**Compounding propositions**

The preceding inferences were not called "syllogisms" because they involve only one premise, and "syllogism" is the Greek word for "a combined statement."

** A syllogism is an inference with two premises.**

As long as I have defined this, here are a couple more terms:

** An enthymeme is a syllogism with one proposition not explicitly stated.**

Enthymemes are often the way we reason in ordinary language, because the statements that are left out are so obvious that
it insults the intelligence of the hearer to state them. In the informal use of logic, we also tend to put the conclusion first (as
I am doing in this sentence), because we want to let the hearer know right away what we are driving at, and *then* give him
the evidence for it. So you would say, "John is going to die, because every human being dies," rather than, "John is a
human being, and every human being is something that will die, and so John is something that will die." You don't need to
say, "John is a human being," because your hearer knows what you are referring to (not to some dog named "John").
Enthymemes can also leave out the conclusion, as obvious. You might say, referring to John's propensity for living
dangerously, "Well, he's human, after all, and all of us have to die sometime." It would be insulting to your listener if you
then said, "and so he has to die sometime too."

** A sorites is a chaining of several syllogisms or enthymemes.**

You might give this hypothetical sorites to someone, for instance: "If you try drugs for fun, then you might start doing drugs, and if you do drugs, then you're going to become an addict, and if you're an addict, you've got nothing to live for but drugs." In the informal use of logic, this would usually be followed by "Then why try drugs for fun?" which points first of all to the omitted conclusion, "If you try drugs for fun, you're going to have nothing to live for but drugs," and the following evaluative inference, "If you don't want to have nothing to live for but drugs, then don't try drugs for fun."

Now then, what are the ways we can combine two propositions so that we can generate a conclusion from their parts?

Let me first state a general rule that can be helpful, now that we are not at the moment getting inside a proposition and looking at its parts:

**Rule: For purposes of combining whole propositions, a statement in any form is taken as a proposition.**

That is, there isn't any special logical form for statements as components of compound propositions. This will not be true
for the categorical syllogism, because it is precisely the way of compounding propositions *because of* the characteristics of
the subjects and predicates of the combined propositions. But other types of syllogisms don't have to worry about how the
components look.

Let me also make a couple of definitions to make what is going on in logic a little clearer.

** The inferential mode of reasoning affirms the compound and affirms or denies one of its components, and
concludes to the affirmation or denial of the other.**

** The refutational mode of reasoning affirms or denies each of the components and concludes to the affirmation or
denial of the compound.**

There may, of course, be more than two components in the compound proposition. "Either you're asleep or you're thinking
of something else or you're stupid" is a perfectly legitimate compound proposition, for instance. In these cases, the rules
for the compound with two components apply *mutatis mutandis,* and so I'm not going to discuss them further.

The reason why I called the second mode of reasoning "refutational" is that, as we will see, the inference from the components to the compound is only valid in proving that the compound is false, because the alleged connection between the components is not what the compound says it is. To understand this, we have to be clear about what the criteria are for a valid inference.

We need a couple of other terms:

** An inference is sound when the premises are factually true statements, and they generate a conclusion which
cannot be factually false. Otherwise, the inference is unsound, even if the conclusion happens to be true.**

This is what we ordinarily mean by the "validity" of an inference, because we see no reason for a person giving premises
which he doesn't think are true (i.e. they may be *negative* statements, but he considers them *true,* or why say them?). But
the validity is something more hypothetical.

** An inference is valid when, if the premises are true, the conclusion cannot not be false.**

So for the logic to be valid, the premises don't have to *be* true, but when they're not, *if* they were, the conclusion would be
true.

** Criterion for a valid inference (contemporary): An inference is valid if when stated as a conditional proposition, it
is true for all truth-values of the components.**

This is another way of saying that in contemporary logic, the inference is valid *its expression is a tautology.* By
"tautology" here is not meant simply "the same term is repeated," which is what we ordinarily mean by "tautology," such
as "a blue bird is blue," or by its definition, such as "a valid inference is a potentially sound inference," but also such
statements as "George Blair is not anything but George Blair." That is, any statement that fits the Principle of Identity we
discussed in the Chapter 8 of Section 1 of the First Part 1.1.8 is what we ordinarily mean by a tautology (it *says* the same thing); but
tautologies also apply to statements which fit the Principle of Contradiction (it *amounts to* the same thing).

Now contemporary logic talks about two kinds of fallacies, which mean that, in their system, the definition of soundness I have given above is not accurate. Contemporary logic's definition of soundness is "If the premises are true, the conclusion cannot be false." But "true" in contemporary logic does not mean exactly the same thing as what I mean by a "factually true statement."

The two kinds of fallacies can clear up what I am talking about. A *formal fallacy* in contemporary logic occurs either with
a false premise or a violation of a logical rule. An *informal fallacy* would be using a word in "two different senses" (taking
the same word as the same *term,* in my terminology, when in fact it is two different terms); or by concluding to something
that was irrelevant to the premises--something that you could only discover by looking at the sense of what you were
saying rather than the form as defined in contemporary logic.

Thus, for instance, to argue from the fact that George Bush is in the White house and the other fact that my feet hurt to
"George Bush is in the White House and my feet hurt" is a *sound argument* in contemporary logic, because, given the truth
of the premises, the conclusion can't be false.

But a person could say, "But Bush's presidency has nothing whatever to do with the state of your feet," meaning not that
either of the two statements was *false,* but that *the fact that each is true does not mean that you can conjoin them.* Hence,
the person would contend, it is *false* to join them into a single statement as if *together* the expressed *a* fact, when they in
fact express two *distinct* facts.

In that sense, the premises can be true and the conclusion false. "But that isn't what we mean by 'false,'" the logician
would say, "because 'and' as we use it does not say that both statements *together* form a statement of a fact, but *merely* that
*each* expresses a fact. And, of course if each expresses a fact, it is sound to say that each expresses a fact (that is, it is a fact
that each expresses a fact)." So the inference is sound.

Now what I am going to try to show in what follows is that in each case, if you take the meaning of the connective to be
*solely* its logical function, then there is no occasion for anyone ever to utter as a statement of fact the proposition using the
connective in this way. So *factually,* the inference is *not* sound.

And what I will conclude from this is that, even if logic as defined in contemporary terms is internally consistent, **it has
no application to statements of fact, **because as statements of fact, its compound propositions (including the statements
of its inferences) are "statements" that no one could have any reason for uttering in the sense contemporary logic intends
them.

Let me here define what I mean by the logical function and the meaning of a connective:

** The logical function of a connective combining statements (or propositions) is the indication of what is to be done
with the statements connected.**

** The meaning of a connective is how the facts stated by the statements are interrelated.**

For various reasons, some logicians who still hold that logic deals with the world "out there," like Bertrand Russell, for
instance, have problems with "connected facts." But since, if you refer back to Chapter 6 of Section 5 of the first part 1.5.6(not to mention what leads up to it), for me a fact *is* a connection among objects (spelled out in terms of knowledge a little more
in Section 3 of the third part), then I am not going to bother with trying to establish that there can be "factual
interrelations."

I gave one example of the difference logical function and meaning with "and," which I will discuss more at length below;
but just to be clear about it, let me say that the logical function says that each component must be affirmed (i.e. accepted as
stating a fact), and the meaning *adds* to this that the two are somehow connected. To take another example, the statement,
"If Chicago is in Illinois, then I am getting gray" illustrates the connective called the "implication." You are obviously
bright enough to see pretty clearly how "if...then..." functions logically as a connective; but the reason why the statement
sounds strange is of course that beyond this logical function, the connective also means "the second statement's being a
fact *depends* in some way on the fact expressed first." Clearly, there is no dependence in the example.

My position is that the logical function of a connective is *not divorced from* its meaning, but *included within it,* so that if
the logical function is violated, the connective is wrongly used (is false). But since the meaning goes beyond mere logical
function, then the connective can be false and still used properly in its logical function.

Further, I contend that the logical function is *derived from* the meaning (i.e. depends on it) and is not just an adjunct to
it;^{(1)} that is, it is *because facts have certain interrelations* that statements have certain connections and not others, and you
can't just stick in any connective you want at any given time and still hope to be describing the real world. For instance, the
implication in statements occurs because effects really depend on causes for their existence, and we know this. That effects
*really* depend on causes is the whole point of the first part of this book from Section 2 on; that we know this is the burden
of Section 5 of that part and Section 3 of the third part. If there were not a connective such as "if...then...," we would have
to invent one.

