LOGIC II:
THE LESSER OPERATIONS
[This subject is treated at greater length (and perhaps more clearly)in Modes of the Finite, Part 4, Section 2, Chapters 4 and 5.]
9.1. Single-proposition operations
There are a few things you can do with one single proposition once it has been translated into Logic.
In the simplest cases, these operations may sound too obvious to bother with, but there are some fallacies that are apt to crop up, so it's useful to know the rules.
A couple of definitions to begin with:
DEFINITION: The CONCLUSION is the proposition that results from a logical operation. It is the proposition which cannot be denied without contradicting one or another of the premises.
DEFINITION: A PREMISE is a proposition from which a conclusion is drawn.
Some logical operations have only one premise, others have more than one. (Note: in some books, especially in England, "premise" is spelled "premiss.")
DEFINITION: An INFERENCE is a logical operation.
DEFINITION: An ARGUMENT is an inference.
DEFINITION: An IMPLICATION is a potential inference.
DEFINITION: An inference is VALID if the logic is correct.
DEFINITION: The conclusion is said to FOLLOW from the premises if the inference is valid.
That is, a statement implies another statement if you can perform some logical operation on it and generate the other statement. When you actually do that operation, you are making an inference. If the inference is valid, then the conclusion can't be denied without denying one of the premises. (Note that if the inference is invalid, the conclusion can be denied without contradicting any of the premises).
The first kind of inference you can make with one single proposition is to interchange the subject-term and the predicate-term.
This is called converting the proposition.
DEFINITION: CONVERSION is the logical operation of interchanging the subject and predicate of a single proposition.
DEFINITION: The CONVERSE of a proposition is the conclusion of a conversion.
RULES: 1. Put the predicate-term in the place of the subject-term, and the subject-term in the place of the predicate-term.
2. The copula's affirmativeness or negativeness remains as it was.
3. The quantities of the terms remain as they were if possible. BUT
4. No term may pass from being indefinite to being definite, but it may pass from being definite to being indefinite.
Let me give a couple of simple examples:
Every horse is an animal [(e)horse - (s)animal]
converts into
Some animals are horses [(s)animal - (s)horse]
Notice that \animal" as the second subject had to be indefinite, because, being the predicate of an affirmative proposition, it was indefinite before. \Horse," on the other hand, was definite, but as the predicate of an affirmative proposition, it had to become indefinite.
\No man is an island" is English.
In Logic
Every man is not an island [(e)man n (e)island]
converts into
Every island is not a man [(e)island n (e)man]
Note that denying the converse of a proposition contradicts the original proposition.
If you deny that some animals are horses, it would be impossible for every horse to be an animal; and if you deny that every island is not a man, then obviously some island is a man, and so it is not the case that every man is not an island (\no man is an island").
Some cats are trees [(s)cat - (s)tree]
converts into
Some trees are cats [(s)tree - (s)cat]
Notice here that it's false that some trees are cats; but if you deny that some trees are cats, then you contradict the proposition that some cats are trees (which, of course, happens&&as we know from extra-logical evidence&&to be a false statement).
Remember, logic is not interested in the truth of the propositions, but only in whether the conclusion can't be denied without contradicting the original proposition.
Some cats are not trees [(s)cat n (e)tree]
cannot be converted.
Why? Because \cat" would pass from indefinite to definite (as the predicate of a negative proposition); and this is not allowed.
So even though it seems obvious that if some cats are not trees, then some trees are not cats, it doesn't follow. And if you take a different proposition, you can see that it wouldn't. Some animals are not horses; but it doesn't follow that some horses are not animals&&and in this case, we know that the inference is invalid, because we can immediately see that the premise is true and the conclusion is false.
COMMON FALLACIES IN CONVERSION:
1. Trying to convert indefinite negative propositions (such as the one above). The subject of the premise would have to pass from indefinite to definite (when it got into the predicate).
2. Converting affirmative propositions and making the new subject definite. For example, it is invalid to say that if every human being is a person, every person is a human being. Even if there were no non-human persons (there actually are), it would still not follow.
Again, this could be seen from trying out something like \if every horse is an animal, then every animal is a horse."
The other thing you can do with a proposition is to change its copula.
That is, create a negative from an affirmative proposition. This is called obversion.
DEFINITION: OBVERSION is the inference which generates a proposition with a negative copula from one with an affirmative copula or vice versa.
DEFINITION: The OBVERSE of a proposition is the conclusion of obversion.
RULES: 1. Leave the subject-term alone.
2. Change the copula from affirmative to negative or vice versa.
3. Add a negative to the predicate term, making it \refer" to the opposite class of objects from the previous predicate.
4. Pairs of negatives cancel each other (not-not's are affirmative).
4. Put the quantity appropriate to the copula on the new predicate term.
Note that in this case, since the predicate is now a different term, it is legitimate for the new predicate to be definite when the preceding one was indefinite. The same term does not pass from indefinite to definite in this case.
