**LOGIC III:**

**THE MAJOR OPERATIONS**

The first of the "major operations" in logic is called traditionally the "disjunctive syllogism."

Actually, it's only a kind of semi-major operation, and it's the last of the "or" ways of combining statements: "either/ or."

**DEFINITION: A SYLLOGISM is an inference involving at least two propositions as premises and a conclusion.**

All of the compound propositions in the preceding chapter were actually syllogisms, because to draw a conclusion, you had to affirm the compound proposition and then affirm or deny one or the other of the propositions that made up parts of the combined one; the conclusion you got was an affirmation or denial of the other part.

Either/or is no different from this. The compound proposition means "not both and not-neither."

**RULES: 1. The compound "either/or" proposition means that one of the parts must be affirmed and the other one denied.**

** 2. An affirmation of either part of the compound demands a denial of the other part.**

** 3. A denial of either part of the proposition demands an affirmation of the other part.**

** 4. The compound is symbolized by / [space slash space], with parentheses around the propositions.**

This compound is the one in which one of the sub-propositions contradicts the other one. You may not know which one of them is "true," but one and only one is "true," and the other one has to be "false."

For example, "Either every Chimera is a lizard or some chimeras are not lizards." [((e)chimera - (s)lizard) / ((s)chimera n lizard)]. You may not know what chimeras are, but intuitively you know that either all of them are lizards or at least one of them isn't a lizard.

Here, if you know that there is a chimera that isn't a lizard (affirming the second part), you have to deny that every chimera is a lizard. If you know that every chimera is a lizard, you have to deny that some aren't lizards. Or if you know that it's false that every chimera is a lizard (denying the first part), then it has to be the case that some chimeras aren't lizards; or if it's false that some chimeras are not lizards (i.e. not even one is not), then every chimera is a lizard. Symbolically, ((p / q) + q) * np; ((p / q) + np) * q; ((p / q) + p) * nq; ((p / q) + nq) * p. I won't trouble you with parentheses involving chimeras and lizards; you can do that for yourself.

The inference is pretty simple. The only time you could get fooled is by multiple negatives. For instance, in English, "It's either raining or it's not raining; but it's not raining, therefore ...?" makes you inclined to draw the "conclusion" "Therefore it's raining," which is absurd (How could it be raining if it wasn't raining?). But if you say, "Either the weather is a rainy-thing or the weather is not a rainy thing" and then you deny the first part (It is false that the weather is a rainy thing) then you have to affirm as true the second part (the weather is not a rainy thing). In other words, if it's false that it's rainy, then it's true that it's not rainy. Of course.

This is one of the cases where knowing logic can help, even though, as I said, in ordinary use of language we do logic that is considerably more complicated than what is taught in logic courses. When in doubt, stop thinking and apply the rules; they always work if you apply them right; and so once you've come up with the correct conclusion, you can think your way through the inference more easily (noticing what the statements mean).

The next compound is the general form of the inference.

Traditionally, it is called the "hypothetical syllogism," because the "if" part is simply put forward for the sake of argument--which is what "hypothesis" means, and the connection means that the person who makes such a statement knows (for whatever reason) that the "then" part follows as a conclusion.

**DEFINITION: The ANTECEDENT is the proposition that forms the "if" clause of the "if-then" compound.**

**DEFINITION: The CONSEQUENT is the proposition that forms the "then" clause of the "if-then" compound.**

**RULES: 1. "If-then" (read if p then q) affirms that the consequent depends on the antecedent.**

** 2. An affirmation of the antecedent demands an affirmation of the consequent, or the compound is contradicted. (This is traditionally called
"modus ponens.")**

** 3. A denial of the consequent demands a denial of the antecedent, or the compound is contradicted. (This is called "modus tollens.")**

** 4. Nothing follows either from a denial of the antecedent or an affirmation of the consequent.**

** 5. The proposition is represented symbolically by * [space asterisk space], with the appropriate parentheses.**

Logicians have a controversy about this, which I will discuss shortly. But first let me give an example.

"If it is raining out, then the cat is in the house." (p * q) or (weather - rainy) * (cat - insider).

Now then, if you happen to know that it is raining out (affirming the antecedent), then the cat has to be inside, or the compound proposition is contradicted. (Cats being what they are, of course, this might happen; but logic doesn't deal with truth, remember.) ((p * q) + p) * q. (((weather - rainy) * (cat - insider)) + (weather - rainy)) * (cat - insider).