So what is at the base of my problem with contemporary logic is its epistemological stance that says that you can refer to
the real world *without taking into account the meaning* of the connectives--or even worse, that the language is simply
self-contained, referring to nothing outside itself, in which case to use it to critique the *logic* of what anyone else says is
like criticizing a statement in French which happens to use words that look like English on the grounds that it doesn't make
sense in English.

Of course, by that token, I would not be "allowed" to criticize what is said in contemporary logic because it doesn't make
sense in my logical system. But that's only forbidden for a person who buys into the idea that a system can't apply outside
itself, and I simply deny this for the same reason that I deny relativism, as I said in Section 1 of the first part. For a person
*within* a self-contained system to issue a "rule" that criticism of his system from outside is invalid or illegitimate obviously
contradicts the self-containedness he demands for his system (because he's criticizing some system outside his).

My contention is that there is *a* logic of *statements,* which may or may not be very complex and only approached by any
known system of formal logic; but formal logic is *an attempt to discover and formulate* this logic. Hence some logics are
better than others because they more accurately express more of how we in fact reason when we connect the expressions of
our acts of understanding to generate what we realize are new *relationships between objects* from old ones.

If someone disagrees with this and

Should say,"That is not what I meant at all.

That is not it, at all."

my answer will be, I dare to eat a peach. Let her go her way, like the skeptics and the relativists of Section 1 of the first part. "And turning to [the reader] he said, 'Do you want to go away too?'"

This is not to say that I find contemporary logic inapplicable. As I said, the logical function of connectives is contained
within their meaning; and so *insofar* as the connections between what is said depend on the logical function of the
connectives, that version of logic will apply to it, and since anything connected depends *at least* on the logical function of
the connective, then what *violates* contemporary logic (what is *in*valid in it) will be invalid for statements also; but there
will be things that are *allowed* in contemporary logic that are fallacies in statement logic. Hence, contemporary logic can
be safely used for refutational purposes only.

Because contemporary logic doesn't really tell you what to do with statements, I will give rules on the permitted and forbidden logical operations based on the meaning and function of the connectives in question. This is very close to Aristotelian logic.

Now then, contemporary logic's criterion for validity above needs some explaining, and in order to do so, I have to give you the truth table of the conditional proposition with its components. I will discuss the conditional proposition later; but for now, its truth table will allow me both to illustrate what a truth table is and show how it works.

The first thing to note is that contemporary logic uses the letters "p" and following to indicate whole propositions in any form; and since we're not now interested in subjects and predicates, as I said, we can do this also. One convention here is that if the same letter appears twice, it stands for the same proposition both times. So "p" means "any old proposition," and "q" means any other proposition you please."

I am not, however, going to use contemporary logic's dots, V's, slashes, and horseshoes and so on that symbolize the connectives, because they make the whole thing terribly confusing to look at; and in a matter like this, unnecessary confusion is something you want to avoid if at all possible. So I will, as above, use the names of the connectives.

Now if you look at the T's and the F's in the first two columns, you will see that they exhaust all the possible combinations
of affirmation and denial there are with two propositions. If there were three, there would be eight lines in the truth table;
but as I said, we are only interested in the basic ideas, so we will stick with two propositions. The T's and F's in the third
column are what the compound proposition turns out to be based on the logical function of the connective and the T's and
F's on the corresponding line of the components' columns.^{(2)} Thus, the first line says that when "p" and "q" are both true,
the compound proposition "p implies q" is also true. For instance, the compound "If dogs are mammals, then dogs are
animals" is true, given that both "dogs are mammals" is true and "dogs are animals" is true.

Don't confuse reading a line of the truth table with an *inference,* however; the truth table is just what you might call the
"logical sense" of the compound proposition: a kind of "truth-definition" of it; it defines its truth-value based on the *logical
function* of the connective, though not its *meaning.* This obviously has to be the case, since logic *defines* the meaning of
the connective to be nothing but its logical function. We will see more of this distinction as time goes on.

But to return to the truth table of the conditional proposition, the inference above about Chicago would be expressed like this, with "Chicago is in Illinois" being "p" and "I am getting gray" "q."

[(p implies q) and p] implies q

The difference between the inference and this proposition is that "p" as an *affirmative proposition* is not *the affirmation of*
"p," but simply a "proposal of 'p,'" one that is "proposed for the sake of argument." But it can be in reality false (and can be
known to be false. That's what the truth tables are for). Similarly, a negative proposition is not a denial, because it is
"proposed as" true and can be denied. Since an inference proceeds by way of affirmations and denials, this is simply the
*expression* of an inference, which can be false in the various ways in which statements can be false, as we saw in Chapter 5
of Section 3 of the third part 3.3.5.

But a statement such as this expresses a valid inference (in contemporary logic) when as a complete statement, it is true all the time, no matter whether the component statements are true or false in themselves. That is, when the connective expressing the main verb (in this case, the "implies" on the right-hand side) is true all the time, no matter what p and q are themselves, then the inference is valid. This is what contemporary logic means by "a tautology."

The way you establish the validity of the inference is this:

First, knowing the truth table for "p implies q," you substitute the last column of *that* truth table for our the column that
represents the parentheses, and at this stage we have

Now we have to look at the truth table for "p and q" (given below under the discussion of "and") to get the next stage; and that gives us (ignore the Ts and Fs in the square brackets for the moment):

*Note that you can't read this table from left to right. You have to read first what has no parentheses or brackets around it,
then what has parentheses, and lastly what has square brackets, then what is in braces.*

since "and" is only true when both components are true. Now we're ready for the last stage, expressed by the Ts and Fs in the square brackets. The column under "and" now is our new "p" and by the truth table for "p implies q" we see that the column for the last "implies" (the letters in square brackets) is all Ts, since it is T when the "p" is true and the "q" is true, and T when "p" is false no matter what "q" is.

Therefore, that inference is, as I said above, a tautology, or is valid, according to the contemporary criterion of validity.

With that out of the way, then, let us go to the first of our connectives, which is called a *conjunction, of propositions,* and
simply asserts the fact that the propositions are connected:^{(3)}

** The logical function of "and" is that each of the component propositions is to be affirmed.**

** The meaning of "and" is that the two facts affirmed are connected somehow; but it does not specify what the
connection is.**

In the logic of statements, this is trivial. It is obvious from the logical function that if the compound proposition with
"and" is to be affirmed (and why would you state it as a fact if you weren't affirming it?), then the only thing you can do is
affirm each part. You can't deny either one, and the affirmation of one doesn't *imply* the affirmation of the other (the
affirmation of the *compound* simply *affirms* both already).

The reason is that you can't affirm it unless you already know the truth of both parts, and so you would already *explicitly*
know the "conclusion" before you drew it. So it isn't *reasoning* to say, "John is tall and John is strong; and John is tall;
*therefore,* John is strong." The second "premise" is a waste of time, and the conclusion doesn't *follow,* because it was
already stated in the compound proposition itself. And if you say, "John is tall and John is strong and John is not tall,"
you've already contradicted yourself explicitly.

In contemporary logic, this is how the truth table for "and" looks:

Now as I said with the conditional proposition above, this is just an assigning of truth values to "p and q" based on the logical function of the connective, and it is not an inference.

You can, however, make inferences based on it, even though when you see them translated back from symbols into sentences, they look silly. We saw one of them above, ("John is tall and John is strong, and John is tall implies John is strong.") which had the form [(p and q) and p] implies q. When all you see is the letters, this looks like an inference.

The truth table check on the proposition looks like what is below. Here, to save space, I have introduced a convention. Again, what is not enclosed in any kind of bracket is the initial stage of "p" and "q"; the result of the second stage is in parentheses; the result of the third in square brackets, and that of the fourth in braces (if there were more steps, they would be in double parentheses, double brackets, and so on).

Since again the last step is all T's, the inference is once again valid. But as I said, no one would ever have any occasion to perform such an "inference."

