BK Writing Lab

Logic Basics - Deductive Reasoning
Paul DiGeorgio, Ph.D.

 

This page will explain the basics of deductive reasoning. Deductive reasoning is a specific type of argumentation where you combine a number of statements (usually referred to as "premises" in logic) in order to arrive at a conclusion that is "logically certain." Just because something is logically certain, however, does not mean that it is actually true, which can be confusing!

In order to see how a logical necessary conclusion is not necessarily true, let's examine validity and soundness in deductive reasoning.

A valid argument is one that has a structure or form that will lead to a certain conclusion. Valid does not mean true!  A valid argument is not necessarily true. Consider the following argument with two premises and a conclusion:

If it's raining outside, then it's March. 

It's raining outside. 

Therefore, it's March.

This is a valid argument, but we know that it's not actually true. It doesn't only rain in March, after all. Validity does not guarantee truth; instead, validity is a feature of an argument that has to do with whether something is logical, and not whether something is true or a fact. When you first start studying logic it can be very difficult to understand this point, since in regular conversation many people use the word "valid" to mean "true."

In logic we use a different word to capture the truth of an argument: a sound argument is one that is valid and all of its premises are true. A sound argument will always be valid and true! Ideally in your writing you will present arguments that are both valid and sound. 

If it's raining outside, then the plants are wet.

It's raining outside.

Therefore, the plants are wet.

This argument sounds much more reasonable, right? This argument is not only valid, but also sound. 

Let's take a closer look at the main two types of valid deductive reasoning: modus ponens and modus tollens

Modus Ponens = If p then q 

Also known as "affirming the antecedent," which means that 'p' is affirmed to be the case. In all of the modus ponens samples on this page, the second premise is the affirmation of the antecedent, i.e., p.

The arguments above take this form. Here is yet another example:

If you ate, then you were hungry. [If p, then q]

You ate. [p]

Therefore, you were hungry. [Therefore q]

 

Modus Tollens = If not q, then not p

Also known as "denying the consequent," which means that 'q' is determined to not be the case. The conclusion from an argument like this will always be 'not p'. This is trickier than modus ponens and it can be harder for students to see why this is a valid type of argument.

Consider this example:

If you ate, then you were hungry. [If p, then q]

You were not hungry. [not q]

Therefore, you did not eat. [Therefore not p]

 

Or another example:

If he has money, then he worked. [If p, then q]

He did not work. [not q]

Therefore he does not have money. [Therefore not p]

 

Common Logical Fallacies

You want to be careful to avoid invalid forms of deductive reasoning in your writing. Some students struggle to see why the two examples below are invalid, but with examples it will be easier for you to understand. The two types of invalid deductive reasoning that we will review are "denying the antecedent" and "affirming the consequent." The point here is that neither of these arguments work.

Denying the antecedent:

If you ate, then you were hungry. [If p, then q]

You did not eat. [not p]

Therefore you were not hungry. [Therefore not q] 

This argument is invalid. Think about the first premise and how it is structured on an "if, then" basis, "if p, then q." If p is not the case, we cannot conclude from that that q is not the case. 

 

Affirming the consequent:

If you ate, then you were hungry. [If p, then q]

You were hungry. [q]

Therefore you ate. [Therefore, p]

This argument is invalid. Think about the first premise and how it is structured on an "if, then" basis, "if p, then q." If q is the case, we cannot conclude from that that p is the case, because we started we "if p, then q" and not "if q, then p." 

 

If you have any questions, send us an email at bkwritinglab@bishopkennyhs.org!

 

Image Credit: Logic by Thuy Nguyen from the Noun Project

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