Today’s Topics
• Limits to the Usefulness of Venn’s Diagrams
• Predicates (Properties and Relations)
• Variables (free, bound, individual,
constants)
Limits to Venn’s diagrams
• We cannot construct a proper diagram (no region
left out, none duplicated, same shape for classes)
for an argument involving 6 or more terms.
• We CAN construct 16 and 32 region “Venn type
diagrams” for both 4 and 5 term arguments, but
they are really hard to use.
• This limit is linked to what mathematicians call
the 4 color problem. Any may can be colored
with 4 colors and have no adjoining regions the
same color.
An Imposter Venn-type diagram for
a 4 term argument
A 16 region Venn-type diagram for a
4 term argument
A 16 region diagram (different shapes
for classes)
A 32 region Venn-type diagram for a
5 term argument
A 64 region diagram (different shapes
for classes)
128 regions
Venn diagrams are good for jokes
Things that aren’t
supposed to be funny
Why I’m
going to
Hell.
Things I find funny.
The Joker “Venn Diagram”
Propositional and syllogistic
logic have serious limitations.
• Some arguments that are clearly valid cannot be
shown valid in our system (remember
DeMorgan’s valid sentence).
• Propositional logic misses the internal structure of
sentences.
• Syllogistic logic cannot deal with more than 4
terms, nor can it deal with relations.
• We need a new, more powerful, tool: Predicate
Logic.
The two central concepts in
predicate logic are:
• The Predicate (Property Constant)
• The Variable
A PREDICATE is either a property
of an object or a relation between a
group of objects.
• A predicate is represented symbolically
by a capital letter followed by one or
more lower case variable letters.
• Hx = ‘x is happy’
• Txy = ‘x is taller than y’
Monadic and Polyadic
(Relational) Predicates
• A predicate that expresses a property
of an object is monadic, it applies to
only one thing.
• A predicate that expresses a relation
between objects is polyadic or
relational, it applies to an ordered set
of objects.
A VARIABLE is a place-holder
in a formulaic expression.
• An individual variable is a true place
holder, expressed symbolically with lower
case letters taken from the end of the
alphabet, t, u, v,...z.
• An individual constant stands for a specific
individual, represented by a lower case
letter taken from the beginning of the
alphabet).
Predicates plus variables allow us to
describe individual and relations
more fully
• Let Hx = ‘x is happy’ and Txy = ‘x is
taller than y’ and ‘a’ be the individual
constant for Alice and ‘b’ the
individual constant for Bob.
• Hb says that Bob is happy.
• Tba says that Bob is taller than Alice
Sentences containing free
individual variables are called
open sentences and have no truth
value.
• Hx says only that x, whoever that is, is
happy. Since the value of x is open, we
can’t assign a truth value to Hx.
• Replacing free variables with individual
constants turns the open sentence into a
closed sentence with a truth value.
Another way to close an open
sentence is to BIND the free
individual variables with
QUANTIFIERS.
• There are only 2 quantifiers in English:
All and Some.
Symbolizing quantifiers
• The universal quantifier, all, is represented
with an upside down A-- -- followed by a
variable letter ( x says ‘for any x’)
– Some systems of logic (the text, LogicWorks)
use a variable in parens—(x)--as the symbol for
the Universal Quantifier. We shall simply be
symbolically polyglot.
• The existential quantifier, some, is
represented with a backwards E---followed by a variable letter ( x says ‘there
exists an x such that’)
Any variable that falls within the
scope of a quantifier is bound by that
quantifier (see pp. 362-372).
• The parentheses following a quantifier mark
its scope.
• xHx says “everybody is happy.”
• x(Txb  Hx) says “someone is taller than
Bob and that person is happy.”
• If all the variables in a sentence are either
bound or individual constants, the sentence
is closed.
• So, while ‘Hx’ is an open sentence,
‘ xHx,’ in which the second ‘x’ is a bound
variable, is closed and means ‘someone is
happy.’
• However, in ‘xHx  Txy’ both the third x
and the y are free, outside the scope of any
quanitfier, and thus the sentence is open.
In which, if any, of the following
WFF’s are there free variables?
1.
2.
3.
4.
5.
6.
7.
x(Fx  (Ga ● Hx))
x(Fx  (Ga ● Hy))
xFx  y(Fy ● Rxy)
x(Fx  y(Fy ● Rxy))
xFx  y(Fy ● Gy)
x(Tx  (Se▼Bxe))
x(Tx  y(Sy▼Bzy))
Answers
•
•
•
•
•
•
•
1.
2.
3.
4.
5.
6.
7.
No free variables
The ‘y’ is free
The second ‘x’ is free
No free variables
No free variables
No free variables
The ‘z’ is free
Symbolizing with Quantifiers:
•
•
The material inside the parenthesis following a
quantifier is called the matrix of the formula.
The dominant operator in the matrix of a
universally quantified proposition will almost
always be the conditional.
1. The word “are” indicates the dominant operator
2. Relative clauses (All ’s who are ’s are ’s)
indicate a compound antecedent.
•
The dominant operator in the matrix of an
existentially quantified proposition will almost
always be conjunction.
Common Errors in Symbolizing
with Quantifiers
• Sentences beginning with “A” do not follow
strict rules:
– ‘A barking dog never bites’ is a universal
claim, but
– ‘A barking dog is in the road’ is an existential
claim.
Common Errors in Symbolizing
with Quantifiers
• Sentences beginning with “A” do not follow
strict rules:
• ‘He who’ sentences are universal claims
Common Errors in Symbolizing
with Quantifiers
• Sentences beginning with “A” do not follow
strict rules:
• ‘He who’ sentences are universal claims
– ‘He who lives by the sword dies by the sword’
is a universal claim
‘Common Errors in Symbolizing
with Quantifiers
• Sentences beginning with “A” do not follow strict
rules:
• ‘He who’ sentences are universal claims
• ‘Any’ and ‘every’ are not synonymous when
following negations
– ‘Hamner is not taller than any NBA player’ is false, but
– ‘Hamner is not taller than every NBA player’ is true.
Common Errors in Symbolizing
with Quantifiers
• Sentences beginning with “A” do not follow
strict rules:
• ‘He who’ sentences are universal claims
• ‘Any’ and ‘every’ are not synonymous
when following negations
• The problem of ‘only’
• The problem of ‘only’
– In English sentences beginning with
‘only,’ the grammatical subject is the
logical predicate. ‘Only freshmen are
eligible’ means ‘All who are eligible are
freshmen.’
Troubling occurrences of ‘and.’
• Sometimes ‘and’ does not signal conjunction.
– ‘Hamner and Peggy are married’ indicates a
relational predicate, not a conjunction
– ‘Women and children are exempt’ says that
whoever is either a woman or a child is exempt,
NOT that whoever is a woman/child is exempt.
– ‘Some dogs and cats do not make good pets’ does
not, the cartoon notwithstanding, indicate that
there are cat-dogs who do not make good pets
Try some on your own.
• Download the Predicate Study Guide from
the Handouts section and review it.
• Download the Handout entitled Predicate
Translation Exercises and try some. Post
and discuss your answers. Be careful to use
quantifiers when necessary, but remember
that we do not quantify across individual
constants—names have a fixed value.