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Relational Calculus

(Rough Draft)

by Vern Crisler

 

Copyright June, 1999

(Last Update: December, 2001).

 

Relational arguments are invalid in formal logic because they commit the fallacy of four terms.  Such arguments are not really “arguments” in a strictly logical sense, since reasoning is limited to the use of three terms, the major, minor, and middle term.  Nevertheless, while relational arguments are not forms of reasoning, and are hence, not properly logical, they are forms of calculation, and are therefore closer to mathematics than to logic.

 

One way of representing relational propositions is by the use of superscripts.  For instance, to say that “Every boy loves some girl,” we could write, “BLaG.”  This is a fairly perspicuous symbolism.

 

In the above proposition, the B is universally quantified by the omnis-operator “a” which means “all” in common or neo-classical notation.  The “predicate” in the above proposition refers to an indefinite part of the class of girls, i.e., “some girl (or other), because in traditional formal logic, the predicate is not usually quantified due to its being a universal.

 

As you may have noticed, the above proposition is in active voice, but how would we symbolize a relation in passive voice, or even a converse relation?  The easiest way would be to symbolize it so that it looks as close to everyday language as possible, which is, “Some girl (or other) is loved by every boy.”  Obviously, this is a converted sentence, so we should look to logical conversion as a way of symbolizing the passive voice.  Hence, “Some girl is loved by every boy” can take the form,

 

(1)  G--LiB

 

The use of the negative sign for passive voice or for the converse of a relation will become clear later when when talk about cancelling terms.  In the above symbolism, the some-operator “i” preserves the meaning of the original sentence, in that only an indefinite part of the class of girls is loved, etc.  But we run into a problem in that there is nothing in the above symbolism that tells us that it’s still every boy who loves some girl.  Now that B is in the predicate place, the symbolism in (1) leaves it ambiguous regarding the quantity of boys involved.  In order to remove this ambiguity, let us introduce a new way of symbolizing universal quantity in the predicate term.  The symbolism for absolute value is here tentatively introduced to symbolize universal predicate quantity.  The relational proposition, “Some girl (or other) is loved by every boy” would be symbolized as follows:

 

(2)  G--Li|B|

 

Here |B| means universal quantity, “all boys,” and also means that in converting the proposition, the term B retains universal quantity even in the converted form, so that the converse of (2) can be symbolized as,

 

(3)  BLaG

 

Or in English, “Every boy loves some girl.”  In general then, any expression of the form,

 

(4)   F--Ri|Y|

 

can be converted to,

 

(5)  YRaF

 

 

Active & Passive Voice

 

Here are some examples of relational expressions in active and passive voice using suggested symbolism:

 

 

Active

Symbolized

Passive

Symbolized

1.  Abelard loves Heloise.

ALaH

 

Heloise is loved by Abelard.

H--LaA

 

2.  Plato taught Aristotle.

PTaA

 

Aristotle was taught by Plato.

A--TaP

 

3.  All the boys married (some of) the girls.

BMaG

Some of the girls were married by all the boys.

G--Mi|B|

 

 

 

You may have noticed that in the first two propositions, the terms represented individuals, i.e., Abelard, Heloise, Plato, and Aristotle, but that in the last proposition, the terms represented groups or parts of groups.  Is there a logically perspicuous way of symbolizing individuals as opposed to groups, and if so, what is the best way to do it?  Sommers speaks of individual terms as having “wild” quantity.  This means that an individual, for instance, “A” for Abelard, can be modified by the operators a, e, i, or o without making any difference to the quantity of the term itself.  It matters little whether you say that “All Socrates is wise,” or “Some Socrates is wise,” because “Socrates” has wild quantity.  What we are really saying is that “[All individuals identical with Socrates] are wise” or “[Some individuals identical with Socrates] are wise,” and since there is only one Socrates in view here, it does not matter if we speak of All or Some with respect to Socrates.  As the logician, Gordon Clark, once wrote, “Socrates is in a class by himself.”

