6.03 Deductive reasoning

Argument

In the real world, we often use the word argument in the sense of disagreement or dispute that can at times end in unhelpful and heated exchanges. In the world of logic, however, the word argument has a very specific meaning.

An argument in logic consists of two parts separated by some symbol (perhaps the "therefore" symbol $\therefore$∴ , or even a line between). The first part consists of a series of propositions called premises and the second part is called a conclusion. 

 

Symbols in arguments

If we denote the premises of an argument $P1$P1, $P2$P2, $P3$P3,... and call the conclusion $C$C, then arguments have the form:

$\left(P1\wedge P2\wedge P3\wedge...\right)\Rightarrow C$(P1∧P2∧P3∧...)⇒C  

The conjunction symbol $\wedge$∧ shows that every premise is required to form the conclusion (although sometimes we can show that certain premises can be simplified). The conditional $\Rightarrow$⇒ shows that we are deriving a conclusion from the premises and not the other way around.  

 

Argument forms

An argument is said to be a valid argument if it has a valid argument form.  A form of argument is valid if and only if the conclusion necessarily follows from the given set of premises.

It is the task of logicians to establish these argument forms using truth tables. 

To explain the idea of argument forms, consider these two simple arguments:

Argument 1

$P1$P1	If I go to school I will make friends
$P2$P2	I went to school
$C$C	Therefore I made friends

 

Argument 2

$P1$P1	If I eat my breakfast I will stay healthy
$P2$P2	I ate my breakfast
$C$C	Therefore I stayed healthy

While the two arguments concern different concepts, they are said to have the same form.

Look at the first proposition $P1$P1 of each row. Each expresses a conditional statement of the form $p\Rightarrow q$p⇒q where $p$p and $q$q are simple propositions (for example, in Argument 1 we have $p$p being "I go to school" and $q$q being "I will make friends"). 

Recall that we regard such propositions $p,q,r...$p,q,r... as variables just like $x$x is a variable in coordinate geometry. We can substitute actual propositions like "I eat my breakfast" or "I will stay healthy" whenever we wish.

Hence if we wish to prove either of these arguments, all we need to do is to prove the validity of the argument form given here using the letters $p$p and $q$q: 

$P1$P1	$p\Rightarrow q$p⇒q
$P2$P2	$p$p
$C$C	$\therefore$∴     $q$q

Once we do that, then any particular argument with the same form will also be valid. This is why we use variables.

This particular argument is in a valid argument form known as the law of detachment or otherwise known as modus ponens or the cut rule.

 

Some important argument forms

We have already met the law of detachment (modus ponens).  Here are other important forms that are always valid. It is important to note that these forms have been proven by logicians using a method known as a truth table.

 

The law of detachment (modus ponens)

$P1$P1	$p\Rightarrow q$p⇒q
$P2$P2	$p$p
$C$C	$\therefore q$∴q

 

The law of contrapositive (modus tollens)

$P1$P1	$p\Rightarrow q$p⇒q
$P2$P2	$\sim q$~q
$C$C	$\therefore\sim p$∴~p

 

The law of syllogism

$P1$P1	$p\Rightarrow q$p⇒q
$P2$P2	$q\Rightarrow r$q⇒r
$C$C	$\therefore p\Rightarrow r$∴p⇒r

 

The law of disjunctive syllogism

$P1$P1	$p\vee q$p∨q
$P2$P2	$\sim p$~p
$C$C	$\therefore q$∴q

 

The law of contrapositive:

$P1$P1	$p\Rightarrow q$p⇒q
$C$C	$\therefore\sim q\Rightarrow\sim p$∴~q⇒~p

 

The law of constructive dilemma

$P1$P1	$p\vee q$p∨q
$P2$P2	$p\Rightarrow r$p⇒r
$P3$P3	$q\Rightarrow s$q⇒s
$C$C	$\therefore r\vee s$∴r∨s

 

The fallacy of the converse

There is a very common fallacy known as the Fallacy of the Converse, otherwise known as a Converse Error. Consider the following argument.

• John: "John, you said that if I get the flu, I'll have a sore throat" 
• Dr Jane "Yes Jane, that's right"
• John "So, guess what, I've got a sore throat!"
• Dr Jane "Ouch, sorry to hear that"
• John "So that means I've got the flu, right?"

It can be proven using truth tables that John's argument is not correct. Here's what it would look like using symbols.

Like a good logician, we will start by defining propositions:

P	$:$:	John has the flu
q	$:$:	John has got a sore throat

 

In logical symbols, we can represent the reasoning part of the discussion as:

$P1$P1	$p\Rightarrow q$p⇒q
$P2$P2	$q$q
$C$C	$\therefore p$∴p

 

It follows that any argument of this form is not valid.

 

Practice questions

Question 1

Question 2

Question 3