If, however, we try to reason the other way, from the components to the compound, this is what we get, indicating the first compound by what is in the parentheses, and the final step by what is in the brackets:

And here we run into the heart of my difficulty. For instance, why is the statement, "George Bush is in the White House
and my feet hurt" funny? Because it connects two statements as if the *facts* were connected; and the humorist expects
people to recognize that *the facts are not in fact connected.*

First note, however, that that's not quite what is being said in the conditional proposition above. That proposition actually
says "*If* (George Bush is in the White House) and (my feet hurt), *then* George Bush is in the White House and my feet
hurt." Well of course. But in making this into one sentence, you have to *make the conjunction* in the antecedent (the "if"
clause), from which the consequent (the "then" clause) trivially follows. In other words, you have turned the inference into
the form "p implies p."

But that isn't what the *statements* say. It is *invalid* to argue from a true *statement* and any other true *statement* to a
*conjunction* of the two statements, because they might be conjoined or they might be totally unconnected. Putting the
inference into a conditional proposition in contemporary logic can't spot that fallacy--and, indeed in contemporary *formal*
logic, it *isn't* a fallacy, and the argument is *sound,* which means that the conclusion is *true.* Of course, it's true that any two
true statements always *can* be conjoined (if under no other guise than that they're both examples of true statements); but
the conjunction may or may not express and actual connection of some sort among the facts, or the statement about the
President wouldn't be funny. Consequently, *it does not follow* in the logic of *statements* that the conjunction must be true
when both of the components are true.

*Contemporary logic's "and" is a weak "and," which does not say that both components (together) express something that
is true, but merely that each is true. That is, it does not say that the two propositions are connected.*

The question I raised above now arises of whether there could ever be a reason for using the connective in the sense defined by contemporary logic.

I don't see how there could be, because of the fact that the two propositions are *connected* into one sentence (one
compound proposition); and it is *bound to be misleading* to connect two things which *explicitly are not to be taken as
connected.*

Now you can *say* if you want, "When *I* connect things, they might be connected and they might not, and so you aren't to
understand them as connected." My answer would be, "If you don't want me to understand them as connected, don't
connect them." Instead of saying "p *and* q" as one proposition, state two distinct propositions: "p" period. "q" period. That
leaves it open as to whether the two are connected or not, since it precisely says nothing about it. It's certainly possible to
do this rather than redefine "and" to be something that no one else would ever use.

In other words, the *very act of connecting the two propositions into one* contradicts the "definition" in contemporary logic
of the connective as something-that-does-not-express-a-connection. You can say, of course, that no facts *are* connected,
and so "and" can't mean anything but the convenience of getting the propositions in convenient shape to be worked on; but
that's the epistemological stance I think is simply silly, or statements like the one about the President wouldn't be funny.
Such statements *recognize* that some statements express connections between the facts represented and some statements
don't. Hence, "and" means something.

Logicians preen themselves on being unambiguous and on saying no more and just precisely what they mean. But I don't
personally see how you can avoid ambiguity when you connect things that may not be connected. Better to reserve "and"
for propositions that express what *is* somehow connected; then you leave no ambiguity in what you are doing. The
logicians would object that they want all the propositions in an inference to be expressed as one single proposition. Very
well, then the ambiguity can't be escaped; but don't claim that you're being unambiguous.

So much for my first argument against the validity of contemporary logic as a system. If we now look at the next connection, said to be a a form of "or," but which I call "is incompatible with" in its clearest formulation, my problem with contemporary logic will be a little clearer.

** The logical function of "is incompatible with" is that at least one of the components must be denied.**

** The meaning of "is incompatible with" is that the facts stated in the components are incompatible with one another.**

This connective is not logically trivial, because all you know by affirming the compound is that *one or the other, and
possibly both* of the components must be denied (is false), but you don't know which one. And as a statement it is not
trivial either, because what you are asserting by the compound is *the fact of incompatibility between the components,* not
necessarily any knowledge of the factuality of either of them.

Generally in common speech, this connection is stated negatively; either as an impossibility, as in "you can't have your cake and eat it," or more often of the form "not p when q" as in "The cat is not outside when it is raining." Or possibly the statements are given as gerundives connected with "is incompatible with" as in "The cat's being outside is incompatible with its being rainy." Note that the second proposition looks as if its first part is a denial; but the "when" shows that the denial belongs to the whole statement. It means "It is not the case that the cat is outside when it is raining," or "It is not, as a general rule, (the force of the "when," as we will see below) simultaneously true that the cat is outside and it is raining."

Here what I am going to do is say what I think is wrong with contemporary logic's approach to the proposition, and then afterwards list the valid inferences that can be made from it. Once again, I think that contemporary logic's ignoring of the meaning of the connective allows it to make "valid" inferences that are fallacies when taken as statements. Let us look at the truth table:

which is just the opposite of "and," you'll notice; and in fact, it is the logical equivalent of "not (p and q)." And here is the
problem. The *statement,* as I will try to show, is *not* merely the denial of a conjunction.

Observe that, if you *affirm* both components, you necessarily *have* to deny the connection and so there is a legitimate
inference this way. For example, if you say, "The cat is not outside when it's raining," you can *prove the connection
inappropriate* by showing an instance when the cat is outside and it is raining (i.e. by affirming both). So this inference
works in both contemporary and statement logic.

But the difficulty with contemporary logic, as I said, is not in its refutational use, but in its use in an affirmative sense; and the reason is below:

** The meaning of "is incompatible with" in contemporary logic is a weak "is incompatible with," which simply denies
that both components are true, but says nothing about whether they are incompatible with each other, but simply that
one or the other or each is false.**

Now it might seem that I've loaded the dice here, because what I call "is incompatible with," contemporary logic (when it
uses this connective at all, which is very seldom) simply calls it "not both." But what I am going to try to show is that to
take the compound in the sense of "The two don't happen simultaneously to be true" produces a statement that there would
be no reason for making.^{(4)}

To begin, then, the meaning of the proposition as contemporary logic would have it could not be expressed as "The cat is
not in the house *when* it's raining", because the "when"makes it a *general* statement (i.e. of what is *always* the case), and so
rules out the statement as *merely* a simple statement of what is going on now. As a simple statement of what's going on at
present (a negative conjunction), it would be stated, "It isn't simultaneously true *at the moment *that the cat is outside and
it's raining." Here, all you would be intending to state is just that the two *happen* not to be the case.

But could you make a statement "It is not simultaneously true that p and q" as a simple statement of fact, totally
unconnected with any general rule? It would be difficult to imagine an occasion for it. First of all, in this case, how could
you know whether it was true as a whole or not without knowing anything of the truth or falseness of at least one of the
components? That is, how could you possibly *assert* as now the case that it's not simultaneously true that the cat is outside
and it's raining without knowing whether either of these were true or not? So you can't make it without knowing something
about the components.

Secondly, if you know one of them is *true*, you still can't assert that the *conjunction* is false (even contemporary logic says
this), because, for all you know, the other one might be true, making your proposition false. That is, if you know that the
cat is outside and that's all, you don't know that its false that the cat is outside *and* it's raining--unless, of course you knew
the general rule that the two are incompatible. But we're not talking now about incompatibility, but simple statements of
fact.

Thirdly, if you already knew that one component was false, why would you conjoin *the opposite of* this false statement
with any other proposition? That is, if all you know is that it's not raining, why would you then say, "It isn't simultaneously
true that it's raining and Lincoln is in the White House.")? Here again we have the problem of connecting two propositions
into a single compound proposition *with absolutely reason to connect them.* The only "grounds" you could give is that you
happen to know that the opposite of one of the components is true; but those "grounds" are *exactly* as good for connecting,
"It's not simultaneously true at the moment that there's life on Mars and the temperature in Miami is ninety degrees."

Fourthly, if you know that one is true and the other false, you would be giving misleading information. The reason is that
if you're trying to convey to someone what *is* the case, and you know that the cat is outside and it's not raining, then to say
"It's not simultaneously true that the cat's outside and it's raining," conveys the information that both *might* be false, which
is impossible as a statement of the present situation, because one is true.

Finally, if you know that both are false, and you want to tell someone what the present state of affairs is, you would say,
"The cat is not outside and it's not raining," denying *each* of them, not denying the conjunction, because that also conveys
that one of them *might* be true, when in fact, neither of them *can* be true because they're both false.

