Introduction to Deductive Reasoning and Mathematics II

Share Cognitive Science Implications for Teaching Mathematics

Inductive reasoning is the process of reasoning that a general principle is true because the special cases you’ve seen are true. I will have a bit more to say about this in a future post.
Deductive reasoning is the kind of reasoning in which the conclusion is necessitated by, or reached from, previously known facts (the premises). If the premises are true, the conclusion must be true.
In this essay we are concerned with Deductive Logic which is also known as Deductive Reasoning, Logical Deduction, Formal Logic, Top-Down-Logic, or The Science of the Formal Laws of Thought.
Deductive reasoning proceeds from general principles to a specific case.
In a typical deductive argument, the premises guarantee the truth of the conclusion, while in an inductive argument, they are thought to provide reasons supporting the conclusion’s probable truth.

A deductive argument is said to be valid or invalid.
Two equivalent definitions of the term valid:

  1. An argument is valid if its conclusion follows with certainty from its premises.
  2. An argument is valid if the truth of all its premises forces the conclusion to be true.

The structure, and only its structure, of a deductive argument determines whether the argument is valid. The actual truth values of individual statements do not affect validity of the argument.

Do not confuse valid with true.

The most famous example of a valid deductive argument is

  Socrates is a man.

All men are mortal.

  Therefore, Socrates is mortal.

The first two statements are called the premises, and the third statement is called the conclusion. By the rules of deduction, if the first two statements are true, the conclusion must be true.

A deductive argument is valid independent of the truth value of the premises and conclusion.

The following is a valid deductive argument

  All ducks play golf.

  No one who plays golf is a dentist.

  Therefore, no ducks are dentists.

I think that we all would agree that both premises are false, and the conclusion is true.

Another valid argument but in this instance both premises are false and the conclusion is also false.

   All toasters are items made of gold.
All items made of gold are time-travel devices.
Therefore, all toasters are time-travel devices.

If we assume that A, B and C each represent statements, then we can generalize each of the above examples as

A is B
C is A
Therefore, C is B

These examples clearly illustrate the following very important fact about valid deductive arguments.

IF the premises are true, THEN the conclusion must be true.

If the premises (one or more) happen to be false, then the conclusion might be true, or it might be false.

If you like the posts of this blog, share them with your friends and associates.  Encourage them to read the post every Tuesday and Thursday.

Use the Social Media buttons to  share this essay with friends and associates.  Use the like button to indicate your approval.

2 thoughts on “Introduction to Deductive Reasoning and Mathematics II

  1. Hi, this is Mark, we met in STL at the pen show. I’m late to the game, but I’m here. It was good to meet you and this place looks amazing.

    1. Glad to hear from you Mark. To get the most out of the blog, you should read post starting in February. Also read the pages. That’s a big assignment, but this stuff is sequential.

Leave a Reply

Your email address will not be published. Required fields are marked *