 

If individuals have Sommers-type wild quantity, this means conversion by limitation will not affect the respective individual terms, just as the absolute value sign proposed above prevents conversion from changing the quantity of the predicate term for non-individuals. Indeed, it is impossible for a term that represents a named individual to be indefinite, and this is the case because every individual is unique, and we say therefore that an individual receives a constant definite term, usually the first letter of his or her name.  Even in a case where we don’t know the name of the individual, or who it is, the individual still receives a constant definite term, though we can substitute a subscript number in place of a name, if we so choose.  Hence, let “I” represent an individual, and let “I1” represent the first individual, and “I2” represent the second, and so on.  This doesn’t mean we cannot place the sign of absolute value around individuals, it’s just that it’s redundant.

 

Of course, if we want to make explicit the fact that the proposition involves an individual who has universal quantity, we can adopt either the absolute value sign, or whatever symbolism that serves this purpose.  I have not come to any hard-and-fast decision regarding what symbol to use for individuality.  The upside-down iota is a traditional sign, but it is not perspicuous.  Some may find In to be a satisfying symbolism, while others may find the symbols I* or İ or Î or even I™, satisfactory.  At this this point, however, I leave it to the discretion of the reader.

 

 

Functional Symbolism:

 

In this relational calculus, I’d like to introduce another symbolism.  Most of you who’ve had some algebra will recognize the traditional function symbolism of mathematics.  It’s usually of the form f(x), though other letters are also used instead of “f”.  What exactly is a function?  In brief, a function is a rule or a command.  The following all pretty much

mean the same thing:

 

function(x)                               f (x)

command(x)                            c (x)

procedure(x)                            p (x)

process(x)                                p (x)

rule(x)                                      r (x)

law(x)                                      l (x)

do(x)                                        d (x)

 

 

The first one is the one that’s used in mathematics texts.  We can also specify what type of function we want.  For instance:

 

if then(x)                                  i (x) 

conjoin(x)                                c (x)

disjoin(x)                                 d (x)

quantify(x)                               q (x)

 

The first thing to note here is that f (x) is not itself the function.  That’s what the “f” stands for.  The expression f (x) is the result of the function.  Or to put it another way, “x” is the input value, or the argument of the function.  The function itself is f, and f (x) is the output value or image of x under f.  Pretend that you are x, and that a camera is f, and that f (x) is the picture of you once the camera has done its work.  Thus, f (x) is the picture or image of you (x), in accordance with, or under, the camera’s process of operation, that is, under f.  The operation is, input, process, output.  A lot of mathematics and computer science makes use of this basic concept in mathematics.

 

If you or a computer comes across the expression,

 

f (x) = 2x + 3

 

what you are being told to do here is to multiply x by 2, then add 3.  It doesn’t matter what number x is.  Whatever it is, multiply it by 2, then add 3.  That’s essentially all that a function is: a rule or command to do something to a specified value.  The x is simply functioning as a placeholder for whatever number you want to stick in there.  If you make x equal to 5, then the result will be 2 (5) + 3 = 13; therefore f (x) would equal 13 if the “argument” is 5.    Similarly, if x were made to be equal to 4, then the image would be equal to 11, (as long as the order of operations specified by the function is followed, i.e., multiply by 2, then add 3).

 

Frege and Russell (hereafter F&R) sought to use the above type of functional symbolism in their logical works.  For F&R, “x” is the argument and “f” is the incomplete function, needing to be saturated by the input value “x”; hence, (x) (f1x É f2x) means that for all x, if x is an f1, then x is a f2.  (The subscripts 1 and 2 are used in order to distinguish between two different functions.)   In my opinion, F&R were mistaken in regarding f1 or f2 as functions in the above argument.  Actually, symbols such as f1 or f2 are merely placeholders for a predicate or a relation, not a function.  There’s not even an analogy here between F&R’s  f (x) and the mathematician’s f (x).  The only similarity that I can see between them is that F&R’s f comes before the (x), just as it does in mathematics; but that is no reason at all to call F&R’s f a function.  It is not anything at all like a mathematical function.