So I submit, therefore, that it is unreasonable to make an "is incompatible with" statement of fact as a mere statement of what is at present the case. If you have no information at all, then you can't make the statement; if you know the truth of one, you can't do it either, because it doesn't follow; if you know the falseness of one, then the statement you make has no connection with the information you have; if you know the falseness of one and the truth of the other, you're conveying the false information that both might be false; and if you know that both are false, you're conveying the false information that one might be true.

This is not to say that the statement might not sometimes be true, as in the case of knowing that one is false and the other
true; but in that case it is unjustified, which means it is made capriciously. It is also possible that the context could be
peculiar enough so that the misleading information in the last two instances would be removed (as, for instance if you
actually gave the information you knew first); but in that case, it would be superfluous, because you would previously have
given *more* information than you give by the statement.

Let me just illustrate this last case. You could say, "The cat is not outside and it's raining out, and it's not simultaneously
true at the moment that the cat is outside and it's raining out." But why would you ever say a thing like that? Even if you
said, "The cat is not outside and it's raining out, and so (i.e. implies) it's false at the moment that the cat is outside and it's
raining out," that's just as bad. In neither of these two cases have you conveyed any more information by the second part of
the statement. Everyone would recognize that what you said was true, but making the statement would be completely
redundant. So *either* such a statement is redundantly repetitious pleonastic superfluity, *or* it is misleading.

A word on ambiguity. Logicians, as I mentioned, like to think that their meaning of the connectives avoids ambiguity. But
the "is incompatible with" proposition is *precisely ambiguous.* That is, it leaves open three possibilities: "p" is false, "q" is
false, and both are false, but does not distinguish among them. Now to leave open three interpretations of the proposition
without picking out one is to leave the proposition ambiguous in its truth value. Granted, the connective is *precise* in its
logical function, because these are the possibilities and there are no more and no fewer; but it isn't speaking precisely (or
unambiguously for that matter) to confuse precision with unambiguity.

So much for that. Now could the "is incompatible with" (in the weak sense of "not in fact both") statement be made as a statement of what has frequently been the case, without implying any incompatibility between the two statements and without grossly misleading your hearer? (Contemporary logicians tend to say, remember, that their way of speaking is the way we "ought" to speak to make ourselves clear.)

That is, can "The cat is not outside when it's raining" convey, "I've never so far seen the cat outside when it's been raining
(but it might happen tomorrow)." In that case, *the use of the present tense* is what is misleading. If all you are trying to
convey is something that so far has invariably happened *and not* that there are grounds for *predicting* anything from this,
then *the present perfect tense must be used.* The present tense is used *only* for present states of affairs or general statements
that occur irrespective of time (and so would also occur, presumably, in the future).

For example, the simple denial as having happened invariably but without predictive implications might be spoken by,
say, the President's wife: "I've never been in this room when George Bush has been in it." Clearly, if she said this, she
would not be intending to convey any hint of what might happen five minutes from now. But if she said, "I'm not in this
room when George Bush is in it," this would be a general statement, and so it would be taken as having a predictive value
also, as implying that for some reason she is not permitted in the room when the President is there (or that she refuses to be
in it when he's there). So to say, that "is incompatible with" means that "it *is* not the case that p and q" *cannot* be taken to
mean, "It has so far not been the case that p and q."

Of course, if you want to adopt Hume's criterion of causality, then of course you could say that *no* general proposition
(except a tautology) has any predictive force, and all are simply summations of what has been observed so far. But I fail to
see how you could make such a general proposition, because it's a general proposition which is not a tautology. In that
case, if it is true, it would only apply to the ones you have seen so far, and would have no bearing on any other one. And
again, your statement of it would be misleading, because all you meant to say was, "So far, I haven't run across any
non-tautological general proposition that has any predictive force, but the next one might be one," like Mrs. Bush's
past-tense statement above. But then why say *no* general statement allows you to know anything beyond what is observed?
You certainly mislead people into thinking that *this* general statement applies beyond the ones *you've* seen.

So to say that the "is incompatible with" statement merely means that so far the two parts have not happened to be
conjoined is to convey by the use of the present tense that they are incompatible, when you don't want to assert that.^{(5)}

Hence, the only way you can be *clear* in what you are trying to say with an "is incompatible with" statement is that it
asserts what you think is the *fact of incompatibility* between the two components. You are not asserting the *grounds* you
have for this, but merely what you consider the fact. Hence, you may know that your cat hates to get wet, and so you say,
"The cat is not outside when it's raining." All your hearer knows is that you are asserting that it is *impossible* for both
components to be true--not that you are asserting that they are not both *in fact* true at the moment.

Where are we, then?

**Conclusion 2: The weak "is incompatible with" statement of contemporary logic has for practical purposes no
occasion to be made as a statement.**

Let me, then, give the rules for the "is incompatible with" compound, given that it expresses the incompatibility of the components:

*Rules for "is incompatible with"*

*Inferential mode:*

** 1. If the compound and one of the components is affirmed it follows that the other must be denied.**

** 2. If the compound and one of the components is denied, no conclusion follows.**

*Refutational mode*

** 3. If both of the components are affirmed, the compound must be denied. This refutes the connection.**

** 4. If one of the components is denied, nothing follows with respect to the compound.**

The next connective is the contrary of "is incompatible with," and is sometimes called the "inclusive or"; it is usually stated "and/or" in informal speech; essentially, it is "not neither."

*The logical function of "and/or" is that at least one of the component propositions must be affirmed.*

* The meaning of "and/or" is that the possibilities referred to are connected in such a way that one of them is in fact
realized, though *which* is realized is not expressed by the statement.*

This will obviously take a little clarifying. First, in ordinary use of this connective, the compound statement can also be stated "One or the other or both," to distinguish it from the disjunction, which we will see after this. The word "or" in English is ambiguous, since it can mean "one or the other" or "one or the other or both," and so clear speakers and writers use "Either...or" and "and/or" (or "(Either)...or...or both") when there is a danger that the context will not distinguish the two.

For instance, a person might say, in reference to some scandal "Either there's something wrong with the corporate structure, or management is corrupt, or both," or "that cat is clever or lucky or both."

What this connection actually asserts as a fact is *the necessity of at least one* of the components, usually because they are
assumed to be *an exhaustive list* of the *explanations* of some affected object (which, if you will recall from Chapter 1 of
Section 2 of the first part 1.2.1, is a contradiction by itself, but which as concrete can have a complicated causer). Explanations
do not necessarily exclude each other (as, for example, the scandal in the corporation might be partly due to a faulty
corporate structure and partly to corrupt management); but there has to *be* at least one; and if you list them all then they
can't all be eliminated.

The proposition is *refuted* by denying both components, because of the fact that the components listed (which may be
more than two, of course) is asserted to be exhaustive. But like the compounds we have seen already, it is not *confirmed* by
affirming one of the components, or even both of them, because there might be another item to the list not taken into
account. For instance, the clever and/or lucky cat above, in order to escape the dangers that occasioned the remark, might
be being watched over by its owner, in which case it might be true that it's neither clever not lucky, but just loved. Or, of
course, it could be all three. So, even though the cat's being clever is *consistent with* "That cat is either clever or lucky or
both," it doesn't *prove* that the statement *has* to be true.

I suppose I should point out here that in contemporary logic, there is a *valid* inference from "p" to "p and/or q," which
suffers from the *informal* flaw that something appears in the conclusion which was not in the premise. Formally speaking,
the argument "The cat is clever. Therefore, the cat is either clever or lucky or both" is sound if the cat is clever (because if
it's clever, obviously it's clever, making the "and/or" proposition true by default). But the compound proposition as a
compound is then irrelevant to the argument, and so even in contemporary logic, it doesn't belong there, but by the
informal fallacy of irrelevance.

Here is the truth table for this compound:

And once again, this says that what contemporary logic means by "and/or" is not and/or, but *a denial that both are false,*
which can be a mere statement of fact. That is, if "p" is true, it is obviously false that both "p" and "q" are false; and that is
what the "inference" above has to mean.