Or if you see the cat outside (denying the consequent), then it can't be rainy out, because if it is rainy out, the cat is inside. ((p * q) + nq) * np. (((weather - rainy) * (cat - insider)) + (cat n insider)) * (weather n rainy). [Here I put the "nots" inside the denied propositions. You have to be careful when you do this, because the denial of a negative proposition is, of course, an affirmative one.]

**10.2.1. "Material implication"**

But the rules above say that a denial of the antecedent doesn't give a conclusion.

The same goes for an affirmation of the consequent. Some logicians don't agree with this. "Anything follows," they say, "from a false premise." What I (and Aristotle, and everyone until the twentieth century) am saying is that nothing follows from a false premise.

The new interpretation arose because modern symbolic logicians approached logic from mathematics, rather than from a linguistics. Mathematics likes closed systems, and abhors things that end up with question marks. So to allow there to be an answer to "If it's rainy, then the cat is in the house; and it's not rainy" they invented the rule that it's okay to draw the "conclusion" "the cat is not in the house" OR to draw the "conclusion" "the cat is in the house."

As an example of the fact that we speak this way, they use statements like, "If you win this bet, I'll eat my hat." The idea here is that "I'll eat my hat" is supposed to "follow from" your winning the bet.

I think, however, that they miss the point of the statement. The consequent is something that the speaker considers, not false but impossible--and the compound is constructed with an impossible consequent "connected" to the antecedent the speaker wants to deny emphatically. Since the consequent is supposed to be seen as manifestly false, (and so must be denied) this "connection" is seen as a graphic way of denying the antecedent.

In other words, "If you win this bet, I'll eat my hat" is another way of saying, "No way you'll win this bet!" An unreal "dependence" is alleged for the purpose of making a kind of "modus tollens" inference--a valid one.

So the example alleged does not in fact show that we ever think that anything follows from a false premise.

Contemporary logic (because the philosophers who deal with it don't want to have anything to do with anything "mystical" like thinking, but want to have a direct bridge between statements and facts) has got itself entangled with actual truth and falseness of propositions, and is very often interested in "proving" the truth or falseness of the compound from what you may happen to know about the truth or falseness of the parts. (For instance, you can prove that an either/or compound is false if you know that both parts of it are true--because this contradicts the compound.)

With "if-then," they want to say that the compound is not contradicted by denying the antecedent; and once you say that, you are forced into saying that the compound is "true" (in the factual sense) when the antecedent is false--no matter what the consequent is.

But this would mean that a statement like, "If Chicago is in Detroit, then George Blair is human" is a factually true statement. I refuse to admit that my humanity depends on Chicago's being in Detroit. We simply do not use language this way.

But the logicians in question don't want to have anything to do with "depends on"--where the person making the "if-then" statement (statement, now, not proposition) would have some evidence that the "then" part actually depends on the "if" part. But I submit that we only use "if-then" when (a) we want to affirm a dependence, or (b) as in the "I'll eat my hat" we create a stupid "dependence" for rhetorical reasons. To take the second case as the paradigm is as silly as taking the "face" of a cliff as the paradigm (the model, primary sense) of what a "face" is.

At any rate, I side with Aristotle in the dispute over "if-then." I think that "material implication" makes a travesty of language, and that it should be
dropped from use in logic, however "powerful" those who like it think it might be. I don't think it reflects in any way either how we use language or how
we reason. (*And* I don't think it's any more "powerful" than the logic I'm describing to you.)

[Actually, things *can* follow from factually false premises. For instance, "If Jupiter were in the orbit of Venus, Jupiter would be hotter than it is." Since we know that proximity to the sun causes a planet to be hot, then obviously the "then" of this proposition follows from the "if," in spite of the fact that Jupiter is actually *not* in the orbit of Venus. On the other hand, "If I were President, I'd be supremely happy" is a false statement, because from what I know of myself, the responsibility would make me miserable. Hence, the "then" doesn't follow. Finally, "If you break up with me, I'll go back to my room and shoot myself" may or may not be valid or true. A student years ago in my class once actually did this; so in this case, you don't know whether the conclusion follows or not.

The upshot of this is that the "if-then" inference is valid *if* (and only if) what is stated in the consequent *actually does in fact depend* on whatever is stated in the antecedent (whether what is stated there happens to be factually true or not).]

