4.01 Logic

The word logic as used in the phrase formal logic is the study of argument form. More specifically, logic is the study of methods and principles used in distinguishing correct from incorrect reasoning. Reasoning itself is the human capacity to make sense of the world. A well reasoned position (or view, or conclusion) is one that is carefully, thoughtfully and unambiguously constructed.  

Over time a precise language and structure was created and refined around logic which allowed for the systematic exposition of the logical form of a valid argument. It is for this reason that we begin our journey by defining certain key words and concepts. 

 

Propositions 

A proposition is any statement that has a definite truth value. That is to say a proposition is a statement that is either true or false, but never both true and false at the same time. We could easily assign a letter, say $T$T or $F$F to indicate a proposition's truth value. Statements to which we cannot assign a truth value cannot be propositions. 

For example, the statement All swans are white is a proposition because it is either true or false. It is not sometimes true and sometimes false, or true and false at the same time. It is a statement that we can assign a truth value to.

The statement Close the door is not a proposition because we cannot attach a truth value to it. The statement Dogs are better than Cats is not a proposition either, because it is based on an opinion - we cannot ascertain whether it is, in fact, true or false. The statement Siblings look alike is not a proposition because it is a statement that is sometimes true and sometimes false.

Naming propositions by variables

We use small letters to label propositions. Sometimes the letter selected relates to a key word in the proposition. Other times consecutive letters are used, such as the letters $p$p and $q$q.

In the strictest sense, the letter is actually not the proposition but is defined as a variable, just as the letter $x$x is used as a variable in coordinate geometry. Think of the letter as a label stuck on the outside of a box into which we could dump any proposition.

As an example we might write:

d	$:$:	The plate is dirty
m	$:$:	The moon is made of green cheese

Why we need variables for argument

By treating letters as variables, logicians can construct valid argument types using just letters rather than continually having to prove specific arguments with specific propositions. This idea is the first step into a powerful symbolic logic world where specific arguments (with specific propositions) can be validated simply by substitution into an established argument form.

By argument, we don't mean emotive argument where, for example, two people might have a disagreement about whether they prefer one sport to another. Rather what we mean is the careful formation of a valid conclusion from a set of other propositions that are assumed to be true (these are called premises).

As a simple illustration of what we mean when we say that letters are variables, consider the following two arguments:

Argument 1:

$p$p	$:$:	All primes greater than $2$2 are odd
$q$q	$:$:	$101$101 is a prime greater than $2$2
Conclusion	$:$:	$\therefore$∴   $101$101 is odd

Argument 2:

$p$p	$:$:	All Polar bears are white
$q$q	$:$:	Fluffy is a Polar bear
Conclusion	$:$:	$\therefore$∴ Fluffy is white

Without going into the details, it is clear that the two arguments fit into an argument type, so it is perfectly reasonable to see the words as substitution instances of the letters $p$p and $q$q. More will be said on argument types later.

Worked examples

Question 1

Question 2

Question 3

Question 4