5. Predicate Logic and Black Cops
1 Fallacy proof by example concerning institutional racism
Firstly, just because an institution, regime, whatever has officers, who
are of the ethnicity that they a suppressing, genociding, murdering, etc.,
doesn’t mean that that regime is not racist. This is not to say that because
some police officers are racist all police officers are racist. A claim like
that would be an inappropriate use of generalization. This is the fallacy proof
by example. This occurs when the validity of a statement is illustrated through
one or more examples or cases as opposed to a full-fledged proof.
I know that K is such.
Therefore, anything related to K is also such.
I know that T, which is a member of group K,
has the property H.
Therefore, all other elements of K must have
the property H.
2 Summary of Predicate Logic
To better understand the claim that all police are racist because a few
are racist (a claim that no one is probably making) we must journey into
predicate logic[i].
Predicate logic, like some that fascist monster Salvador Dali, has many names
including First-order logic, quantificational logic. First-order predicate
calculus uses quantified variables over non-logical objects, and allows the use
of sentences that contain variables. Instead of propositions like as
"Brianna is a woman",
We can now have expressions in the form
"there exists x such that x is Brianna
and x is a woman",
where "there exists" is a
quantifier,
while x is a variable.
(Brianna is the subject and woman is the
predicate)
Propositional logic
does not use quantifiers or relations in the same way. But propositional logic
is the foundation of first-order logic, or Predicate logic.
Predicate logic[1]
integrates the most powerful features of categorical and propositional logics,
thereby allowing for a more extended scope of argument analysis than either of
the two can achieve individually.
Predicate logic contains
all the components of propositional logic, including propositional variables
and constants. In addition, predicate logic contains terms, predicates,
and quantifiers. Terms are typically used in place of nouns and
pronouns. They are combined into sentences by means of predicates. For example,
in the sentence " T'Challa loves
Kiki",
the nouns are " T'Challa " and "Kiki",
and the predicate is
"loves".
The same is true if this
sentence is translated into predicate logic, except that " T'Challa " and "Kiki"
are now called terms. Predicate logic uses quantifiers to indicate if a
statement is always true, if it is sometimes true, or it is never true. In this
sense, the quantifiers are used to correspond words such as "all",
"some", "never", and related expressions. There are two
types of statements in predicate logic: singular and quantified. These include;
A singular statement is about a specific person, place, time,
or object. A quantified statement is about classes of things.
Such statements are either universal or particular. There are two elements in a
singular statement: predicate and individual constant. The predicate of
a singular statement is the fundamental unit, and is translated with a capital
letter, A-Z. The subject of a singular statement is called an individual
constant, and is translated with a lowercase letter, a-w: Individual
variables, the lowercase letters, x, y, and z, are
enlisted as placeholders in quantified statements. That’s because quantified
statements do not specify things, only classes of things. Things are included
in, or excluded from, classes:
All dogs are mammals.
No dogs are cats.
Some dogs are beagles.
Some cats are not friendly animals.
Notice that quantifiers
and classes are features of predicate logic borrowed from categorical logic.
What is borrowed from propositional logic are the logical operators, ~, •, v, ⊃, ≡:
Ordinary Language Statement
Function
All dogs are mammals. Dx ⊃ Mx
No dogs are cats. Dx ⊃ ~Cx
Some dogs are beagles. Dx •
Bx
Some cats are not friendly
animals. Cx • ~Fx
The statement
functions, above, are expressions that do not make any universal or
particular assertion about anything; therefore, they have no truth value. Their
variables are free, which means we don’t know how many things we’re
talking about. “All” and “no” are universal quantifiers. They are translated as
follows: (x). “Some” is a particular quantifier, and is translated
as follows: ($x).
Ordinary Language Predicate
Logic Translation
All dogs are mammals. (x)(Dx ⊃ Mx)
No dogs are cats. (x)(Dx ⊃ ~Cx)
Some dogs are beagles. ($x)(Dx
• Bx)
Some cats are not friendly
animals. ($x)(Cx • ~Fx)
Notice that the
appearance of the quantifiers includes parentheses around what are otherwise
statement functions. These parentheses tell us the domain of discourse,
which is the set of individuals over which a quantifier ranges. The variables
in the statement function are bound by the quantifier:
“For any x, if x is
a dog, then x is a mammal.”
“For any x, if x is
a dog, then x is not a cat.”
“There is at least one x that
is a dog and a beagle.”
“There is at least one x that
is a cat and not a friendly animal.”
There are four quantifier
rules of inference that allow you to remove or introduce a quantifier:
including Universal Instantiation (UI),
Universal Generalization (UG), Existential Generalization (EG) and Existential
Instantiation (EI).
In
Universal Instantiation (UI) we consider what a
universally quantified statement asserts. It asserts that the entirety of the
subject class is contained within the predicate class. Therefore, any instance of
a member in the subject class is also a member of the predicate class.
