1/15/201412.1: truth and validity in logical arguments expectations: l3.2.1: know and use the terms...

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.


Logical Argument Statements

Conclusion – final statement

Premises – all statements preceding the conclusion


Valid Argument

An argument is considered valid if the conclusion follows logically from the premises .


Valid Reasoning

If the premises are all true, then the conclusion will be ______.


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.


Is this a valid conclusion?

Some triangles are isosceles.

ABC is a triangle.

Conclude: ABC is an isosceles triangle.


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.


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.


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



If a triangle is a right triangle, then it has a right angle.

ΔABC is a right triangle.

Conclude: ΔABC has a right angle.


More Types of Arguments

Modus Tollens: Law of the Contrapositive

If p => q and ~q are true, then __________________

This is a ________ form of reasoning.


Denying the Antecedent

If p => q and ~p, then _____________.

This is a ___________ valid form of reasoning.



If x = 4, then x2 = 16.

x2 ≠ 16

Conclude: x ≠ 4



If x = 3, then x2 = 9.

x ≠ 3

Conclude: x2 ≠ 9


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.


The ___________ is a sufficient condition for the conclusion and the ___________ is a necessary condition of the hypothesis.


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.


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.


“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.


Notice simply saying AM ≅ MB does not guarantee that M is the midpoint of AB, so it is not a sufficient condition.


“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.


“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.


“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


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.



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.


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.


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.


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.


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.


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.


Assignment

pages 772 – 774,

# 7-15 (odds), 21-34 (all)

1/15/201412.1: truth and validity in logical arguments expectations: l3.2.1: know and use the terms...

Documents

logical arguments necessary

logical arguments expectations

conclusion slide

valid conclusion

true statement

statement p

following conclusion

true conditional statement