Some examples:
Every horse is an animal [(e)horse - (s)animal]
obverts into
Every horse is not a non-animal [(e)horse n (e)non-animal]
\Non-animal" here means anything at all that doesn't have the relationship of similarity-in-animality, or in other words what is \referred to" is the \class" of \everything except animals." So \animal" did not go from indefinite to definite; a new term was introduced.
Every horse is not a cat [(e)horse n (e)cat]
obverts into
Every horse is a non-cat [(e)horse - (s)non-cat]
Some horses are animals [(s)horse - (s)animal]
obverts into
Some horses are not non-animals [(s)horse n (e)non-animal]
And finally
Some horses are not cats [(s)horse n (e)cat)]
obverts into
Some horses are non-cats [(s)horse - (s)non-cat]
To give a more complex example, \Fourscore and seven years ago,...etc" was, in Logic
\Every father of ours is something that fourscore and seven years ago brought forth on this continent a new nation, conceived in liberty and dedicated to the proposition that all men are created equal." Symbolically: (e)father - (s)bringer-forth
This obverts into
\Every father of ours is not something that fourscore and seven years ago did not bring forth upon this continent..." [(e)father n (e)non-bringer-forth]
Here what you do is stick a \not" in the relevant clause in the predicate, making the predicate negative. You have to be careful, in a complex predicate like this, where you put the \not," or you might negative the wrong part of it. For instance, \...something that fourscore and seven years ago brought forth on this continent a new nation, not dedicated..." negatives the modifier of nation, and not the predicate as a whole. It's easy to see what you have to negative if you symbolize; you have to make the predicate \refer to" non-bringer-forths.
COMMON FALLACIES IN OBVERSION
1. Not keeping the negatives straight. If the predicate is already negative, this can cause confusion. If the subject-term contains a negative, this can be confused with a negative copula (even though the proposition itself is affirmative).
2. Not canceling out double negatives. It then looks as if the proposition is negative when it isn't.
3. Changing the quantity of the subject. Leave the subject-term strictly alone.
9.1.3. Multiple conversions and obversions
You may have wondered whether you can convert the converse of a proposition or obvert the obverse.
Of course you can; these are new propositions, and they don't know they've been arrived at by conversion or obversion. In fact, you can convert the obverse and obvert the converse, and keep doing this until you get tired or get back something you already had earlier.
Here's a simple case of converting twice:
Every horse is an animal [(e)horse - (s)animal]
converts into
Some animals are horses [(s)animal - (s)horse]
which converts into
Some horses are animals [(s)horse - (s)animal]
Note that you can't get back the original proposition by converting a definite affirmative proposition twice, because the definite subject becomes indefinite when it moves to the affirmative predicate, and then has to stay indefinite.
With double obversion, however,
Every horse is an animal [(e)horse - (s)animal]
obverts into
Every horse is not a non-animal [(e)horse n (e)non-animal]
which obverts back into
Every horse is an animal [(e)horse - (s)[non-non-animal]
and you're back where you started.
Actually, you can get back where you started in converting too, if you convert a definite negative or an indefinite affirmative proposition twice.
Now what happens if you convert and obvert alternately?
Every horse is an animal [(e)horse - (s)animal]
obverts into
Every horse is not a non-animal [(e)horse n (e)non-animal]
which converts into
Every non-animal is not a horse [(e)non-animal n (e)horse]
which obverts into
Every non-animal is a non-horse [(e)non-animal - (s)non-horse]
which converts into
Some non-horses are non-animals [(s)non-horse - (s)non-animal]
which obverts into
Some non-horses are not animals [(s)non-horse n (e)animal]
which can't be converted. So you have to stop.
Depending on whether you alternate with the conversion and obversion, or whether you do two conversions in a row and then obvert (two obversions don't get you anywhere), you can generate all the propositions dealing with horses, animals, non-horses and non-animals which follow from \every horse is an animal." Because of the confusions caused by multiple negatives, it isn't always obvious if these follow. For instance, does it follow from \Every horse is an animal" that some non-animals are not non-horses? I leave it to the reader to figure that one out.
There are fancy names for (at least some) of these multiple conversions and obversions, but I see no reason for burdening you with them.
There's more, actually, that can be done with horses and animals.
But to show this, you have to know the logic of the way we combine propositions.
Note that, since whole propositions are compounded intact, it is not necessary to translate them from English into Logic to perform these manipulations.
Since we're going to dealing with propositions which are compounds whose parts are whole propositions, there are a couple of things to note. First, some definitions.
DEFINITION: A proposition is AFFIRMED when it is accepted as it stands or \taken as |true' for the sake of argument."
DEFINITION: A proposition is DENIED when it is not accepted, or is \taken to be |false.'"