All dogs are mammals.
Joe is a dog.
Therefore, Joe is a mammal.
In Universal
Generalization (UG) we move from a universally quantified
statement to a singular statement. Although it may seem like we can go in the
other direction, we can’t. Consider the following statement: “Sam the horse is
a Clydesdale.” We cannot infer from this statement that all horse
are Clydesdales. When we want to make an inference to a universal
statement, we cannot do so from an individual constant. The generalization must
be made from a statement function (expressions that do not make any universal
or particular assertion about anything; therefore, they have no truth value),
where the variable, by definition, could be any entity in the relevant
class of things.
The next rule is the Existential
Instantiation (EI). Just as we have to be careful about
generalizing to universally quantified statements, so also we have to be
careful about instantiating an existential statement. When you instantiate an
existential statement, you cannot choose a name that is already in use. That’s
because we are not justified in assuming that the individual constant is the
same from one instantiation to another. So, if you have to instantiate a
universal statement and an existential statement, instantiate the existential
first. The universal instantiation can then assert the same constant as the
existential instantiation, because there are no restrictions on UI.
When
this rule is not used correctly we run into a fallacy called the existential fallacy. This occurs when one presupposes
that a class has members when one is not supposed to do so. For example:
·
All haters will be smacked.
·
Therefore, some of those smacked will have been haters.
This is a fallacy because the first statement does not require the
existence of any actual haters (stating only what would happen in the event
that some do exist), and therefore does not prove the existence of any. Note
that this is a fallacy whether or not anyone has hated.
You
can infer existential statements from universal statements, and vice versa,
without having to instantiate first. In ordinary language, the phrase
equivalences are as follows:
“All are,” is equivalent to, “It’s
not the case that there is one that is not.”
“It is not the case that all are not,”
is equivalent to, “Some are.”
“Not all are,” is equivalent to,
“Some are not.”
“It is not the case that there is
one,” is equivalent to, “None are.”
A
fallacy that occurs when the quantifiers of a statement are erroneously
transposed is the quantifier shift.
For every A, there is a B, such that C. Therefore, there
is a B, such that for every A, C.
1.
Every person has an organ
that is their brain. Therefore, there is an organ that is the brain of every
person.
∀x∃y(Px → (Wy & M(yx))) therefore ∃y∀x(Px → (Wy & M(yx)))
It
is fallacious to conclude that there is one organ who is the
brain of all people. However, if the major premise ("every
person has an organ that is their brain") is assumed to be true, then it
is valid to conclude that there is some organ that is any
given person's brain.
2. Everybody has something to believe in. Therefore, there is something
that everybody believes in.
∀x∃y Bxy therefore ∃y∀x Bxy
It
is fallacious to conclude that there is some particular concept to
which everyone subscribes.
The
last rule we will look at is Existential Generalization. Existential
Generalization (EG) like UI, is a fairly
straightforward inference. Take the sentence “Sam the horse is a Clydesdale
horse.” The sentence specifies an existing Clydesdale horse. So, if Sam is one,
it follows that at least one Clydesdale horse exists.
It
is from this rule that we come to the fallacy proof by example. Although the flaw is pretty clear
arguments like,
I've
seen police kill black people.
Therefore,
police must be killers.
can
seem convincing. Unfortunately this fallacy has been used to make statistically
insignificant examples appeal credible. This is why we must look at these cases
on an individual basis.
This
is why we can’t say every police officer is racist, but can we say the same
about the institution? Getting into the individual predicates and claiming some
universality is problematic, however claiming that the institution, both
historically and currently is racist is plausible. But more so saying that
because a there are black police the institution isn’t racist is another
fallacy.
3 The Fallacy of Necessity, Black Cops and
Jewish soldiers in the Wehrmacht
If I
were to claim that;
Institutions
with minorities necessarily aren’t racist
The
institution that is the police has minority officers
That
institution cannot be racist.
This is also a fallacy of
necessity. Because the
conclusion presumes the second premise will always be the case. As the history
of police shows us that institution, atleast in America and depending on what
part of the country, was founded on a public service that had the express purpose
of protecting property, in the South mostly meant slave (also across the
nation, to be fair).
There is another example coming for Germany. This is my favorite counter
to the “Institutions
with minorities necessarily aren’t racist” argument.
Saying that institution
isn’t racist, because there are black officers, is like saying the Nazis aren’t
anti-Semitic because of Jewish soldiers in the Nazi army.
[1] Predicate language consists of the
Syntax, the alphabet and rules for formation.
Semantics: Questions
connecting models: determining truth values, interpretation. How to translate
certain sentences into natural language, and the exact conditions for their
truth. Proof theory: The principle of calculus, axioms (the basic truths of the
system) and inference rules. Theorems are deduced from axioms by means of
inference rules.
[i] Definition
1.1.