** The "and/or" of contemporary logic is a weak "and/or" which simply means that one proposition is true, and says
nothing about whether one has to be true or not.**

We must again discuss whether we can ever sensibly make such an "and/or" as a statement of fact. Clearly, with no
information about either component, it can't be asserted. If all that is known is that one component is *false,* it doesn't follow
that the compound is true, because the other proposition could be false, making the compound possibly false. If all that is
known is that one proposition is *true,* then this does not constitute sufficient grounds for connecting it with the other
proposition, because *any* other proposition, true or false, would on these grounds fit just as well. Why would you, knowing
that George Bush is President, convey information to someone by saying, "George Bush is President, and/or there is life on
Mars"? It's true, but you have no reason for saying it. If you know that one is true and the other false, you are misleading
your hearer into thinking that both might be true when they can't be--as a mere statement of fact, because what is in fact
false can't be true if it's false. And similarly, if you know that both are true you would be conveying the false information
that one might be false when it can't be.

This also is subject to the same sort of qualifications as with the "is incompatible with" statement as a mere statement of
fact. It *could* be said, and it would not be false, but it would be either misleading or capricious to say it. And with that, we
can draw the following conclusion.

**Conclusion 3: There is for practical purposes no occasion where contemporary logic's "and/or" could be uttered as
a mere statement of fact.**

But if "and/or" means that at least one component *must* be true, then it would be self-contradictory if the list of
possibilities wasn't exhaustive, because you would then be asserting that one must be true when both could be in fact false.

Here again we have a *logical* aspect of statements that is not covered by contemporary logic, which does not recognize
"That's not all the alternatives" as a denial--as in fact it is in every case of the use of the "and/or" proposition. But of
course, that refutation doesn't involve anything *within logic,* which is what contemporary logic wanted to avoid. *But in that
case, what it should have said is that no conclusion can be drawn from knowing the truth of one component,* not construct
the logical system in such a way that the conclusion is valid by making up this "weak" sense of "and/or" which never has
been used and never will be. Why not rule out the "formally valid but not necessarily always the case" with its *ambiguous*
use of "true," by stating a rule that the statement is meaningless as a mere statement of fact and is to be used when there is
an exhaustive list of possibilities? Then the logical function would be allowed to do its work properly.

And this is precisely what the rules below do. Recognizing that "and/or" as used as a statement implies the necessity of one component's being true, here are the logical things you can do with it:

*Rules for "and/or"*

*Inferential mode:*

** 1. If the compound and one of the components are affirmed, no conclusion follows.**

** 2. If the compound is affirmed and one of the components is denied, it follows that the other must be affirmed.**

*Refutational mode:*

** 3. If one or both components are affirmed, nothing follows with respect to the compound.**

** 4. If both of the components are denied, the compound must be denied. This refutes the connection.**

The next connection, "either/or," is given the name "the exclusive 'or'" in contemporary logic; and in Aristotelian logic, the inference made from it is called the "disjunctive syllogism," because it is a more common way of reasoning than either of the two we have discussed. Actually, the commonest fallacy dealing with both "is incompatible with" and "and/or" is that (since they can be stated using simply "or") they are apt to be confused with this one (while in fact, "either/or" is another way of saying "not both and not neither").

** The logical function of "either/or" is that one of the components must be affirmed and the other one denied.**

* The meaning of "either/or" is that the two facts referred to contradict each other.*

Here is the truth table for the proposition in contemporary logic:

Once again (and I will again leave you to take my word for it or do it out yourself) we have a case of the fact that you can conclude to a denial of the compound (and so a refutation of the connection) by either affirming both or denying both of the components.

But you can't confirm *the fact of* the compound by affirming one component and denying the other (because the actual fact
might be either a not-both or a not-neither compound, both of which are compatible with one component's being true and
the other false). Thus, for example, "Either you're in New York or you're in Chicago" can't be established by saying that
you are in fact in New York and not in Chicago--because clearly the proposition is actually an "is incompatible with"
proposition that's disguised by the use of the wrong connective (i.e. as stated, it would be refuted if you were in
Cincinnati).

Note, however, that doing out the truth table check will show you that the proposition "p and not q implies either p or q" is a valid inference in contemporary logic. Here there is not, as in "and/or," a rule of irrelevance to eliminate this fallacy. In contemporary logic, it is valid, and if the premises are true, sound.

** Either/or as used in contemporary logic is a weak "either/or" which simply asserts that one component is true and the
other false without saying anything about whether this has to be the case or not.**

There is again something funny going on, which is masked in the example, because being in New York and Chicago are
incompatible. But if we take two propositions that are compatible but don't have to be simultaneously the case, we can see
how strange the meaning of the connective in contemporary logic is. It would be odd to say, "You are healthy and you are
not six feet tall" *implies* that you are *either* healthy or six feet tall, because "you are either healthy or six feet tall" (the
"exclusive" use of "or") *seems* to be saying that you *can't* be both. Well of course you can't be both if you're not in fact six
feet tall; but to offer this is to use "can't" in two senses. One means "being healthy is in itself incompatible with being six
feet tall," while the other means "both are not the case, and so at the moment they happen to be incompatible." But if you
want to say simply "One and not the other" why not say that instead of saying "Either one or the other"? That is, why open
up the ambiguity by using "Either/or" when "p and not q" would be clear?

Here again, logic is *not* being unambiguous. The "exclusive or" in contemporary logic does *not* exclude either "is
incompatible with" or "and/or" in two out of the four cases in its truth table. Hence, the following proposition expresses a
valid inference:

p and not q implies (p is incompatible with q) and (p and/or q) and (either p or q).

If we fill in the p's and q's, we get, "The fact that you're healthy and not six feet tall implies simultaneously that you're healthy when you're not six feet tall, you're healthy and/or six feet tall, and you're either healthy or six feet tall." That sounds really peculiar, but you can make sense out of it if you say, "If you're healthy and you're not six feet tall, then you're not both healthy and six feet tall, you're not neither healthy nor six feet tall, and you're one or the other."

But of course, when you put it this way, it is trivial as a statement. If you put it the way it was originally stated, then even
ignoring the impression someone might get from the first conclusion that you're trying to say that you're healthy *only* when
you're six feet tall (i.e. that your being healthy is incompatible with your being six feet tall), why would you say "you're
healthy *and/or* six feet tall, *and* you're *either* healthy or six feet tall"?

That is, "and/or" includes the *possibility* of both, while "either/or" *excludes* that possibility. And here is where the two
senses of "can't" mentioned above come in. In contemporary logic, "and/or" is true in this proposition because *in itself* it is
possible for both to be true, but "either/or" is true, because *as it happens* they can't both be true because one is in fact false.
And then there's the fact that "and" in logic doesn't mean "and."

So you have to be very, very careful if you're going to apply contemporary logic to statements. They don't mean what they seem to mean.

**Warning: When contemporary logic talks about something being "impossible," this can mean simply that it is not
the case.**

This is the celebrated unambiguity of contemporary logic?

Further, as this example shows, to call "either/or" the *exclusive* "or" is to give the impression that it is *incompatible* with
"is incompatible with" (in which both can be true) and also with "and/or," (which, after all, is called the "inclusive or").
But it isn't; it's *inclusive* of both of them in the sense in which the intersection of a set is inclusive of the two sets that
intersect in it (i.e. it is the set that includes part of each of them).

And once again the problem is in the fact that contemporary logic does not recognize the additional information in the
meaning of the connective. "*Either*/or" is *in fact* used *only* when the two propositions in question *contradict* each other, or
only when you have grounds for saying that one *has* to be true and the other false, not merely when one *happens* to be true
and the other *happens* to be false. If you say, "Well, it *can* be used simply as meaning "one and not the other," then I say
that contemporary logic's goal to be clearer than ordinary speech has been violated. If you're stating it as a mere fact and
not based on some internal contradiction, then you have to know *which* one is true and which one is false, and so why do
you make the ambiguous statement, "One or the other of these is true and the other one is false" instead of telling what you
know? Or why not use "and/or," if you want to leave open the possibility that both can be true?

I'll tell you this much. There are plenty of contemporary logicians who are not clear about the distinction I have been making, and who think that it's English that conveys inexact information and logic that always means exactly what it says. If it does, it certainly doesn't convey to the unsuspecting what it's saying.