 

Nevertheless, there are functions that are operative in F&R symbolism.  One is explicit though not stated to be a function, and the other is implicit.  Take a look again at (x)( f1x É f2x).  Where exactly is a command being given to do something?  Well, if you know anything about quantifiers, you can see that the (x) standing before the function expression is telling us to do something.  It’s telling us to quantify over x.  This is then a specific type of function, quantify(x) or q(x).  If we want to make F&R symbolism more mathematical, we would write it like this:

 

q ("x) [( f1x É f2x)]

 

Here then we are saying that for whatever x is, quantify it universally, such that whatever x stands for in the argument, you’re dealing with all x.  But there is another function at work in the argument, though it’s not shown anywhere in F&R symbolism.  What about the “if-then” process?  The symbol É is telling us that the first expression f1x implies the second expression f2x.  Hence, we would need to use if-then(x) or i(x), as in the following expression:

 

i (x) {q ("x) [( f1x É f2x)]}

 

 

I would like to introduce functional symbolism here as an alternative to my earlier symbolism, but I want to make it clear that I’m only using the word “functional” because of a similarity in symbols, not because there is any real mathematical functionality involved.  If we take the expression “Every boy loves some girl,”  let’s use functional symbolism and combine it with arithmetical calculus, hence:

 

~b + l (g)

 

If we wanted to express the passive voice, we could use the traditional inverse sign for functions, f--1, and since the “function” in the expression is l, we would have:

 

~b + l--1 (g)

 

And this would mean, “Every boy is loved by some girl (or other).”  If we wanted to say that “some boys loved every girl, we could write it this way:

 

b · (l (|g|))

 

Or leaving out the sign of conjunction:

 

b (l (|g|))

 

The absolute value sign, of course, gives us universal quantity, and its absence means particular quantity.

 

Logicians normally define relations as symmetrical, reflexive, or transitive, etc.  Let’s look at each of these types of relations, and provide some symbolism to help make explicit what types of relations are involved.

 

Symmetrical

 

The general form using prenex symbols is (x)(y)(Rxy É Ryx), meaning “for all x and for all y, if x is R to y, then y is R to x.”  Here are English equivalents (following Copi):

 

“is married to”

“is next to”

“is the same weight as”

 

Probably the best way to symbolize this using a combination of relational calculus and logical algebra is:

 

(6)  [«R]  ARaB  É  BRaA                  [~a + r (b)] É [~b + r (a)]

 

Here the double arrow placed before the relational symbol shows that the proposition can be read forward and backward without changing the nature of the relation.  For instance, if the relation is in active voice for the first relational proposition, its symmetrical proposition will also be in active voice, and so on.

 

Asymmetrical

 

An asymmetrical relation is described in prenex form as (x)(y)(Rxy É ~Ryx), meaning “for all x and for all y, if x is R to y, then y is not R to x.”  This would include such expressions as the following:

 

“is north of”

“is a parent of”

“weighs more than”

 

Perhaps the best way to symbolize this in this relational calculus is by simply using a one way arrow, hence,

 

(7)  [®R]  ARaB  É  BReA                  [~a + r (b)] É [~b + ~ r (a)]

 

 

Or, A is R to B, but B is not R to A.  Notice the use of the “e” operator, meaning, “no” in the second relational proposition, and the negative sign before the r in the functional symbolism.

 

Non-symmetrical

 

This is simply a condition of neutrality with respect to symmetry.  To say that A loves B does not require that B love A, or that B not love A.  The statement is neutral.  We could perhaps state this in prenex form as follows: (x)(y)[Rxy É (Ryx Ù ~Ryx)].  We have used the exclusive disjunction symbol “Ù” because the consequent relation can be one or the other but not both, for it would make very little sense to say that if x is R to y, then y is R to x, and y is not-R to x.  In English this could be,

 

“If Tom loves Susan, then either Susan loves Tom or she doesn’t.”

 

But of course, we would not say she loves Tom and doesn’t love Tom at the same time, unless we are delving into the psychology of indecision, so the use of exclusive disjunction is appropriate for non-symmetrical relations.  In this relational calculus, we could represent it thus,

 

(8)  [® ÙR]   ARaB  É   (BRaA Ù BReA);       [~a + r (b)] É [~b + r (a) Ù ~b + ~ r (a)]

 

This would read, “If A is R to B, then either B is R to A or B is not-R to A.”  The use of the symbol  Ù for exclusive disjunction does not seem to be widespread, but I can see no other symbolism that could work better for the time being.

 

Reflexive

 

There are roughly two types of reflexivity, absolute and property reflexivity.  In the first case, absolute reflexivity, the prenex form is (x)Rxx, and means “for all x, x is R to x” and it does not matter which way we read it.  This type of reflexivity is one of absolute identity and basically has the English form,

 

“is identical with”

 

In relational calculus, this can be symbolized easily just by using the equal sign.  Identity of this type is not primarily a logical form of identity, which is predicative, but rather an operational kind of identity, which is the type of identity used most often in mathematics and science.