But since this cannot be known inside logic itself, we will therefore assume, for purposes of this text, at least, that nothing follows from denying the antecedent or affirming the consequent.

For example, "If it is raining, then the cat is inside; and it is not raining; and so ?" Or "If it is raining then the cat is inside; and the cat is inside; and so ?" Symbolically, with words (((weather - rainy) * (cat - insider)) + (weather n rainy)) * ?; or (((weather - rainy) * (cat - insider)) + (cat - insider) * ?

[The reason behind this is that the assumption in logic is that the argument that is being examined contains (at least in the proposer's mind) true statements. (Why else would he state them and draw a conclusion from them?) Of course, if you can prove that one of his statements is false, then sometimes (but only sometimes) you can destroy his argument. But enough of this.

**10.2.2. Some English compounders**

There are a couple of words in English that are logical connectors of propositions, but which do not appear in logic.

The reason is probably partly because they're rather complex. The logic we use is actually more complicated than what you have seen here.

**"Because" means that the statement which precedes "because" is implied by the statement which follows (reversing "if-then's" order), AND that
the statement which follows is true.** "John is in the house because his mother won't let him come out" is logically the same thing as "If John's mother
won't let him come out, then John is in the house, and his mother won't let him come out; therefore, John is in the house." It's the whole "modus ponens"
inference, then.

That is,"q because p" means "((p * q) + p) * q."

**"But" means that the statement which follows is true and is the opposite of what you would think is implied by the statement which precedes.**
"The cake is on the table, but I'm not going to eat it" means "The cake is on the table implies that I'm going to eat it, and I'm not going to eat it, and so the
inference is invalid."

That is, "p but q" means "((p * nq) + n*) + q."

**10.3. The "square of opposition"**

Let us go back, now, to our propositions with subjects, copulas, and predicates.

Interestingly enough, the various propositions that can be made using a given subject-term and a given predicate-term (and changing the quantities and the affirmativeness and negativeness of the propositions in every possible combination) turn out to be related in all the ways we have seen of the compounding of propositions--with the exception of the trivial "and."

**10.3.1. The great "some" controversy**

Unfortunately, here we also run into a place where modern logic has "advanced" backwards. It has left once again the traditional logic and has got itself into a mess.

Since modern logic wants to have some connection with "truth" in the factual sense (and to deal with statements and not propositions, using my terminology), it has made the following allegation about language.

Modern logic asserts that the definite proposition (the one using "every" as the quantity of its subject) is actually an "if-then" inference in disguise, and
does not necessarily affirm the *truth* of what it seems to be talking about. That is, "Any horse is an animal" is taken to be the equivalent of "If the object
you are talking about is a horse-thing, then the object you are talking about is an animal-thing--for any object you want to talk about)."

Symbolically, (e)horse - (s)animal becomes in modern logic (using my notation and square brackets [x] to symbolize "for any x" where "x" is that thing the mathematicians love, a "variable.") [x](x - horse) * (x - animal).

The trouble is that this proposition is true only if it is true now, not for every *horse,* but for anything you want to name. What they tried to avoid is the
problem of induction, where you've leaped from knowing that some horses are in fact animals to the general statement that every horse is an animal.
They think they've solved the problem by making the generalization an "if-then" statement; but in doing so, they don't apparently realize that now to find
out if the connection is valid, you have to test every x you can think of in order to find if there are some that are not both horses and animals. They made
the problem worse, in other words.

On the other hand, modern logicians call the "some" quantifier, the "existential quantifier" and symbolize it with a backwards E (which, since I don't have a backwards typewriter, I won't attempt to duplicate). "Some unicorns have four legs" is a false statement, because there aren't any unicorns.

But putting the two together, "every unicorn has four legs" is true, because it means "If there is such a thing as a unicorn--and there might not be, but *if
*there is--then it has four legs," while "some unicorns have four legs" is false, because it means "There actually is a unicorn and it has four legs."

Frankly, I think this is plain silly; and it shows what a tangled web you can weave if you try to mix propositions and statements, or to talk about facts in logic.

Not only silly, but it would make the whole of mathematics a tissue of false statements, because mathematics is talking about numbers (which don't actually exist as such) and points and lines and sets and whatnot, which as mathematics talks about them are abstractions, not realities. Hence, any "some" statement mathematics makes would automatically be false.