We are not, as I have said so often, interested in whether the propositions are in fact true or false, but in what happens when you accept them as true or refuse to do so.
Notice that a denial of a proposition does not necessarily mean the affirmation of the proposition that has the opposite copula. To deny that \Every horse is a maverick" is not to affirm that \Every horse is not a maverick." (The denial of a proposition&&as we will see later&&affirms its contradictory, not as in this case its contrary. The denial of \Every horse is a maverick" affirms that \Some horses are mavericks [i.e. at least one horse is a maverick]"; but we will see this in the next chapter.
When working with compound propositions, it is assumed that the compound is affirmed; the inference then deals with what happens to one of the constituent propositions when the other or others are affirmed or denied.
That is, you assume that the compound proposition is \true"; then, depending on how the parts are connected, you try to find out what happens when one part is taken as \true" or affirmed, or \false" or denied. When there are inferences, the affirmation of one part may force (for example) the denial of the other or the compound will be contradicted. For instance, to say \This is a page and you are reading it" is contradicted if you deny \You are reading it." Hence, \You are reading it" must be affirmed (because we are assuming that the whole proposition \This is a page and you are reading it" is \true").
The first compound proposition is the \and" combination.
It turns out to be logically trivial.
RULE: 1. A compound proposition formed by putting \and" between two (or more) propositions means that each of the propositions is to be affirmed.
2. \And" is symbolized by + [space + space]. Parentheses are put around the propositions when symbolizing. For convenience, a complete proposition may be symbolized by p [lower-case \p"] and the letters following [i.e. q, r, etc.]. A lower-case \n" before one of the letters means the denial of that proposition.
Rule 1 is another way of saying that \and" means that both of the propositions are to be taken as \true"; but of course in Logic we aren't dealing with truth, but with affirmation and denial.
Hence, when you say, \It's raining out and Chicago is in Detroit," you contradict the proposition when you say, \Chicago is not in Detroit," even if it's actually raining out. Now Chicago actually isn't in Detroit, so the compound proposition is actually false. But we aren't interested in its truth, but merely in whether it's contradicted if you deny some part of it.
You can symbolize the general \and" proposition by p + q; and the one above by (weather - rainy) + (Chicago - in Detroit), without worrying too much about whether they're in strict Logical form.
The whole inference can be symbolized this way (using * to mean \implies") ((p + q) + nq) * n(p + q); or, using words, (((weather - rainy) + (Chicago - in Detroit)) + (Chicago n in Detroit)) * n((weather - rainy) + (Chicago - in Detroit)) .
As I say, that's trivial. Things get less trivial, however.
The next way you can combine propositions is a way that some logicians call a kind of \or," where you mean that one or the other of the parts of the proposition might be \true," or both might be \false," but they both can't simultaneously be affirmed (i.e. \true"). I don't like to use \or," however, because there are various senses of the word, and the ordinary one is \either/or" (one part \true," the other part \false"), so it's confusing. I'm going to stick with an awkward expression that is at least clear: \not-both." (NOTE: If you recast the sentence a bit, you can usually make more sense out of "is incompatible with," because that's the idea: the fact referred to in the first clause is incompatible with the fact in the second one; so while they might both be false, they can't both be true.)
There isn't any decent way to express this compounding in English. The statement \The cat is outside not-both the weather is rainy" can only be expressed by some sort of statement like, \The cat is never outside when it's raining," or \It's never true that it's raining out and the cat is outside." But notice that this statement is compatible with the cat's being inside when it's not raining out.
This is the black-not-white kind of thing, where you're affirming that something can't be black and white at once, but you're not denying that there are shades of gray that are neither black nor white.
RULE: 1. The compound proposition \not-both" means that at least one of the propositions must be denied, or the compound is contradicted.
2. An affirmation of one part demands a denial of the other.
3. From a denial of one part, nothing follows with respect to the other.
4. The symbolic representation of \not-both" is V [space capital \V" space]. Parentheses go around the propositions combined.
Take the proposition above as an example. \The cat is outside not-both it is rainy" [p V q, or alternatively (cat - outsider) V (weather - rainy).
Now if you happen to see the cat outside (which affirms the first proposition), then you have to deny that it is rainy. ((p V q) + p) * nq; or (((cat - outsider) V (weather - rainy)) + (cat - outsider)) * (weather n rainy).
Similarly, if you know (i.e. affirm) that it is raining out, then you have to deny that the cat is outside. In either of these two cases, if you don't deny the other part, you have contradicted the compound proposition. I will skip the symbolism here.
On the other hand, if you see the cat inside (denying the first part), you don't know whether it's rainy or not, because both parts can be denied without contradicting the compound. Symbolically, ((p V q) + np) * ?. The question mark indicates, of course, that nothing follows.
There's another kind of combination that exists in Logic and isn't really common in English.