 

Property reflexivity has the prenex form (x)(y)[Rxy É (Rxx · Ryy)], meaning, “for all x and for all y, if x has R to y, then x has R to x and y has R to y.”  Copi’s examples are:

 

“has the same hair color as”

“is the same age as”

“is a contemporary of”

 

Here the identity involved is some property or characteristic that two individuals have in common.  Therefore, we shall choose a sign that is close to the equal sign, something suggesting congruence but not absolute equality.  Hence, in relational calculus, we could symbolize a property-reflexive relation as follows:

 

(9)  [ @ R]   ARaB  É   ARaA ·  BRaB              [~a + r (b)] É [~a + r (a) · ~b + r (b)]

 

That is,  if A is property-reflexive with B, then A and B both have the same property.

 

Transitive

 

The prenex form is (x)(y)(z)[(Rxy · Ryz) É Rxz], meaning, “for all x, for all y, for all z, if x is R to y, and y is R to z, then x is R to z.”  English sentences are:

 

“is north of”

“is an ancestor of”

“weighs the same as”

 

Thus, if x is north of y, and y is north of z, then x is north of z, and so on.  In relational calculus, we could symbolize this type of relation thus,

 

(10)  [ ³ R]   ARaB  +  BRaC  =  ARaC            ~a + r (b) · ~b + r (c) É ~a + r (c)

 

We would read it as A is R to B and B is R to C, therefore, A is R to C.  (Note that in the functional symbolism, conjunction has priority over implication, thus allowing us to avoid too many parentheses.)  The transitive relation communicates its quality all down the line, from A to Z, so that each term in the series can be described as involving the same qualitative relation.  An example would be:

 

1.  Noah is the ancestor of David

2.  David is the ancestor of Mary

3.  Therefore, Noah is the ancestor of Mary.

 

Or, in relational calculus form:

 

(11)  [ ³ R]   NAaD  +  DAaM  =  NAaM

 

 

This is easily solved by the method of logical algebra, crossing out the linking term D, hence,

 

(12)   NRe~D  +  DRe~M  =  NRe~M  (or NRaM)

 

In cancelling the term D, make sure that you don’t cancel the relation R.  We will see later that the relation R can only be cancelled by ~R, which is the opposite voice or converse of R.  In function symbolism, we have

 

1.  ~n + a (d)

2.             ~d + a (m)

--------------------------

3.  ~n + a             (m)

 

That is, ~n + a (m), “Noah is the ancestor of Mary.”  Note that d is cancelled, but not the relation a, ancestor.  The relation remains in the conclusion for the above argument.

 

Intransitive

 

This relation is similar to the above except that it does not communicate its quality to the next term in line, but rather its quantity.  Copi provides some English examples:

 

“is father of”

“is mother of”

“weighs twice as much as”

 

In relational calculus, we can symbolize this in general as,

 

(13)  [ > R]  ARaB  +  BRaC  =  ARaC             ~a + r (b) · ~b + r (c) É ~a + r (c)

 

In other words, A is the father of B, and B is the father of C, therefore, A is the father of C (or A is the grandfather of C).  In general, an intransitive relation is additive, and achieves cardinality-R, the latter here defined as the number steps that are in the relational series.

 

With an additive relation, we can mix the relations in the series.  For instance, if we take the above example and change the first father in the series to mother, then A would be the grandmother of C.  In this case, A, the mother, would have cardinality 2, because she is the second relation from the last term in the series, i.e., A, B, C.  The father has cardinality 1, since he is only 1 relation away from the final term.  Here are some examples:

 

1.  Abraham was the father of Isaac;

2.  Isaac was the father of Jacob;

3.  Jacob was the father of Joseph.

4.  Therefore, Abraham was the great-grandfather of Joseph

 

Dictionary: A=Abraham; I=Isaac; J=Jacob; Y=Joseph; F= father of

 

(14)  [ > R]   ARaI  +  IRaJ  +  JRaY  =

 

We solve (14) by means of logical algebra:

 

(15)  AFe~I  +  IFe~J  +  JFe~Y  =  AFe~Y  (or, AF3aY) 

 

Since we can cancel the I and J linking terms, we have A and Y left, as well as the relations.  As you can guess, F3 in the superscript sign means “great-grandfather,” (i.e., father, grandfather, great-grandfather).  The conclusion is “Abraham was the great-grandfather of Joseph.”

