# Logical Argument

Pronunciation: /ˈlɒdʒ.ɪ.kəl ˈɑr.gju.mənt/ Explain

A logical argument consists of one or more premises followed by one or more conclusions. The conclusion of a logical argument may be valid or invalid.

A premise is a statement that helps support a conclusion. Examples of premises are:

• The sun is yellow.
• Socrates is a man.
• Two distinct lines in a plane either intersect or are parallel.

A conclusion is a statement that may follow from the premises. If the logic in the argument is used correctly, the conclusion is valid. If the logic in the argument is not used correctly, the conclusion is invalid. Examples of conclusions are:

• The sun is a class G star.
• Socrates is mortal.
• Two distinct lines in a plane intersect exactly zero or one times.

The letters p and q are often used to represent premises when discussing logical argument in general. The statement:
if p then q
means, "If the statement p is true, then it follows that the statement q is true."

There are five types of logical arguments that, if used correctly, will have a valid conclusion. If these types of logical arguments are used incorrectly, any conclusions will be invalid.

NameDefinitionExample
Direct Argument If p is true then q is true. If a shape is a square, then it is a rectangle.
p is true. HIJK is a square.
Therefore q must also be true. Therefore HIJK must also be a rectangle.
Indirect Argument If p is true then q is true. If a shape is a square, then it is a rectangle.
q is not true. HIJK is not a rectangle.
Therefore p can not be true. Therefore HIJK can not be a square.
Chain Rule if p is true, then q is true. If a shape is a square, then it is a rectangle.
if q is true, then r is true. If a shape is a rectangle, then it is a parallelogram.
Therefore, if p is true, then r is true. Therefore, if a shape is a square, then it is a parallelogram.
Or Rule Either p is true or q is true. Figure A is a circle or a square.
p is not true. Figure A is not a circle.
So q must be true. So figure A must be a square.
And Rulep and q are not both true. Figure A is not both a circle and a square.
q is true. Figure A is a square.
So p must be false. So figure A can not be a circle.
Table 1: Five types of logical arguments

### Understanding Check

Examine each logical argument. Identify the type of logical argument in the list in table 1. Click on Type of logical argument to see if your answer is correct. Then decide if the conclusion is valid or invalid. Click on Valid or invalid to see if your answer is correct.

Logical argumentType of logical argument Is argument valid or invalid
if y = 3 then y2 - 4 = 5.
y2 - 4 ≠ 5.
Therefore y ≠ 3.
Type of logical argument Valid or invalid
Child A is either a boy or a girl.
Child A is not a boy.
Therefore child A is not a girl.
Type of logical argument Valid or invalid
Table 2: Understanding check
1. McAdams, David E.. All Math Words Dictionary, logical argument. 2nd Classroom edition 20150108-4799968. pg 110. Life is a Story Problem LLC. January 8, 2015. Buy the book

McAdams, David E. Logical Argument. 4/24/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. https://www.allmathwords.org/en/l/logicalargument.html.

### Revision History

4/24/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)