Let's face it; in any rational system of logic, it is *improper* to use either/or when both *can* be true, even though one in fact
happens to be true; that is "and/or," not "either/or," no matter how much you may be able to justify it using that etiolated
sense of "can't." And the same goes for using "either/or" as in the New York and Chicago statement above, where the
proper connective is "is incompatible with."

From this it follows that reasoning to "either/or" from "p and not q" *ought* to be forbidden. And that, in fact, is what
traditional logic has done for millennia. If it is a convention to forbid it, then that convention is closer to the way
statements are made (in Greek, Latin, English, Chinese, and Swahili) than contemporary logic's convention of taking
"either/or" to mean no more than "one and not the other."

Then let me list the rules for how the disjunctive syllogism actually works:

*Rules for the disjunctive syllogism*

*Inferential mode*

** 1. If the compound and one of the components is affirmed, it follows that the other must be denied.**

** 2. If the compound is affirmed and one of the components is denied, it follows that the other must be affirmed. **

*Refutational mode*

** 3. If both of the components are affirmed, it it follows that the compound must be denied. This refutes the connection.**

** 4. If both of the components are denied, it follows that the compound must be denied. This refutes the connection.**

** 5. If one component is affirmed and the other denied, nothing follows with respect to the compound.**

The final basic way logic combines propositions is the one we saw proleptically, called the "implication," or the
"conditional proposition." or in traditional Aristotelian logic, the "hypothetical syllogism" (from *hypo-thesis,* a "putting
under" or "supposition," because "q" "supposes" "p" "underneath" its intelligibility somehow, as an effect "supposes" its
cause. It is stated either "if p then q" or "p implies q"; it is the general form of the inference, according to contemporary
logicians, although it is not as they use it itself an inference because in order to *make* an inference you have to affirm or
deny the p's and q's; but as the form of the inference, it's supposed to reflect the kind of thing you're doing when you make
an inference.

Well it doesn't, as I think has been made clear. But before I discuss it further, let me state the function and meaning of the compound:

** The logical function of "if then" is that an affirmation of the antecedent (the contents of the "if" clause) demands an
affirmation of the consequent (the contents of the "then" clause), and a denial of the consequent demands a denial of
the antecedent.**

** The meaning of "if then" is that the consequent depends somehow on the antecedent. **

It is here that contemporary logic really takes flight into never-never land. Since contemporary logic wants to have
nothing to do with things like question marks and wants its truth tables filled with either Ts or Fs, then this proposition
elevates the *non sequitur* into a legitimate implication.

I already gave the truth table for this type of proposition when I introduced this section on compounding propositions, so I refer you back there if you want to look at it.

Now it is true that, since formal logic deals with the form under which statements go together, the contents of the
statements don't enter into it. There was no problem with this in Aristotelian logic, because it always used the inferential
mode of reasoning ("sophistical refutations" was an area that wasn't strictly formal); and in the inferential mode you
always first *affirm* the compound and *then *argue to the components. Hence, if the compound is some *non sequitur* like, "If
I schedule an outdoor party, then it rains," it doesn't matter; because, *assuming that compound statement to be true,* then
you can make it rain by scheduling a party, or you can guarantee that there'll be no party by noticing that the day is sunny.

But in the real world, if you're going to argue from the truth of the components to the truth of the *compound,* then the
meaning of the connective can't be ignored; because the truth of the compound implication *as a factual statement* depends
on whether the consequent in fact depends on the antecedent. That is, what you would be saying with respect to the
compound above is that scheduling a party and having it rain *proves* that scheduling a party *implies* having it rain, which
of course is ridiculous. That's the first problem, and it's the same one as we had with other compounds as treated by
contemporary logic.

** The implication in contemporary logic's conditional proposition is a weak implication (called "material implication,")
because all it means is that it is false that simultaneously the antecedent is true and the consequent false, and says
nothing about whether the consequent follows from the antecedent.**

There has been a firestorm about this particular deviation of contemporary logic from the way we use statements, because
it is obvious that "implies" as a word *means more than* a denial of "p and not q."

I said that it is a valid inference in contemporary logic from "p and not q" to "p is incompatible with q, p and/or q, and
either p or q." But now (and you can work this out on the truth tables if you want), it is a *valid inference* from "not p and q"
to "p is incompatible with q, p and/or q, either p or q, and p implies q."

Come again? It's a *valid* inference from "*not p* and q" to "p implies q"? Knowing that the Cincinnati Reds are losing their
game today and that it's raining out *proves that* if the Reds are *winning,* then it's raining? (Yes. It also proves, by the way,
that if the Reds are winning, then it's *not* raining.)

I think you can see why this has caused something of a problem.

**Warning: Even though material implication uses the word "implies," it is compatible with the absolute
independence of the two components from one another.**

The *only* requirement for saying that "p implies q" is that the combination of "p's" happening to be true and "q's"
happening to be false is forbidden. Why? Because the inventors of this logic decreed that it is, not because of anything
about "p" and "q," certainly, and not because the connective "implies" *means* this. Everyone who has ever struggled with
contemporary logic has to spend a great deal of time erasing any meaning to "implies" that has anything to do with a
sequence or a dependence or anything else; and of course, once they do this, they think they've finally "mastered"
something very difficult, and they fight tooth and nail for how "powerful" material implication is and how much more
"accurate" it is than the messy way we talk, and all the rest of it.

Now "formal implication" (that is, "implication" in the sane sense of the word, expressing dependence) *also* makes it
impossible for "p" to be true and "q" false, so *some* of the invalid inferences in "formal implication" are also invalid in
material implication, and all of the invalid inferences in material implication are also invalid in formal implication. But
there certainly are valid inferences using material implication that are invalid by any rational standard of when something
follows. For instance, let us try to unpack the inference above about the Reds, to see if head or tail can be made of it in any
sense. First of all, I can eliminate some of the confusion by saying that the conclusion also follows from the mere fact of
"p's" being false (you don't also have to know that "q" is true).^{(6)}

So in this slightly easier form, what it says is "It's always true to say that if it's false that the Reds are winning, then it's
true that *if* the Reds are winning, then it's raining."

But that's still a little difficult; let's take a simpler example: If I am here, then I am at home. Now when it's *false* that I am
here, it is legitimate in contemporary logic to infer that if I am here I am at home--no matter where I actually happen to be.
(To avoid quibbles, I mean by "here" a certain address I could give you.) Granted, it *happens* to be true that here is my
home, so if I am here, I am at home. But it certainly doesn't *follow *that *if* I am in Chicago, then if I am here I am at home
(that is, that my being in Chicago establishes where my home is).

Now then, the first thing to remember is that this is a *negative* statement in contemporary logic, not an affirmative one. So
"If I am here, I am at home," is actually the statement, "'I am here and not at home' is false"; *this* is what is said to "be
implied" by my not being here. Now what *that* says is not an implication, but another negative statement; so the whole
thing becomes "'"I am not here" and "I am here and not at home is false" is false' is false." The last "is false" is the denial
which is the basic "implication." The next-to last is because is the denial of the "q" part of the basic implication (it is,
remember, not (p and *not q*). The third from last is the denial which forms the embedded "implication," and the "not at
home" is of course the denial of the "q" in that embedded implication.

But two of these last three "is false's" cancel each other out, so simplifying, we get (putting, for clarity, the last "is false"
first now), "It is false to say that "I am not here *and* I am here and not at home."

Well of *course* that's false, because it's false to say that I'm not here and I'm here, wherever my home is. So this
proposition is going to be false because hidden in it is "not p and p," not because anything depends on anything else.

**Warning: "Implications" which use material implication are really just negative propositions.**

You know, I don't think much of a logical system that says it is "making an argument" when in fact it is making a *denial.*
If *this* is what these people mean by "speaking precisely," and "saying exactly what you mean," I would hate to hear them
speak imprecisely and say what they mean inexactly.

The justification for taking "p implies q" to mean "not (p and not q)" rather than "q depends on p," (which only *implies*
"not (p and not q)") is that there are allegedly ambiguous uses of "p implies q" in ordinary speech, where sometimes causal
dependence is meant, sometimes rule-following dependence (as in logic), sometimes mere sequential dependence (as in
winter's implying spring to follow), and so on; and then there's the statement, "If you win this bet, I'll eat my hat," where
something absurd is made to "depend on" what the speaker considers a false statement. Obviously, that can't be
*dependence,* these people say; and so the "common core of meaning" is the negative proposition above.

