The Math Forum

Ask Dr. Math - Questions and Answers from our Archives
Associated Topics || Dr. Math Home || Search Dr. Math

Valid Arguments

Date: 12/17/2002 at 22:06:31
From: Siamak
Subject: Valid Arguments

What are the real-life applications for valid arguments?

I understand the concept of valid arguments and the theorems 
associated with them but I can't see the big picture. It seems like 
we use it subconsciously, but what are the specific uses in the modern 

Date: 12/18/2002 at 12:53:19
From: Doctor Achilles
Subject: Re: Valid Arguments

Hi Siamak,

Thanks for writing to Dr. Math. That's a very rich question.

Valid arguments allow us to start with a set of statements and reason 
our way to a conclusion or to a set of conclusions.

The statements we start with can be knowledge that we already have, 
such as conclusions from other arguments or statements that we know 
are true. The starting statements can also be assumptions that we 
pretend are true for the time being (a sort of logical fantasy to see 
what our assumptions might mean if they were true). Or the starting 
statements can be pieces of scientific ("empirical") data such as 
observations about certain phenomena. Often (perhaps always) the set 
of statements we start with consists of a mix of these three types.

The rules we use to get from starting statements to conclusion(s) 
depend somewhat on the field we are in. The basic rules of logic - see 
the Dr. Math crash course in symbolic logic: 

are used in all arguments. There are often other rules associated with 
reasoning in other fields. What makes an argument valid (or invalid) 
is the proper (or improper) application of the rules of reasoning.

The conclusion(s) we reach from our starting statements is(are) 
guaranteed to be true if (1) the starting statements were true and 
(2) the argument was valid (i.e. the rules of reasoning were properly 

I would assert that every field of work requires the ability to make 
valid inferences. In fact, in order to get through life, one must be 
comfortable reasoning. Consider the following exchange:

  Mother: Go mow the lawn.
  Child: I already did the dishes, so my brother should mow the lawn.

This may sound like a natural and obvious exchange. But there is 
actually some fairly involved reasoning going on on the part of the 
mother and child. Let's just consider what the child had to reason 
through in order to produce her reply. First, she had to recognize 
that mowing the lawn was part of a set of household chores. Then she 
had to go from that abstract set (household chores) back to a 
different concrete example (doing the dishes). She then brought in an 
assumption that chores should be split evenly between children, as 
well as the knowledge that there is one other child in the family, to 
reason that in order to make the chores even, the brother should mow 
the lawn. Notice that the mother could use the same set of starting 
statements together with a few others to turn the argument around:

  Mother: Your brother already did laundry, vacuumed the house, 
          and walked the dog; you should mow the lawn instead of him.

Notice also that neither mother nor child ever explicitly states that 
the number of chores each child does should be equal, but this 
assumption is clear in both of their reasoning.

Aside from daily life, valid reasoning is vital to all fields of work.  
I am a biologist. There is a certain set of facts about how cells work 
that is taken as true. From these facts and the experimental 
observations in the laboratory, I have to infer conclusions about 
other details of how cells work that have not yet been discovered by 
anyone. All the sciences (including social sciences like polital 
science and psychology) use an analogous process in making conclusions.

Valid inferences are vital in law and medicine as well. As a lawyer, 
you are given the laws themselves, the history of all the cases that 
have happened so far, and the particular facts of a case. From that, 
your goal is to convince a judge (and/or a jury) to come to a certain 
conclusion about what action the law requires. In medicine you are 
given the set of knowledge about how the human body works, the set of 
treatments that have been used and how they have worked, and the 
particular symptoms a patient has. From that, you must make valid 
inferences about what treatments will most help the patient (this 
always requires first inferring what is wrong with the patient - which 
can be difficult - and then deciding the best treatment).

Valid inferences are necessary for us to get through life. Any time we 
ever need to convince or persuade another person, the ability to make 
valid arguments - to understand what is taken as given and to explain 
how a certain conclusion must be true - is vital.

- Doctor Achilles, The Math Forum 
Associated Topics:
High School Logic
Middle School Logic

Search the Dr. Math Library:

Find items containing (put spaces between keywords):
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

Math Forum Home || Math Library || Quick Reference || Math Forum Search

Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.