Now if you want to make a "science" of "reasoning" which makes half of mathematics false, then go ahead, I suppose. But don't claim that you're doing anything but playing nonsense games.

(Incidentally, mathematics, not surprisingly given the "mathematical" nature of contemporary logic, has accepted the "existential quantifier" as its "some";
but *uses* it in its mathematics, on the assumption that 'in *mathematics* these things exist." That is "Some imaginary numbers are even" is taken as a *true*
statement in mathematics, meaning "There exists at least one imaginary number which is even." Talk about begging the question!)

My contention is that if you can talk about things like imaginary numbers as existing (in mathematics), then you can talk about anything else as existing (in fantasy, in joking, in cartooning, or what have you); and so the "existential" sense of "some" as meaning "there in fact is one" is meaningless; and if it's false that some unicorns have four legs because there aren't any unicorns, then what mathematics does with logic is a tissue of falsehoods OR in the world of fantasy there "actually are" unicorns with four legs. You can't have it both ways. Or in other words "either every 'some' proposition in mathematics is false or the 'some' proposition about unicorns is true. There's a disjunctive-hypothetical syllogism for you.

This, by the way, is what is called a *dilemma*, according to the following definition.

**DEFINITION: A DILEMMA is an either/or proposition, each branch of which is the antecedent of an if-then proposition; and the conclusion
of each if-then proposition is something your opponent does not want to admit.**

Hence, I will solve the "every" and "some" controversy by the following assertions:

** The subject and predicate of every proposition are taken, for logical purposes, to have "objects" they refer to, whether there are actually such
objects or not in fact.**

** Definite propositions' subjects refer to each member of the class of objects named, and are not a substitute for an "if-then" proposition.**

Thus, "Every unicorn has four legs" refers to every individual in the "class of unicorns" irrespective of whether there are any in actual fact; and "Some unicorns have four legs" assumes four-legged unicorns in the same sense that the definite proposition does.

With that out of the way, then, consider horses and animals.

How many propositions can you make with "horse" as subject and "animal" as predicate? Clearly, four: "Every horse is an animal"; "Every horse is not an animal"; "Some horses are animals"; and "Some horses are not animals."

You notice that, taken as statements, not all of these are true. In fact, the four are related in interesting ways, with these traditional names:

**DEFINITION: CONTRARIES are propositions that are related as "not-both."**

**DEFINITION: SUBCONTRARIES are propositions that are related as "not-neither."**

**DEFINITION: CONTRADICTORIES are propositions that are related as "either/or."**

**DEFINITION: SUBALTERNS are propositions that are related as "if-then."**

These definitions can also be used to describe relations of terms. For example, black and white are contraries (because there's gray), while black and non-black are contradictories (because anything has to be either black or non-black). The other two definitions don't really apply to terms as such.

Schematically, the square looks like this:

The two definite propositions are contraries: "Every horse is an animal not-both every horse is not an animal" That is, one or the other of these must be false; but both might be false. You could see this intuitively if you took "Humans" as subject and "pianists" as predicate. "Every human is a pianist not-both every human is not a pianist." has both false. The two indefinite propositions are subcontraries. "Some horses are animals not-neither some horses are not animals." One of these must be true, but both might be. In this case, of course, "some horses are not animals" is false because there isn't even one horse that isn't an animal. But with humans and pianists, both are true.

The definite affirmative and the indefinite negative proposition contradict each other; and the indefinite affirmative and the definite negative contradict each other. "Either every horse is an animal or some horses are not animals"; and also "Either some horses are animals, or every horse is not an animal." This also works with humans and pianists: "Either every human is a pianist or some humans are not pianists"; and also "Either some humans are pianists or every human is not a pianist."

Finally, the two affirmative propositions are subalterns, and so are the two negative propositions. "If any horse is an animal, then some horses are animals"; and also "If any horse is not an animal, then some horses are not animals." Again this works with humans and pianists.

**10.4. The categorical syllogism**

There remains one major operation in logic, which was first formalized by Aristotle.

It is called the "categorical syllogism" (that is, the syllogism involving predicates). Here, two propositions are combined with "and" to generate a conclusion based on what can be done with subjects and predicates.

The general form of the categorical syllogism is this ((term 1 . term 2) + (term 2 . term 3)) * (term 1 . term 3), where the periods stand either for affirmative or negative copulas.

For example "Every horse is an animal, and every animal is a living thing; and so every horse is a living thing."