But in the last case, that statement is a *premise* of an *enthymeme,* which the speaker thinks the hearer is too intelligent for
the speaker to have to flesh it out. For the benefit of our logicians, let us do so. What is *conveys* to anyone with any sense
is, "If you win this bet, I'll eat my hat, and I'm not going to eat my hat; and so it's impossible for you to win this bet."
Knowing that a false consequent *refutes* the antecedent, a proposition which someone wants emphatically to deny is made
to *imply* something known to be false. It is a rhetorical device, and the first proposition was not "proposed" as false, but as
*problematic* until the false "conclusion" was stated. So it isn't that "anything follows from a false statement"; it is the
perfectly legitimate reasoning that anything that *implies* a false statement has to be false. So that usage does not by any
means mean that in ordinary discourse we ever use "implies" without the notion of *some* kind of dependence.

Now even if every case of "p implies q" makes "not (p and not q)" true, it is a far cry from that to say that "p implies q"
*means* (or even "really means" or "ought to mean, if you want to be accurate") "not (p and not q)." This is about as
intelligent as saying that since every human being can talk, then what is "really meant" by "human being" is "talker."
Humans can do a lot besides talk.

Then why did the logicians get into that convoluted way of reasoning by turning the process into negative propositions that have negative propositions embedded inside them? The basic reason was that they didn't want to have compound propositions with anything but Ts and Fs in the truth tables, and in actual implications, you'd have to put a question mark in a couple of places.

But if you take this ploy, then to follow what is going on in the simplest inference then becomes an enormously tedious task involving negations of negative negations, trying to keep them straight as above.

Now of course, you don't have to *follow* an argument using contemporary logic. You can simply state it and then set up
your computer to do the truth tables and wait until it spits out the truth table for the final result. Then you know that your
conclusion is valid or not *by the standards of contemporary logic*; but of course, if it turns out to be valid, you still don't
know whether it's only "formally" valid and not--shall we say?--existentially so without going back over it and trying to
spot the "informal" fallacies.

Note that there is nothing that is *invalid* in contemporary logic that is not invalid in traditional Aristotelian logic, and so
nothing is gained by using contemporary logic to spot fallacies. But there are conclusions that are valid in contemporary
logic that are simply nonsense as statements of fact.

I will now consider that I have proved that contemporary logic is a waste of time; and this applies both to "propositional
logic" (where you don't care what the proposition looks like but just combine whole propositions) and "predicate logic,"
(where you care about subjects and predicates), because the definite proposition in contemporary "predicate logic" has the
*form of the implication* and so is infected with the disease of material implication.^{(7)}

But why did logicians develop this system? It was partly to get out of a Hegelian idealism, whose logic, you will remember, was the logic of contradictions, and was also "metaphysics" in his sense of the term. That was the main incentive for purging logic of all Hegel stuffed into it.

But it was also true that the inventors of the system (particularly Boole) were mathematicians, and, like mathematicians, they wanted to develop a system of logic that was "closed and complete." A system is closed if all the conclusions from premises within the system are still inside the system, and it is complete if every operation on meaningful statements within the system results in a meaningful statement. Thus, addition is closed and complete on the set of the natural numbers, because any number added to any other number yields a natural number (natural numbers are {1,2,3,...}). Subtraction is not closed on the set of the natural numbers, because 2 - 3 gives a result (-1) which is not a natural number. Division is closed over the rational numbers (the integers {...-2,-1,0,1,2...} plus all the fractions) but division is not closed over the rational numbers, because division by zero doesn't yield any result at all.

You'll notice that mathematics seeks closure by simply inventing numbers that fit; and we will see in the next section why
this is legitimate in mathematics. But it doesn't follow from the fact that mathematics can do this that it is legitimate to do
this in logic, if logic is supposed to either reflect or apply to the way we reason--and if it doesn't, it's mathematics, not
logic. Note, however, that mathematics *uses* logic (in the ordinary sense), and so presupposes it and is not the same as it.
That's one difficulty in trying to make a "mathematical logic." You are taking something that is a particular example of a
logical system and trying to use it as a *model* for the system it is only one particular example of. It wouldn't be surprising if
a mathematical logic would work in some cases (those in which the reasoning was similar to what is done in mathematics),
but not in others (those cases of logical reasoning that are not in mathematics).

But the real problem in taking mathematics as a model is trying to make logic complete, so that *every* logical operation on
something that is true or false results in something that is true or false. Propositions themselves can be true or false and
nothing else; but *conclusions* can be *true, false, or problematic,* because inference deals with the *necessary* truth or
falseness of the conclusion based on the truth or falseness of the premises and the type of reasoning involved. Hence, logic
as we actually reason *necessarily will be incomplete,* because, while the conclusion as a proposition has to be *either* true or
false, you can't *generate* one or the other always from the truth or falsity of the premises.

And if you try to make it complete by simply filling in the truth tables à la mathematics, arbitrarily declaring as "True"
certain things which should have question marks, then you get something which does *not* have an application outside itself,
and which (as in the case with the definite and indefinite propositions) has some glaring inconsistencies within itself.

It was a noble effort, but it was doomed to failure, because to model logic on mathematics is a classic case of "reasoning
from the particular to the universal," (or from the indefinite to the definite). Of course, there are a lot of contemporary
logicians who aren't going to swallow this, because they have Ph. D.s in the field of logic, and have studied it for years and
years, and can perform operations using symbolic logic that would make your head spin. And it *looks* so mathematical!
And as everybody since Descartes knows, mathematics is the source of all knowledge and truth.

But let's face it. Contemporary logic is to logic what astrology is to astronomy. You can spend dozens of years studying
astrology and can draw charts and all that sort of thing by a very complicated and intricate system that is very difficult to
learn; but when all is said and done, astrology rests on the foundation that the earth is at rest in the center of the universe
and that the spheres of heaven moving around it cause all the changes on it--and this happens to be false.^{(8)}

By the same token, no matter how complex modern symbolic logic may be, it depends on a radically false epistemology (which is no less strong because it wants to avoid epistemology and thinks you can--which is an epistemological stance in itself) plus the false notion that logic can be modeled after mathematics.

The fact that contemporary logic works well so often is simply a reflection of the fact that mathematical reasoning is a very large subset of logical reasoning; and it isn't surprising that those with a mathematical turn of mind (and who but a person with a mathematical turn of mind would attempt to get into the field of contemporary logic?) would not notice the cases where the applications of their logic were absurd. Hence, they would have no reason to suspect the unsoundness of the logical system itself.

So much for my attack on contemporary logic. I have shown (a) that it doesn't work as applied to statements, (b) why it doesn't work, (c) why it was developed, and (d) why those who are in the field would think that it was a good theory.

To return, then, to the implication as we actually use it, here are the rules:

*Rules for the hypothetical syllogism*

*Inferential mode*

** 1. If the compound and the antecedent are affirmed it follows that the consequent must be affirmed. **This valid
process is called

** 2. If the compound is affirmed and the antecedent is denied nothing follows with respect to the consequent.**

** 3. If the compound is affirmed and the consequent is affirmed, nothing follows with respect to the antecedent.**

** 4. If the compound is affirmed and the consequent is denied, it follows that the antecedent must be denied. **This valid
process is called

*Refutational mode*

** 5. If the antecedent and the consequent are affirmed, nothing follows with respect to the compound.**

** 6. If the antecedent is affirmed and the consequent is denied, it follows that the compound is false. This refutes the
implication.**

** 7. If the antecedent is denied, nothing follows with respect to the compound.**

This is more complex than the other forms of compounding propositions, because here the *order* in which the two
components are placed is significant. Hence, in the inferential mode of reasoning, the first two rules deal with proceeding
from the antecedent to the consequent; and the only valid one is *modus ponens* (the "putting mode" which could be
translated as the "affirmational mode," since it goes from affirmation to affirmation). The second two go backwards from
the consequent, and in this case, the valid one is the *modus tollens* (the "taking mode" or "denial mode") of a denial's
implying a denial.^{(9)}

The invalid mode "arguing" from the truth of the consequent to the truth of the antecedent is the reason why no scientific
theory can be *verified.* Every scientific theory is of the form "p implies q," because it is giving the cause of the effect in
question; and obviously all the theory's predictions are the result of reasoning from the *truth* of the theory to the necessary
truth of the results predicted (*modus ponens*). But by observing that these predictions are in fact *true,* when you test the
theory, you are now in the mode of reasoning from the consequent to the antecedent, and arguing from the truth of the
consequent is invalid. You can *refute* a theory by showing that its predictions are *false,* but you can't verify one by its
predictions.