This is what logicians call the \weak |or,'" which means that both parts can't simultaneously be \false"&&or in other words, that at least one of them must be \true."
RULES: 1. The connective \ means that the compound is contradicted when all parts are denied.
2. From a denial of one part, it follows that the other part must be affirmed, or the compound is contradicted.
3. From an affirmation of one part, nothing follows, since both parts may be affirmed.
4. The symbolic representation of \and/or" is v [space lower-case \v" space], with parentheses around the propositions.
(By the way, that "v" is the first letter of "vel," which is Latin for "or."
For example, \Some people are pianists and/or some people are not pianists." This is the \either one or the other or both" proposition, and it isn't all that easy either to express in English or to grasp intuitively. In English, we would have to say, \Either some people are pianists or some people are not pianists, or both," which is still pretty awkward.
Be that as it may, if you know that it's not true that some people are pianists, then you are denying \Some (i.e. at least one) people are pianists," in which case, you have to affirm that at least one person is not a pianist. Symbolically ((p v q) + np) * q; or ((((s)person - (s)pianist) + ((s)person n (e)pianist)) + n((s)person - (s)pianist)) * ((s)person n (e)pianist). Note that in the symbolization the denial first occurs with the whole proposition, and then there is the affirmation of the negative proposition as what follows.
If you did it the other way, (leaving out the (s)'s and (e)'s for clarity) (((person - pianist) v (person n pianist)) + n(person n pianist)) * (person - pianist)
But if you know that some people are pianists, you can't say that it's false that some people are not pianists (as you can see intuitively; because in fact some people are not pianists). But it isn't so obvious if you take \some horses are animals and/or some horses are not animals"&&remember, the Logic meaning of \some" is \at least one," and not \some are and some aren't"). That is, symbolically, ((p v q) + p) * ?
These, then, are the lesser operations in logic. Either/or, if-then, and the \categorical syllogism" remain to be explored.
An implication is a potential inference, and an inference or argument is a logical operation; logical operations draw conclusions from the propositions called "premises." The conclusions "follow" from the premises. Premises imply their conclusions.
The first logical operation with a single proposition is conversion, in which the subject-term and predicate-term are interchanged, leaving the copula as it was, and making sure that no term passes from indefinite to definite.
Obversion changes the quality of the copula, and is done by adding a "not" to the copula and a negative to the predicate (making it "refer" to the class of "all objects but" the original. Double negatives cancel each other out. Conversions and obversions can be repeated with the former conclusions, generating new propositions.
To "affirm" means to "take as it stands" or "accept as 'true'," while "to deny" means to "refuse to accept (as 'true')." With compound propositions, the whole proposition is affirmed; the inference is what happens to one part when the others are affirmed or denied. If the part must be affirmed (or denied) to avoid contradicting the compound, the inference is valid.
The compound proposition that uses "and" (p + q) is logically trivial, since it says that both sides are to be affirmed. It is contradicted if either constituent is denied.
The compound that is formed with "not-both" (p V q) means that one or the other or both of the constituent propositions must be denied; it is contradicted only if both are affirmed.
The compound that is formed with "and/or" (p v q) means that one or the other or both of the constituent propositions must be affirmed; it is contradicted only if both are denied.
Translate first, when necessary, into logical form.
1. Convert the first four propositions in the exercises of the preceding chapter.
2. Obvert the second four propositions in the exercises of the preceding chapter.
3. Obvert, then convert, etc. the ninth proposition in the exercises of the preceding chapter.
4. Convert, obvert, etc. the following proposition: "No one with any sense believes in horoscopes.
5. Is it valid to say, "I know I can't have my cake and eat it, but I don't have it; so I must have eaten it."? If it is, tell what kind of inference it is, and if not, tell what rule is violated.
6. Is it valid to say, "Since I know that coughs are signs of colds or coughs are signs of emphysema, and I have a cold as I now cough, then I can't have emphysema."? If it is, tell what kind of inference it is, and if not, add what rule is violated.
7. Is it valid to say, "Every cat hates dogs or [not-both? and/or? Think!] every cat loves cream, and this cat loves cream, and so it can't hate dogs."? If it is, tell what kind of inference it is, and if not, add what rule is violated.
8. Jesus said, "You can't be the slave of God and property." Logically, this would be, "You are the slave of God or [not-both? and/or? Think!] you are the slave of property." You know you are not the slave of property; therefore, you must be the slave of God, right? If so, show what argument is used, and if not, what rule is violated.
9. Students are in school for love of learning, for the sake of the degree, because their parents expect it, because they have no job, or because they like the company. This person just said she was in school because she liked the company; and so she really doesn't care about learning. Correct? If so, what argument is used; if not, why not?
10. The sun is shining, it's cloudy, or it's night. It's not night and it's not cloudy; therefore, the sun has to be shining. Correct? If so, what argument is it; if not, why not?