There are actually various ways in which the propositions can be arranged. In fact, Aristotle arranged the basic general figure by reversing the two premises (I.e. his arrangement would be (term 2 . term 3) + (term 1 . term 2) * (term 1 . term 3). "Every animal is a living thing and every horse is an animal; therefore every horse is a living thing."

The difference in the Aristotelian arrangement and mine is that Aristotle saw the syllogism in terms of "class-inclusion," and the logic went this way: If a smaller class is included inside a larger class, and another class is included inside the smaller one, then every member of the smallest class is included inside the largest one. That is certainly true.

My own arrangement says that there are times when the function of "predication" (putting a predicate to a subject) is "transitive" (i.e. allows you to attach the last predicate to the first subject). Since predicates do not actually refer to classes of objects, I think this arrangement is closer to the way we use language. My arrangement also has the advantage of putting the "middle term" in the middle, which shows more obviously its function of connecting the extremes.

Some terminology:

**DEFINITION: The SUBJECT-TERM of a categorical syllogism is the term that forms the SUBJECT OF THE CONCLUSION, whether it is
the subject of the premise it is in or not.**

**DEFINITION: The PREDICATE-TERM of a categorical syllogism is the term that forms the PREDICATE OF THE CONCLUSION, whether
it is the predicate of the premise it is in or not.**

**DEFINITION: The MIDDLE TERM is the term that DOES NOT APPEAR IN THE CONCLUSION.** It "mediates" between the subject-term and
the predicate-term.

**DEFINITION: The SUBJECT-PREMISE is the premise in which the subject-term appears.**

**DEFINITION: The PREDICATE-PREMISE is the premise in which the predicate term appears.**

Traditionally, the subject-premise was called the "minor premise," because it dealt with the smallest class (the subject); and the predicate-premise was called the "major premise" because it dealt with the largest class (the predicate). Again, that was due to the theory of class-inclusion as an explanation of why the syllogism works.

**10.4.1. Rules of the categorical syllogism**

The rules of the categorical syllogism are just the statements of the conditions under which predication is transitive.

You can attach, in other words, a new predicate to a subject under the following conditions:

**RULES: 1. There must be three and only three propositions (two premises and the conclusion).**

** 2. There must be three and only three terms.**

** 3. The middle term must be definite at least once.**

** 4. If a term is definite in the conclusion, it must be definite in its premise.**

** 5. Both premises may not be negative.**

** 6. If one premise is negative, the conclusion must be negative.**

** 7. If both premises are affirmative, the conclusion must be affirmative.**

Traditionally, there are other "rules" that are actually corollaries of the preceding seven. For instance, at least one premise must be definite (or else either both will be negative, the middle term will be indefinite twice, or the predicate would be indefinite in the premise and definite in the conclusion). Also, if one premise is indefinite, the conclusion will have to be indefinite (or else either the subject-term will pass from indefinite to definite or there will be two indefinite middle terms). If it pleases you, you may learn these other two rules also. (There is also the "rule" that the middle term must not appear in the conclusion; but that would mean either that it was used three times or that it wasn't the middle term.)

Some comments on the rules:

1. There are inferences like the categorical syllogism that contain more propositions (and act like chains of syllogisms); such a chain is called a ** sorites**; it
must have the same number of terms as propositions. For example, "Every maverick is a horse, and every horse is an animal, and every animal is a living
being, and every living being is an active being; and therefore every maverick is an active being."

There are also hypothetical sorites (the plural is the same as the singular): "If you study hard in this course, then you will pass it, and if you pass it, you will graduate, and if you graduate, you will get a better job; therefore, if you study hard in this course, you will get a better job.

There is also in ordinary usage a kind of informal syllogism that *leaves out* some premise that is so obvious as not to be worth stating (or which does not
explicitly draw a conclusion which obviously follows). Such a truncated syllogism is called an ** enthymeme**. For example, "Every human being dies, and
so you will die" leaves out "and you are a human being." Most "because" statements are actually enthymemes. Here is a hypothetical enthymeme: "If you
study hard, you will get an A, and you

2. This rule of having only three terms is violated by the "four-term syllogism," where the same word is used in two different senses (and hence is two terms). Obviously, then, there can be no mediation between the subject and predicate: "Hercules is a hero and hero is a four-letter word; and therefore Hercules is a four-letter word."