Alphabet of Predicate Logic consists of
connectives (familiar from propositional logic) ¬, ∧, ∨, → and ↔and parentheses
(, ) and dot ,. Other symbols
belonging to alphabet are
individual constants: a, b, c, . . .
variables: x, y, z, . . .
predicate symbols P, Q, R, . . .
function symbols f, g, . . .
identity symbol =
quantifiers ∀ (Truth for all thing
“universal qualifier) and ∃ (is
the existential quantifier and its means something is true for at least one
thing; "There exists an x such that A").
In predicate logic, conditional and
indirect proof follow the same structure as in propositional logic:
1. Assume
a statement, P, derive another statement, Q, then discharge the assumptive
proof by asserting, P ⊃ Q.
2. Assume
a statement, ~P, derive a contradiction, then discharge the assumptive proof by
asserting, P.
In predicate logic, however, there is one
restriction on UG in an assumptive proof: when the assumption is a free
variable, UG is not allowed from the line where the free variable occurs. That
is because the assumption names an individual assumed to have the property
designated by the predicate. It does not, therefore, act as an arbitrary
individual from which we may generalize to a universal statement.
When demonstrating invalidity there are
two methods one can use in predicate logic: Counterexample Method and Finite
Universe Method.
The counterexample method follows the
same steps as are used in Chapter 1: replace the premises with another set we
know to be true; replace the conclusion with one we know to be false. Each
replacement must follow the same form as the original:
All oranges are fruits.
Some vegetables are not fruits.
Some oranges are not vegetables.
(x)(Ox ⊃ Fx)
($x)(Vx • ~Fx)
($x)(Ox • ~Vx)
All citizens are eligible to vote.
Some people are not eligible to vote.
Some citizens are not people.
The finite universe method enlists
indirect truth tables to show, truth-functionally, that a predicate logic
argument is invalid:
3. Suppose
a universe that contains only one member.
4. Instantiate
the premises and conclusion to the same constant.
5. Construct
an indirect truth table to determine whether or not the argument is invalid.
6. If
the argument does not prove invalid with a single-member universe, try two
members.
Note: if you do not prove the argument is
invalid assuming a three-member universe, double-check your work and then
consider using the inference rules to construct a proof. It may be that the
argument is, in fact, valid.
Predicate logic notation allows us to
work with relational predicates (two- or more place predicates), rather than
only single-place predicates:
Joe is to the right of Stew: Rjs
Joe and Stew like each other: Ljs • Lsj
Everyone likes someone: (x)(Px ⊃ ($y)Lxy)
Relational predicates include a number of
different types:
Symmetrical relationship:
7. If
A is married to B, then B is married to A.
Asymmetrical relationship:
8. If
A is the father of B, then B is not the father of A.
Nonsymmetrical: When a relationship is
neither symmetrical nor asymmetrical. For example: If Kris loves Morgan, then
Morgan may or may not love Kris.
Transitive relationship:
9. If
A is taller than B, and B is taller than C, then A is taller than C.
Intransitive relationship:
10.
If A is the mother of B, and B is the mother of C, then A is not the mother of
C.
Nontransitive relationship:
11. If
Kris loves Morgan and Morgan loves Terry, then Kris may or may not love Terry.
Proofs involving relational predicates
require an additional restriction on UG:
12. Universal
generalization cannot be used if the instantial variable is free in any line
that was obtained by existential instantiation (EI).
One of the central and most important
ideas in logic is identity. Identity is the relation between two named
individuals.
T'Challa = Black Panther
(p=q)
There is also negates Identity (“ ~ “ is
the negate sign)
T'Challa is not Black Panther
~(p=q)
With Identity we can translate almost
statement into a logical proposition.
Blank is the only President of the United
States
Pb&( ∀x) (Px > (x=b))
(“>” is a rule of implication. “If
this..then that..”)
Lower case letters are often used as
non-specific individuals
a b c d
There are Properties of Relations (or
rules of inference for Identity
Equivalence
Reflexivity
Symmetry
Transitivity
Equivalence
Reflexivity
All things are identical to themselves
Symmetry
If some a is identical to b, then b is
identical to that a
Transitivity
If “a” fulfills some function “F” and “a”
is identical to “b”, then b fulfills F
Fa
a = b
Fb
This only works in extensional contexts.
Not in intensional contexts
Example
Kiki knows that T'Challa was born in
Wakanda
Kjm
T'Challa is Black Panther
m=s
Kiki knows that Black Panther was born in
Wakanda
Kjs
Kxy is x knows that y was born in Wakanda
This is an invalid statement, because
Jane might not know T'Challa is Black Panther. She might not even know who
Black Panther is. If we don’t have the qualifier of “Kiki knows on the second
statement then we can’t conclude that our conclusion that Kiki Knows that Black
Panther was born in Wakanda.
Statements about knowledge, or beliefs,
are going to be intensional contexts
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