(Rough Draft)
by Vern Crisler
Copyright June, 1999
(Last Update: March 30, 2002).
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.
There
is more than one way to represent a relation, but the trick is to make one’s
symbols perspicuous and easy to use.
One way of representing relational propositions is by the use of the
symbolism from logical algebra with the additional device of superscripts. For instance, to say that “Every boy loves
some girl,” we could write, “B^{L}aG.” This is a fairly perspicuous symbolism. The L superscript is the relation “loves”
while the “a” means “every.” Another
way to represent relations would be to use functional symbolism, as in ~B +
L(g). This latter symbolism will come
in handy later. (I assume that the
reader is probably already familiar with current textbook symbolism for
relations.)
In
the superscript symbolism, 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^{--L}iB
The
use of the negative sign for passive voice or for the converse of a relation
will become clear later when we 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 in mathematics is here introduced to symbolize universal predicate
quantity. The relational proposition,
“Some girl (or other) is loved by every boy” would thus be symbolized as
follows:
(2) G^{--L}i|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) B^{L}aG
Or
in English, “Every boy loves some girl.”
In general then, any expression of the form,
(4) F^{--R}i|Y|
can
be converted to,
(5) Y^{R}aF
Here
are some examples of relational expressions in active and passive voice using
suggested symbolism:
Active |
Symbolized |
Passive |
Symbolized |
1. Abelard loves Heloise. |
A^{L}aH |
Heloise
is loved by Abelard. |
H^{--L}aA |
2. Plato taught Aristotle. |
P^{T}aA |
Aristotle
was taught by Plato. |
A^{--T}aP |
3. All the boys married (some of) the girls. |
B^{M}aG |
Some
of the girls were married by all the boys. |
G^{--M}i|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 “I_{1}”
represent the first individual, and “I_{2}” 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 I_{n} to be a
satisfying symbolism, while others may find the symbols I* or İ or Î or
even I™, satisfactory. At this point,
however, I leave it to the discretion of the reader. (My basic criterion of symbolic “fitness” would be whether the
symbol is easily duplicated in email formats.)
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 refer to the same type of 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 jam 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).
Bertrand
Russell sought to use the above type of functional symbolism in his logical
works. In this type of logic (hereafter
called F&R logic, for Frege & Russell), “x” is the argument and “f” is
the incomplete function, needing to be saturated by the input value “x” – like
a sponge needing to be saturated by water.
Hence, (x) (f_{1}x É
f_{2}x) means that for all x, if x is an f_{1},
then x is a f_{2}. (The
subscripts 1 and 2 are used in order to distinguish between two different
functions.) In my opinion, F&R
were mistaken in regarding f_{1 }or f_{2} as
functions in the above argument.
Actually, symbols such as f_{1} or f_{2}
are merely place holders 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)( f_{1}x É f_{2}x). 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 anything that “x” stands for. 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)
[( f_{1}x É
f_{2}x)]
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 f_{1}x implies the
second expression f_{2}x.
Hence, we would need to use if-then(x) or i(x), as in the
following expression:
i
(x) {q ("x)
[( f_{1}x É
f_{2}x)]}
This would seem to bring F&R symbolism more in line with mathematical symbolism. I would like to introduce functional symbolism in this essay as an alternative to my earlier symbolism of logical algebra, 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 express “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|))
In
this case, I’ve adopted the & sign to stand for conjuction. It’s a lot easier to duplicate the & in
email formats than it is to duplicate the dot sign. Leaving out the sign of conjunction would give us the following:
b
(l (|g|))
But
this would be somewhat ambiguous, so let’s proceed with the & sign for
conjunction. 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.
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”
In
symbolizing this, I’ll give the symbolism in logical algebra first, then in
arithmetical symbolism.:
(6) [«R] A^{R}aB É B^{R}aA [~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.
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] A^{R}aB É B^{R}eA [~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.
