1/15/201412.1: truth and validity in logical arguments expectations: l3.2.1: know and use the terms...
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04/10/23 12.1: Truth and Validity in Logical Arguments
12.1: Truth and Validity in Logical Arguments
Expectations:L3.2.1: Know and use the terms of
basic logic L3.3.3: Explain the difference between a necessary and a sufficient condition within the
statement of a theorem.
04/10/23 12.1: Truth and Validity in Logical Arguments
Logical Argument Statements
Conclusion – final statement
Premises – all statements preceding the conclusion
04/10/23 12.1: Truth and Validity in Logical Arguments
Valid Argument
An argument is considered valid if the conclusion follows logically from the premises .
04/10/23 12.1: Truth and Validity in Logical Arguments
Valid Reasoning
If the premises are all true, then the conclusion will be ______.
04/10/23 12.1: Truth and Validity in Logical Arguments
What conclusion follows from the premises:
If a polygon is a square, then it is a rectangle.
If a polygon is a rectangle, then it is a parallelogram.
ABCD is a square.
04/10/23 12.1: Truth and Validity in Logical Arguments
Is this a valid conclusion?
Some triangles are isosceles.
ABC is a triangle.
Conclude: ABC is an isosceles triangle.
04/10/23 12.1: Truth and Validity in Logical Arguments
Types of Arguments
Modus Ponens: The Law of Detachment
If (p => q) is a true conditional statement and p is a true statement, then ___________________________.
This is a valid form of reasoning.
04/10/23 12.1: Truth and Validity in Logical Arguments
Affirming the Consequent
If (p => q) is a true conditional statement and q is a true statement, then p must be true.
This is _____ a valid form of reasoning.
04/10/23 12.1: Truth and Validity in Logical Arguments
Determine if the following conclusion is valid or invalid.
If 2 lines are parallel, then they do not intersect.
l does not intersect m.
Conclude: l is parallel to m
04/10/23 12.1: Truth and Validity in Logical Arguments
Determine if the following conclusion is valid or invalid.
If a triangle is a right triangle, then it has a right angle.
ΔABC is a right triangle.
Conclude: ΔABC has a right angle.
04/10/23 12.1: Truth and Validity in Logical Arguments
More Types of Arguments
Modus Tollens: Law of the Contrapositive
If p => q and ~q are true, then __________________
This is a ________ form of reasoning.
04/10/23 12.1: Truth and Validity in Logical Arguments
Denying the Antecedent
If p => q and ~p, then _____________.
This is a ___________ valid form of reasoning.
04/10/23 12.1: Truth and Validity in Logical Arguments
Determine if the following conclusion is valid or invalid.
If x = 4, then x2 = 16.
x2 ≠ 16
Conclude: x ≠ 4
04/10/23 12.1: Truth and Validity in Logical Arguments
Determine if the following conclusion is valid or invalid.
If x = 3, then x2 = 9.
x ≠ 3
Conclude: x2 ≠ 9
04/10/23 12.1: Truth and Validity in Logical Arguments
Necessary and Sufficient Conditions
In the statement of a theorem in “if- then” form, we can talk about sufficient conditions for the truth of the statement and necessary conditions of the truth of the statement.
This is really just another way of looking at the Law of Detachment and Affirming the Consequent.
04/10/23 12.1: Truth and Validity in Logical Arguments
The ___________ is a sufficient condition for the conclusion and the ___________ is a necessary condition of the hypothesis.
04/10/23 12.1: Truth and Validity in Logical Arguments
Necessary
Consider the statement p => q. We say q is a necessary condition for (or of) p.
Ex: “If if is Sunday, then we do not have school.”
A necessary condition of it being Sunday is that we do not have school, but it is not sufficient to say it must be Sunday if we do not have school.
04/10/23 12.1: Truth and Validity in Logical Arguments
Sufficient Condition
A sufficient condition is a condition that all by itself guarantees another statement must be true.
Ex: If you legally drive a car, then you are at least 15 years old.”
Driving legally guarantees that a person must be at least 15 years old.
04/10/23 12.1: Truth and Validity in Logical Arguments
“If M is the midpoint of segment AB, then AM ≅ MB.”
Given that M is the midpoint, it is necessary (true) that AM ≅ MB.
This means that M being the midpoint is a ____________ condition for AM ≅ MB.
04/10/23 12.1: Truth and Validity in Logical Arguments
Notice simply saying AM ≅ MB does not guarantee that M is the midpoint of AB, so it is not a sufficient condition.
04/10/23 12.1: Truth and Validity in Logical Arguments
“If a triangle is equilateral, then it is isosceles.”
A triangle having 3 congruent sides (equilateral) guarantees that at least 2 sides are congruent, so a triangle being equilateral is sufficient to say it is isosceles.
04/10/23 12.1: Truth and Validity in Logical Arguments
“If a person teaches mathematics, then they are good
at algebra.”Because Trevor is a math teacher, can we
conclude he is good at algebra. Justify your answer.
04/10/23 12.1: Truth and Validity in Logical Arguments
“If a person teaches mathematics, then they are good
at algebra.”Betty is 32 and is very good at algebra. Can
we correctly conclude that she is a math teacher? Justify.
Which is not a sufficient condition for 2 lines being coplanar?
A. they are parallel
B. they are perpendicular
C. they intersect
D. they have no common points
E. they have 2 common points
04/10/23 12.1: Truth and Validity in Logical Arguments
Which of the following is a necessary but not sufficient condition for angles to be
supplementary?
A. they form a linear pair.
B. their angle measures add to 180.
C. they are both right angles.
D. their angle measures are 135 and 45.
E. none of the above.
04/10/23 12.1: Truth and Validity in Logical Arguments
04/10/23 12.1: Truth and Validity in Logical Arguments
Bi-Conditional Statements
If a statement and its converse are both true it is called a bi-conditional statement and can be written in ________________ form.
04/10/23 12.1: Truth and Validity in Logical Arguments
Ex:“If an angle is a right angle, then its measure is
exactly 90°” and “If the measure of an angle is exactly 90°, then it is a right angle” are true converses of each other so they can be combined into a single statement.
04/10/23 12.1: Truth and Validity in Logical Arguments
Necessary and Sufficient
If a statement is a bi-conditional statement then either part is a necessary and sufficient condition for the entire statement.
Remember all definitions are bi-conditional statements.
04/10/23 12.1: Truth and Validity in Logical Arguments
A triangle is a right triangle iff it has a right angle.
Being a right triangle is necessary and sufficient for a triangle to have a right angle and possessing a right angle is necessary and sufficient for a triangle to be a right triangle.
04/10/23 12.1: Truth and Validity in Logical Arguments
Necessary, Sufficient, Both or Neither
Given the true statement:
“If a quadrilateral is a rhombus (4 congruent sides), then its diagonals are perpendicular.”
Is the following statement necessary, sufficient, both or neither?
The diagonals of ABCD are perpendicular.
04/10/23 12.1: Truth and Validity in Logical Arguments
Necessary, Sufficient, Both or Neither
Given the true statement:
“A quadrilateral is a rhombus if and only if its 4 sides are congruent.”
Is the following statement necessary, sufficient, both or neither?
The sides of ABCD are all congruent.
04/10/23 12.1: Truth and Validity in Logical Arguments
Assignment
pages 772 – 774,
# 7-15 (odds), 21-34 (all)