3. Not having the middle term definite at least once (it may be definite twice) results in the "demagogue's fallacy." "Every Republican is an investor of money, and every miser is an investor of money; and therefore every Republican is a miser." If the middle-term is the predicate of two affirmative propositions, it is indefinite twice. The idea here is that the "class" of investors is larger than either Republicans or misers (which form an indefinite part of it), so that you can't guarantee that the two extreme classes overlap.

4. The reason the term that is definite in the conclusion must be definite in the premise is that you can't conclude to more than you started with. The
middle term may be indefinite once and definite the other time, because you are drawing no conclusion *from one premise to the other*; but you are doing
so from the premise to the conclusion.

5. The reason both premises can't be negative is basically that two exclusions do not force either an exclusion or an inclusion. "No horses are pigs and no pigs are whales; and so no horses are whales" has a true conclusion; but that it doesn't follow can be seen from "No horses are lizards and no lizards are mammals; and therefore no horses are mammals."

6. The rule that a negative premise needs a negative conclusion can be seen from what happens if you obvert the negative premise. I will let the reader do this.

7. The fact that affirmative premises generate affirmative conclusions basically says that you can't argue to a disconnection by connecting.

There are, as I said, several ways of arranging the terms of the propositions.

Traditionally, these are called "figures," of which there are four. Traditionally also, there are rules for each of the figures; but they are all applications of the above seven rules, and I won't burden you with them.

Here are the four figures, with the propositions arranged above each other:

I II III IV

S . M S . M M . S M . S

M . P P . M M . P P . M

S . P S . P S . P S . P

Notice that the subject-term and the predicate-term appear in different places in their own premises. The first figure is the clearest, because the subject-term is the subject of both the conclusion and its premise, and the predicate-term is the predicate of both; and the middle-term is in the middle. "Every horse is an animal and every animal is a living being; and so every horse is a living being."

In the second figure, the middle-term appears as the predicate of both premises (with the predicate-term appearing as the subject of its premise--though it is the predicate of the conclusion). In this figure, one of the premises must be negative, or the middle term is indefinite twice; and hence this figure can only generate negative conclusions. "Every horse is an animal, and every typewriter is not an animal; and therefore every horse is not a typewriter."

In the third figure, the middle term is the subject both times, meaning that the subject-term is the predicate of its premise. "Every animal is a living being, and every animal is not a typewriter; therefore some living beings are not typewriters."

The fourth figure, with the middle term on the "outsides" of both premises, is the most confusing of all. "Every horse is an animal, and every maverick is a horse; therefore, some animals are mavericks." [When I wrote the original version of this, I myself became confused and "concluded" "Some horses are mavericks," which uses the middle term three times. Don't use this figure; convert some proposition so that it gets into a clearer form.]

Symbolic logicians like to say that their logic (which, as you can see, I don't like) is more "powerful" that what is here.

It can, they assert, allow them to make inferences which are "forbidden" in traditional logic. It is true that it obviously follows that if Frank loves Mary and Mary is a woman, then Frank loves a woman. It is equally true that in Aristotelian logic as it stands, there is no way to put this into syllogistic form.

Not one to be daunted by the fact that giants of intellect (which quality I am perfectly willing to concede to the founders of modern logic) disagree with me, let me make the following rule to "save" traditional logic--instead of throwing out the baby with the bath and succumbing to material implication and the idiocy that "every" does not imply "some."

**RULE OF SUBSTITUTION: If a term appears as part of a more complex term, then every term predicable of the part can be substituted, in its
indefinite form, for the term of which it is a part.**

What does that mean? In logical form, the "Frank loves Mary" syllogism would go this way:

"Every Frank is something that loves Mary, and every Mary is a woman; therefore every Frank is something that loves some women." Mary might not like that if she doesn't realize that "some" here means "at least one" and is compatible with "only one" even though grammatically it uses the plural form.

Similarly, to put "A horse is an animal, and so the head of a horse is the head of an animal" into logical form, we have to do this:

"Every head of a horse is a head of a horse, and every horse is an animal; and therefore every head of a horse is a head of some animal." Here it doesn't make grammatical sense to say "some animals"; but the meaning is the same. In symbolic logic, it takes twelve steps to reach this conclusion--so I think my rule actually is "more powerful."

There are other variations on this; but this will be enough, I think, to show how logic basically works.