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] A^{R}aB É (B^{R}aA Ù
B^{R}eA); [~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. (If anyone can suggest a better symbol,
please contact me.)
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] A^{R}aB É A^{R}aA · B^{R}aB [~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.
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] A^{R}aB + B^{R}aC = A^{R}aC ~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] N^{A}aD + D^{A}aM = N^{A}aM
This
is easily solved by the method of logical algebra, crossing out the linking
term D, hence,
(12) N^{R}e~D + D^{R}e~M = N^{R}e~M (or N^{R}aM)
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.
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] A^{R}aB + B^{R}aC = A^{R}aC ~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]
A^{R}aI + I^{R}aJ + J^{R}aY =
We
solve (14) by means of logical algebra:
(15) A^{F}e~I + I^{F}e~J + J^{F}e~Y = A^{F}e~Y (or, A^{F3}aY)
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) R^{M}aJ + J^{F}aY =
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) R^{M}e~J + J^{F}e~Y = R^{M2F1}e~Y (or R^{M2F1}aY)
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.”
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.” W_{1}iL; 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|.
Another
way of symbolizing cross reference would be to use “footnotes,” that is,
subscripts that keep track of the subjects involved. For instance, to borrow a sentence from Sommers, we could
symbolize the expression, “Some person is a sailor and it is not the case that
some person is a child and some person fails to give some person a toy” as:
(P_{1}
& S) & ~[(P_{2} & C) & (P_{1} (~G (P_{2}
(T)))]
Notice
how the subscripts 1 and 2 make it impossible to confuse one person with
another person in the same sentence. In
my opinion, the use of subscripts and absolute value signs for universal
quantity eliminates the need to worry about pronoun cross referencing, or about
the scope of the quantifier. In the
above sentence, if either P_{1} or P_{2} had been universally
quantified at least once in the sentence, i.e., as |P_{1}| or |P_{2}|,
they would have been universally quantified throughout.
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) J^{TCB}aP ~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) P^{L}aH 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, (though from now on, in arithmetical symbolism, I’ll
leave universal quantification of individuals as implicit rather than explicit,
and will leave out absolute value signs--but just for arithmetical symbolism).
As
we have seen, the passive voice, “Helen is loved by Paris” can be expressed in
the following way:
(20) H^{--L}aP 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) P^{L}iH + HaV =
Solving,
we have,
(22) P^{L}o~H + He~V = P^{L}o~V (or, P^{L}iV)
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) S^{G}iM + MaT =
Solving
by logical algebra, we have:
(24) S^{G}o~M + M^{G}e~T = S^{G}o~T (or, S^{G}iT)
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 + P^{D}iC =
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 + P^{D}o~C = P^{D}o~F (or, P^{D}iF)
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) B^{GR}iG + RaF +
GaC =
By
cancelling the R’s, and solving the first part of the equation, we have:
(28) B^{GR}iG + RaF = B^{GF}iG
Or,
“Some boy gave a flower to a girl.”
Combining this conclusion with the remaining proposition, we have,
(29) B^{GF}iG + GaC = B^{GF}iC
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) B^{L}aG + G^{A}a|C| + (P^{A}a|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 G^{A}a|C| is an instance of P^{A}a|C| and
thus we have the resulting proposition:
(31) G^{A}a|C| + P^{A}a|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) G^{A}a|C| + C~^{A}a|P| =
We
then use logical algebra to solve the equation, cancelling both the C’s and the
A’s:
(33) G^{A}e~|C| + C^{--A}e~|P| =
Ge~|P| (or, Ga|P|)
(34)
Ga|P| + PaF = GaF
Once we have GaF, we can then combine it with the
first proposition in (30), giving us:
(35) B^{L}aG +
GaF = B^{L}aF
Cancelling the G’s, the conclusion is, “Every boy
loves (some) fool.”
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) H^{F}iD + DaM =
(52) H^{F}o~D + De~M = H^{F}iM
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. r &
B
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, r &
B, “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.