 

To use arithmetical calculus, we have:

 

1.  ~a + f (i)

2.             ~i + f (j)

3.                       ~j + f (y)

--------------------------------

4.  ~a + f 3                    (y)

 

The conclusion, ~a + f 3(y)  “Abaham was the great-grandfather of Joseph.”

 

Here is an example from W. W. Bartley:

 

A.  Rebecca is the mother of Jacob;

B.  Jacob is the father of Joseph;

C.  The mother of the father is the paternal grandmother.

\ Rebecca is the paternal grandmother of Joseph.

 

We can recognize 3 as a definition, and it would seem the best way to symbolize this argument is as follows:

 

Dictionary: R=Rebecca; J=Jacob; Y=Joseph; M=mother of; F=father of.

 

(16)  RMaJ  +  JFaY  =

 

We can count up the relations in the argument to give us the “exponents” of m and f, respectively.  The relation m involves two relations, whereas f only involves 1.  We will therefore symbolize the conclusion accordingly:

 

(17)  RMe~J  +  JFe~Y  =  RM2F1e~Y  (or RM2F1aY)

 

The conclusion is “Rebecca is the grandmother of Joseph through Joseph’s father Jacob,” or to paraphrase, “Rebecca is the paternal grandmother of Joseph.”

 

In arithmetical symbolism, where p (y) equals the paternal grandmother of Joseph, we have:

 

1.  ~r + m (j)

2.             ~j + f (y)

3.      ~[m      ( f)]    + p (y)

--------------------------------

4.  ~r                       + p (y)

 

Hence, ~r + p (y), “Rebecca is the paternal grandmother of Joseph.”  In the above, we were able to cancel two of the relations (“mother of” and “father of”), leaving the last relation, “paternal grandmother of.”

 

Propositions

 

Here are some possible ways to represent propositions, and pronominal cross-reference.  In the following, let P=person, or persons.

 

A.  “A woman is loved; she is kind.”  W1iL; P(w1) aK

B.  “Some women are loved; they are kind.”  WiL; P(w) aK

C.  “All women are loved; they are kind.”  WaL; P|w| aK

 

The difference between 2 and 3 with regard to cross-referencing the plural pronoun (they), is that we made use of the sign of universality |w| in 3.  In 1 and 2, we’ve used the parenthesis signs for individual or particular quantity, though to be consistent, we ought to have left the “w” standing alone in both cases.  Nevertheless, I think it’s probably safer in order to avoid confusion to place parentheses around the “w1” or “w” due to their proximity to the “all” operator in both sentences.   In the first proposition, the woman who is loved is a specific woman (i.e., “she”), and this is symbolized by the W and 1 subscript in the first clause.  The pronoun “she” cross-references back to this specific woman by the use of the numerical subscript (w1) in the second clause.  If we wanted to be more specific, we could symbolize the following sentence in this way,

 

D. “Julia Roberts is a pretty woman; she is an actress.”  JRa(PW); P(JR)aA 

 

Here, we’ve actually identified the pronoun “she” with a named individual by use of the absolute value subscript with the initials of the named person, i.e., |JR|.

 

Here is a proposition from Copi, and it involves both a relation and an object within the relation:  “Jack traded his cow to the peddler for a handful of beans.”  Here we have the agent, Jack, the direct object, the cow, the handful of beans that modifies the direct object, and finally the indirect object, the peddler.

 

(18)   JTCBaP                  ~j + t (c (p (b)))

 

I grant that the more information one tries to capture by means of relational symbolism, the more chance one has of losing one’s middle terms in a morass of bad grammar.  Be that as it may, the above symbolism seems to represent Copi’s proposition fairly well, though for purposes of logical simplification, information such as the purchase price of a cow, or of other information related to a direct object in a relation, should be saved for a subsequent proposition.

 

Here is a simple relational proposition (from the logician George Englebretsen)

 

“Paris loves Helen.”

