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Arguments with Quantified Statements
Lecture 10
Section 2.4
Thu, Jan 27, 2005
Universal Modus Ponens
The universal modus ponens argument form:
x S, P(x) Q(x)
P(a) for a particular a S
Q(a)
Example
Let F be the set of all functions from R to R.
f F, if f is differentiable, then f is continuous.
The function f(x) = x2 + 1 is differentiable. Therefore, f(x) is continuous.
An Argument within in Argument
f F, if f is differentiable, then f is continuous.
f F, if f is a polynomial, then f is differentiable.
The function f(x) = x2 + 1 is a polynomial. Therefore, f(x) is differentiable. Therefore, f(x) is continuous.
Universal Transitivity
The previous example could have been handled differently using the argument form of universal transitivity:
x S, P(x) Q(x)
x S, Q(x) R(x)
x S, P(x) R(x)
Universal Transitivity
Equivalently,
P(x) Q(x)
Q(x) R(x)
P(x) R(x)
Universal Modus Tollens
The universal modus tollens argument form:
x S, P(x) Q(x)
~Q(a) for a particular a S
~P(a)
Diagrams
The statement
x S, P(x) Q(x)
means that the truth set of P is a subset of the truth set of Q.
The statement P(a) means that a is in the truth set of P.
Therefore, a must be in the truth set of Q.
Diagrams
Therefore, we can represent these statements by using Venn diagrams.
a
Truth set of Q
Truth set of P
A Diagram for Universal Modus Ponens
x S, P(x) Q(x)
P(a) for a particular a S
Q(a)
a
Truth set of Q
Truth set of P
Example
Continuous functions
Example
Continuous functions
Differentiable functions
Example
Continuous functions
Differentiable functions
Polynomial functions
Example
Continuous functions
Differentiable functions
Polynomial functions
f(x) = x2 + 1
Diagrams
Recall the example that showed that
x R, x/(x + 2) 3 x -3.
Example
{x R | x -3}
{x R | x/(x + 2) 3}
Example
A better representation:
0 1 2 3 4-4 -3 -2 -1
Statements with “No”
Rewrite the statement
“No HSC student would ever lie”
using quantifiers. ~(x {HSC students}, x would lie) x {HSC students}, ~(x would lie) x {HSC students}, x would not lie Thus, this is a universal statement.
Arguments with “No”
Which arguments are valid?No HSC student would ever lie.
Joe is an HSC student.
Therefore, Joe would never lie.No HSC student would ever lie.
Buffy is an RMC student.
Therefore, Buffy would lie.
Arguments with “No”
People
HSC Students Liars
The diagram shows that Joe cannot be a liar.
Joe
Statements with “No”
Note that the following two statements are equivalent.No HSC student is a liar.No liar is an HSC student.
Arguments with “No”
People
HSC Students Liars
Where would we place the oval for RMC students?
Joe
Arguments with “No”
People
HSC Students Liars
Where would we place the oval for RMC students?
RMCStudents
?
Arguments with “No”
People
HSC Students Liars
Where would we place the oval for RMC students?
RMCStudents
?
Arguments with “No”
People
HSC Students Liars
Where would we place the oval for RMC students?
RMCStudents
?
Arguments with “No”
People
HSC Students Liars
Where would we place the oval for RMC students?
RMCStudents
?
Arguments with “No”
People
HSC Students Liars
Where would we place Buffy?
RMCStudents
Arguments with “No”
People
HSC Students Liars
Where would we place Buffy?
RMCStudents
Buffy
Arguments with “No”
People
HSC Students Liars
Where would we place Buffy?
RMCStudents
Buffy
Arguments with “No”
People
HSC Students Liars
Where would we place Buffy?
RMCStudents
Buffy
Arguments with “No”
Which fallacy is committed in the “Buffy” argument?
A Logical Conclusion?
Is the following argument valid? x, y, z, if x is better than y and y is
better than z, then x is better than z. A peanut butter sandwich is better than
nothing. Nothing is better than sex. A peanut butter sandwich is better
than sex.