And since this is just a kind of sketch of knowledge from the point of view of how we get it and how we express it, let us end this book here.

"Either/or" or the "disjunctive syllogism" (p / q) means that one of the constituent propositions must be affirmed and the other denied. An affirmation of one concludes to a denial of the other, and a denial of one concludes to an affirmation of the other.

"If-then," also called the "hypothetical syllogism" (p * q), has two parts: the "if" or "antecedent" and the "then" or "consequent"; it is the general form of inference. It means that the consequent depends (in some way known outside of logic) on the antecedent. Affirmation of the antecedent concludes to affirmation of the consequent ("modus ponens"), and denial of the consequent concludes to denial of the antecedent ("modus tollens"). Affirmation of the consequent or denial of the antecedent leads to no conclusion.

Contemporary logic uses "material implication," where the inference is assumed valid if the antecedent is denied, whether the consequent is affirmed or denied. This does not reflect the way we think or use language.

"Because" means that the statement which precedes is the consequent and that which follows is the antecedent, and in addition affirms the antecedent (thus establishing the "truth" of the statement preceding the "because"). "But" means that the statement which follows is true, and is the opposite of what would seem to be implied by what preceded (denying, thus, the inference itself).

Contemporary logic treats definite propositions as if they were "if-then" inferences with a variable (so that they need not be factual), while indefinite ones are assumed to be "false" if the subject does not in fact exist. This position does not reflect how we think or, in fact, how symbolic logic is used in mathematics. We will assume that for logical purposes terms have referents, and that definite propositions refer to every member of the class named, and are not in fact "if-then" propositions.

The "Square of Opposition" includes the four possible combinations of propositions with a given subject-term and predicate-term. The two definite propositions (contraries) are related as "not-both"; the two indefinite ones (subcontraries) as "not-neither"; the two affirmative propositions are related as "if-then," with the definite implying the indefinite; and this also goes for the negative propositions. The definite affirmative and indefinite negative are contradictories, related as "either/or"; and this also goes for the indefinite affirmative and definite negative.

The categorical syllogism (syllogism involving predicates) consists of two premises and a conclusion. The subject-premise contains the subject-term (the subject of the conclusion); the predicate-premise contains the predicate-term (predicate of the conclusion); and both premises also contain a "middle term." Conclusions are valid when the relation of predication is transitive through the middle term to the conclusion.

Predication is transitive if the following rules hold: Only three terms and three propositions; middle term definite at least once; definite term in conclusion demands that it be definite in its premise; both premises not negative; if one premise negative, negative conclusion; and if both premises affirmative; affirmative conclusion.

The way terms are arranged in the propositions is not relevant, as long as the rules are followed; but there are four possibilities, called the "four figures," of which the first (S . M, M . P; S . P) is clearest.

To allow for certain operations not permitted by the above, if a term is part of a more complex term, anything predicable of the part may be substituted for the term which is the part.

A sorites is a chain of syllogisms, where from the first premise of the chain, the final conclusion follows. An enthymeme is an informal syllogism, where an obvious premise or conclusion is left unsaid.

*Are the following valid? If so, what kind of syllogism are they, and if not, what rule do they violate?*

1. If I were king, I would give you half my kingdom; but I'm no king, so tough luck.

2. It's either not a bargain at all, or it's not something he got legitimately; and I happen to know that he didn't get it illegitimately, so it must be quite a bargain.

3. You're either lying or you're stupid; but if you're stupid, they you wouldn't have been there in the first place. But since I know that you were there, you must be lying.

4. If there is a life after death, then it's either neutral or there is something like a heaven and hell; but the evidence leading to a life after death is contradicted if it's a neutral state, so there's either no life after death at all or there's something like a heaven and a hell.

5. John is a very intelligent student, and practically all very intelligent students can do logic well; therefore, John can do logic well.

6. A dog is not something that can make a free choice; and nothing that can make a free choice is something that can morally be tortured; therefore, a dog is not something that can morally be tortured.

7. Every harpist is a musician, and some women are harpists; therefore, some women are musicians.

8. Some musicians are harpists, and no musician has a tin ear; therefore, no harpist has a tin ear.

9. Every person who discriminates is doing something wrong; and John, when engaged in wine-tasting, is a person who discriminates; therefore, John, when engaged in wine-tasting, is doing something wrong.

10. Every hard-drug user started using marijuana; you started using marijuana; therefore, you are a hard-drug user.