 

(19)  PLaH               or             ~p + l (|h|)

 

Both Paris and Helen are individuals, so in accordance with the traditional logic of terms, they will be accorded Sommers-type wild quantity, i.e., they will be universally quantified.  Hence, as we have seen, the passive voice, “Helen is loved by Paris” can be expressed in the following way:

 

(20)  H--LaP              or             ~h + l--1(|p|)

 

 

Arguments:

 

Here are some arguments from Englebretsen, Sommers, and Copi:

 

A.  Someone loves Helen

B.  Helen is a vamp

C.  Someone loves a vamp.

 

We will use the letter P to symbolize the term “Someone,” i.e., some individual or other.”  Then we solve the equation in the same way that we solved either regular form or transposed syllogisms discussed in my earlier essay “Logical Algebra.”  In the following equation, the H  is polarized and can be cancelled.  The remaining terms are v and i.  We read from left to right to get our conclusion, so i is in the subject place, and v is in the predicate place.

 

(21)  PLiH  +  HaV  =

 

Solving, we have,

 

(22)  PLo~H  +  He~V  =  PLo~V  (or, PLiV)

 

Cancelling the H’s, we have the conclusion: “Someone loves a vamp.”

 

Arithmetically, we have:

 

1.  x  l (|h|)

2.        ~h + v

-------------------

3.  x l          v

 

Or,  x (l (v)); i.e., “Someone loves a vamp.”  Note that the v becomes absorbed into the place where the h had been before it was cancelled.

 

A.  Some senator gives away some money.

B.  All money is tained.

C.  Some senator gives away something tainted.

 

(23)  SGiM  +  MaT  =

 

Solving by logical algebra, we have:

 

(24)  SGo~M  +  MGe~T  =  SGo~T  (or, SGiT)

 

Cancelling the M’s, the conclusion is: “Some senator gives away something tainted.”

 

Arithmetically,

 

1.  s (g (m))

2.        ~m  + t

------------------

3. s  g            t

 

Or, s (g (t)), “Some senator gives away something tainted.”

 

A.  Every circle is a figure.

B.  Someone draws a circle.

C.  Someone draws a figure.

 

Letting P represent someone (or other), i.e., some person, we can write the premises as follows:

 

CaF  +  PDiC  =

 

Proposition (25) is in regular first figure form, rather than transposed form, so we solve it in the same way we solve regular syllogisms, i.e., with the middle terms away from the + sign, crossing out the C’s in this case, leaving P, F, and the relation D:

 

Ce~F  +  PDo~C  =  PDo~F  (or, PDiF)

 

Conclusion: “Someone draws a figure.”  Or to combine the whole argument: “Since every circle is a figure, someone who draws a circle is someone who draws a figure.”

 

Or, arithmetically, remembering that ~c + f is also f + ~c:

1.   f  +          ~c

2.            x (d (c))

----------------------

3.  f         x d

 

In this case, the f is absorbed into the place where the c was before it was cancelled, thus becoming, x (d (f)), i.e., “Someone draws a figure.”

 

A.  Some boy gave a rose to a girl.

B.  Every rose is a flower.

C.  Every girl is a child.

D.  So, some boy gave a child a flower.

 

(27)  BGRiG  +  RaF  +  GaC  =

 

By cancelling the R’s, and solving the first part of the equation, we have:

 

(28)  BGRiG  +  RaF  =  BGFiG

 

Or, “Some boy gave a flower to a girl.”  Combining this conclusion with the remaining proposition, we have,

 

(29)  BGFiG  +  GaC  =  BGFiC

 

Cancelling the G’s (though not the G in the superscript, since there it means “gave” rather than “girl”), we have the conclusion: “Some boy gave a flower to a child” or stated somewhat differently, “Some boy gave a child a flower.”

 

1.  b (G (r (g)))

2.         ~r         + f

3.             ~g           + c

-----------------------------

4.  b G                 f      c

 

Note the absorption of f into r, and c into g.  The conclusion would then be b (G (f (c))), “Some boy gave a flower to a child.”