And of course, this is why I called arguing from the components to the compound the "refutational" mode of reasoning,
because if you analyze what you are doing, you are supposing that the compound's relation to the components in it is an
implication (the inferential, normal mode), and you are arguing backwards--in which case, the only valid reasoning is the
*modus tollens.* Contemporary logic has surreptitiously assumed that the components can "generate" the compound as well
as the compound "generating" the components, precisely because the refutational mode of reasoning works; but what they
didn't see is that the form of the relation of the compound to the components is in all cases "compound implies something
about components," not "compound if and only if something about components" (p implies q and q implies p).^{(10)}

Now then, if you *want* to use contemporary logic as a guide for making inferences, and you want your conclusions to
express facts if your premises do, then you *have to add to the truth table* a column indicating *whether the connective is
"true" or not:* that is, whether it is applicable to these propositions. Just as the truth of the propositions is verified
extra-logically, so whether the connective belongs is also verified extra-logically. But once this is done, symbolic logic
will follow the rules I have given above.

For those who are interested, here are a couple of examples of what the truth tables will look like:

I think you can see what happens. The compound proposition now is false on the last four lines of the table, which is to say whenever the connective is inappropriate. Since contemporary logic ignores these last four lines, it makes the compound true in all these cases, when in fact it isn't.

The truth table for "implies," however, is somewhat peculi *can't depend on something false,* because there's nothing to
depend on if that's the case. Hence, the connective is inappropriate *whenever* "p" is false, as well as inappropriate
sometimes when "p" is true. Therefore, the truth table needs only six lines to cover all the possibilities and make symbolic
logic conform to *formal* implication.

Now of course, since truth tables are not used except at the most elementary level, all of the logical transformations (the "short cuts" and theorems) would have to be worked out taking this extra information into account. But that is not the point of this chapter, and I rather suspect it's not something for me to do. I simply propose it as a suggestion if anyone wants to do logic that looks like contemporary logic, and still guarantee that his conclusions will have to state facts if his premises state facts.

In any case, these are the basic ways of combining propositions. Other connectives we use are more or less complicated combinations of them.

There is one, however, that is quite interesting, because it seems so simple and yet is so complex: "but." I am not talking
of "but" as it is usually used in traditional logic (All men are mortal *but* John is a man, therefore), because that "but" is
simply "and" in disguise and is not used that way in ordinary speech.^{(11)}

I am talking about "but" in the sense of "The sun is shining but it is raining."

What does "but" mean? It means, "The statement I am about to utter would seem to be the opposite of what would follow
from what I just said *and* what I just said is true *and* the statement to follow is true" (which of course implies that the
inference you were going to draw from what I just said is invalid). For instance, "The sun is shining, but it's raining"
means, "The sun is shining and you might infer that it wouldn't be raining; and (despite that), it's raining." Or, "The sun is
shining and you might infer that it's raining, and the inference is not valid, because it's raining."

Schematically, "but" means the following: "p and not-q and not (p implies q)"'--using the symbols from now on to reflect statement logic, not contemporary logic. The thing that keeps it from being a simple "and" connecting an affirmative and negative proposition is the implicit implication that is refuted by it. You wouldn't connect two statements with a "but" unless you expected your hearer to disagree with the second one because of what he thought followed from the first one. So "but" is a way of steering the hearer away from an invalid inference, and preventing him from leaping to a false conclusion. But it isn't treated logically as a separate connective (he said, connecting this statement with the preceding by "but") because (he said giving the antecedent to this consequent) it can be described as a combination of simpler connectives.

I don't suppose it is amiss also to mention "because" as a connective. This gives the conclusion of an inference first, and
then the fact that implies it; so the form is "q and (p implies q) and p." So, "I am home because I am not at work" says, "I
am home, and if I am not at work, then I am home; and I am not at work." This actually gives what is dependent first and
then what it depends on, which reflects the way we reason when we go from effect to cause, and so is more natural actually
than showing the relation of dependence (the compound), and then arguing from the independent in that relation to what
depends on it--not that this is illegitimate, nor that it might not be clearer in *revealing* the reasoning process.

**Notes**

1. I owe this insight to a suggestion from my son.

2. I should point out that some logical notations of the truth table list the column under "p" as I did (TFTF) while others
list it (TTFF) and use the former for "q." For practical reasons, I happen to think that the way I did it is preferable, because
if there *is* to be an "r," then the eight lines under the "p" is (TFTFTFTF), that under the "q," (TTFFTTFF), and under the
"r," (TTTTFFFF). If there is also to be an "s," then the sixteen lines of these three will just repeat, and you have
(TTTTTTTTFFFFFFFF) for "s," and so on. It just makes things easier to write, which, believe me, can be a blessing if you
have to go on to "t," "u," and beyond.

3. By the way, conjoining propositions is called "logical multiplication" in contemporary logic, while "and/or" is called "logical addition." Now that you've heard the terms, forget them. This profusion of technical words is to me another way of making the smoke screen contemporary logic is hiding behind thicker.

4. Of course, if I wanted to talk à la contemporary logic, I could get picky and say, "If two statements don't happen simultaneously to be true then they're in fact incompatible with each other (because one is true and the other is false or both are false). But what I'm going to try to establish is that this "factual" sense of incompatibility is never intended by "not both p and q" as a statement.

5. Interestingly, the statement about general statements above is a "is incompatible with" statement, which, if it is couched
in the present tense precisely *asserts* the incompatibility of a statement's being non-tautological with its having predictive
force. (And it's obvious from the context that Hume meant this.) But of course incompatibility would *allow* you to make
predictions from it (which the context shows Hume also intended)--which would make it a non-tautological general
statement with predictive force. I've never been able to figure out how Hume has been able to get away with some of the
things he's said.

6. Of course, here we would run into the informal fallacy of irrelevance, because "q" is in the conclusion and not in the
premises. But the inference is *formally valid* in contemporary logic in any case, and we can permit this in order to
eliminate clutter.

7. For some reason, logicians don't seem to like the idea of applying propositional logic to predicate logic. I have made
statements about how, based on what I have been saying, statements with subjects and predicates go together (as in the
Square of Opposition) below, and have been met with, "Now don't go confusing propositional and predicate logic." For
heaven's sake! Don't the propositions in propositional logic have subjects and predicates? In that case, *everything* that is
said in propositional logic will *have* to be true in predicate logic; though the converse is, of course, not true.

8. Note, by the way, the fact that this is the foundation of astrology means that the science as a science *depends* on it ("If
the earth is at the center, etc., then there is a science of the influences of heavenly bodies on our lives.)--and so by
contemporary logic, the fact that this foundation is *false* implies that the science is a *true science.* That dilemma in the
philosophy of science was pointed out by Carl Hempel.

9. If you want the traditional names for the invalid modes, they are *ponendo tollens* and *tollendo ponens.* Even if you don't
know Latin, I assume that you're clever enough to figure out what they mean and therefore which they apply to.

10. Note once again that in symbolic logic, the "if and only if" proposition is not the same as "and" because it is true both when both components are true and both are false. Hence, you can "prove" that John is at home if and only if no one else is at home by finding an instance of nobody's being at home. To put it another way, nobody's being at home implies that John is always alone when he's at home.

In actual logic, "if and only if" differs from "and" in that "and" simply asserts *some* connection in which both have to be
true, while "if and only if" means that there is *interdependence* between the facts indicated by the statements.

11. In fact, it's a translation of the Greek word *de,* which simply means, "what follows adds to what preceded," rather than
*alla,* the adversative "but," which is our only use of the word "but." We use "and" for additional information as well as for
information that is just in general connected with what preceded.