six.
Breaking Rule One
Consider: All humans are mortal. Socrates is human. Therefore, Socrates is mortal. This is valid, unless Socrates refers to two different people, which would then give you four terms. In that case, the conclusion has nothing to do with the premisses and the argument is invalid. This is referred to as the “Fallacy of Four Terms”
Breaking Rule Two
Consider this argument: All dogs are animals. All cats are animals. Therefore, all cats are dogs. It is obvious this term is invalid. The middle term is animals and it is the prediciate term of both premisses. Since the A proposition does not distribute its predicate term, this argument does not distribute its middle term in at least one premiss, which is required for Rule Two. We call this the “Fallacy of the Undistributed Middle Term”
Breaking Rule Three in the Minor Term
Consider the following syllogism: No cats are dogs. All cats are animals. Therefore, no animals are dogs. The minor term is animals. It is distributed in the conclusion, since the E proposition distributes both its terms. But it is not distributed in the minor premiss (All cats are animals) since the A proposition does not distribute its minor term. This argument is invalid since it breaks Rule Three. We call it the “Fallacy of the Illicit Minor Term”
Breaking Rule Three in the Major Term
Consider this argument: All cats are animals. Some pets are not cats. Therefore, some pets are not animals. Here, the major term is animals. It is distributed in the conclusion, since the O proposition distributes its predicate term. But it is not distributed in the major premiss (All cats are animals) because the A proposition does not distribute its predicate term. We call this the “Fallacy of the Illicit Major Term”
Breaking Rule Four
Consider this argument: No dogs are cats. No cats are animals that bark. Therefore, no dogs are animals that bark. The negative propositions are E and O. Since both premisses are E propositions. This argument is invalid because it breaks Rule Four. We call this the “Fallacy of Exclusive Premisses” because the negative propositions are the ones that exclude part or all of one class from part or all of another class. If two people are arguing and both say what they will not do, no agreement can be reached. There is a quick way to discover, which categorical propositions break Rule Four. Remember, the mood of a syllogism is a list of the letters of the Major premiss, minor premiss and the conclusion in that order. Any mood beginning with EE, EO, OE, or OO will break Rule Four.
Breaking Rule Five
Consider this argument: No dogs are cats. Some dogs are barking animals. Therefore, some barking animals are cats. In this argument, the major premiss is a negative proposition (E). But the conclusion is affirmative (I). So this argument is invalid because it breaks Rule 5. We call this the “Fallacy of Drawing an Affirmative Conclusion from a Negative Premiss.” The mood of this syllogism is EII. Some of the moods that would break Rule Five are EIA, AEI, AAO, EEI, etc.
Breaking Rule Six
Consider this argument: All unicorms are animals. All animals are living things. Therefore, some unicorns are living things. Here, both premisses are universal propositions (A), but the conclusion is a particular proposition (I). Therefore, this argument is invalid since it breaks Rule Six. We call this the “Existential Fallacy” The conclusion seems to imply that some unicorns exist.