 

A.  Every boy loves some girl.

B.  Every girl adores every cat.

C.  Whoever adores every cat is a fool.

D.  So, every boy loves a fool.

 

(30)  BLaG  +  GAa|C|  +  (PAa|C|  É  PaF)

 

In C, we have the expression, “Whoever adores every cat is a fool,” and this is represented as a conditional proposition, “If P adores every cat, then P is a fool” or, “If anyone P adores every cat, then P is a fool.”  In order to solve the above equation, note that the subject of GAa|C| is an instance of PAa|C| and thus we have the resulting proposition:

 

(31)  GAa|C|  +  PAa|C|  =

 

From this we convert the second relational proposition, remembering that conversion of an active voice relational proposition puts the relation in the passive voice.  The negative sign before the relation represents this change of voice:

 

(32)  GAa|C|  +  C~Aa|P|  =

 

We then use logical algebra to solve the equation, cancelling both the C’s and the A’s:

 

(33)  GAe~|C|  +  C--Ae~|P|  =  Ge~|P|  (or, Ga|P|)

 

In (33), the relations cancel because one of the propositions was converted, thus turning an active voice to a passive voice.  When two relations of the same type are polarized, they both cancel, leaving only a subject-predicate conclusion.  This is how we obtained Ga|P| as the conclusion, and this conclusion can then be combined with the remaining part of the conditional clause of (30), giving us:

 

(34)  Ga|P| + PaF = GaF

 

Once we have GaF, we can then combine it with the first proposition in (30), giving us:

 

(35)  BLaG  +  GaF  =  BLaF

 

Cancelling the G’s, the conclusion is, “Every boy loves (some) fool.”

 

Arithmetically,

 

1.  ~b + l (g)

2.           ~g  + a ( |c| ) 

3.                ~[a ( |c| )] + f

-------------------------------

4.  ~b + l                        f

 

 

The following example is from Sommers:

 

A.  All censors withhold some books from every minor.

B.  All books are literature.

C.  Some minors are female.

D.  Therefore, All censors withhold some literature from some females.

 

Solving arithmetically, we have:

 

1.  ~c + w (b ( |m| ))

2.             ~b            + l

3.                     m             f 

---------------------------------

4.  ~c + w l                       f

 

That is, ~c + (w (l (f))), “All censors withhold some literature from some females.”

 

From Copi,

 

A.  Some horses are faster than some dogs.

B.  All dogs are faster than some men.

C.  Therefore, some horses are faster than some men.

 

(51)  HFiD  +  DaM  =

 

(52)  HFo~D  +  De~M  =  HFiM

 

Cancelling the D’s, the conclusion is, “Some horses are faster than some men.”

 

Or, arithmetically,

 

1.  h (f (d))

2.        ~d   + f (m)

----------------------

3.  h  f               m

 

The conclusion is h (f (m)), or “Some horses are faster than some men.”

 

From Sommers:

 

A.  No Englishman buys anything from any native except Riffians.

B.  Some English tourist bought a fez from a blond native.

C.  All English tourists are Englishmen.

D.  All fezes are things.

E.  Therefore, Some Riffians are blond.

 

Dictionary: E = Englishman; b = buys; t = thing; n = native, r = Riffian; B = blonde; eT = English tourist.

 

1.  ~E +              ~b (~t  (~nr))]

2.               eT     (b (F      (n B)))

3.    E  + ~[eT]

4.                          ~F + t

-------------------------------------------

5.                                          rB

 

 

The first premises was originally ~E + ~[b (t (n~r))], “No Englishman buys a thing from a non-Riffian native,” but multiplying the negative sign through, we have ~b (~t (~nr)), which allows us to come to the correct conclusion, rB,  “Some Riffians are blonde.  Note that the conclusion must be particular since premiss 2 is particular.

 

A.  All supporters of Nixon will vote for Reagan.

B.  Avery will vote for none but a friend of Harriman.

C.  No friend of Kruschev has Reagan for a friend.

D.  Therefore if Harriman is a friend of Kruschev, Avery will not support Nixon.

 

Dictionary:  s = supporters; n = Nixon; v = vote; r = Reagan; a = Avery; f = friend of; k = Kruschev.

 

1.  ~[s (n)] +   v (r)

2.                             ~a + v (f (h))

3.                ~f ~(r)                      + ~[f (k)]

4.                                           ~h  +    f (k)

--------------------------------------------------

5. ~ [s (n)]  +           ~a      

 

That is, ~a + ~ [s (n)], or

 

a  É ~ [s (n)],  i.e., “Avery will not support Nixon.”  Note that in premiss 3, ~f ~(r) was originally ~[f (r)] before multiplying the negative through, and the ~[f (k)] is really the start of the premiss before converting it to its present form.

 

Finis for now.