## Inference, Validity and Invalidity

Recall the primary goal of philosophical logic is to be able to distinguish between correct and incorrect reasoning. As it turns out, reasoning is exemplified by arguments, in other words, when one creates an argument one is participating in the process of reasoning.

Suppose we have an argument. Then by definition, we have a set of premises and a main conclusion. Let P be the conjunction of all of the premises (meaning we just take each individual premise, connect them each with the word "and" and form a larger proposition which we are calling P), and let C be the conclusion. We will call the truth functional relationship between P and C the argument's inference.

Definition: The truth functional relationship between the conjunction of an argument's premises and the argument's conclusion is called the argument's inference.

In Section 1 we explored three distinct and types of relations between statements, namely implication, relevance and independence. Since an argument is something created by humans (for now at least) then we can ignore any cases where all of the premises of an argument are actually independent of the conclusion. Since an argument's inference is the relationship between the argument's premises and its conclusion, we will classify arguments by whether the argument's inference is that of implication or relevance.

Let's illustrate by starting with an example.

Example 2.2.1

Suppose Emily notices the following pattern:

1 = 12

1+3 = 4 = 22

1+3+5 = 9 = 32

1+3+5+7 = 16 = 42

1+3+5+7+9 = 25 = 52

Emily points out that the pattern suggests that the sum of 1 odd number is 12 and the sum of the first 2 consecutive odd number is 22 and the sum of the first 3 consecutive odd numbers is 32 and so forth. Emily then tells you that she has a proof showing that the sum of the first n odd numbers is n2 for any whole number n. She then gives you that proof, which consists of a set of premises that lead to the conclusion that the sum of the first n odd numbers is n2. What does Emily mean when she says she has a proof?

In logic and mathematics (and systems which use formal reasoning) to say one has a proof of some proposition C is to say that if the premises one presents in the proof are true, then the conclusion C of the proof must be true. In other words, to prove a claim C is to offer reasons (premises) such that if those premises are true then the conclusion C must be true. To recast the notion of proof in terms of inference and relations between statements we introduce a key term used in logic which is more common than the term "proof".

Definition: Let P be the conjunction of each premise for a given argument, and let C be the argument's conclusion. If P implies C, then we say the argument is valid.

Equivalently, if an argument has the property that it is impossible to have all true premises and a false conclusion, then the argument is said to be valid.

In other words a valid argument is an argument whose inference between the conjunction of its premises and the conclusion is one of implication.

When the inference of an argument is one of implication, then the inference is said to have the validity property and the argument is said to be valid.

Note that an argument's inference is valid or not. We will use this fact as our first step in classifying arguments. As a result our first step in analyzing arguments will be to decide whether the inference of an argument is valid or not. We represent this step visually the following way:

The general question as to how one determines whether an argument's inference is valid is a question of such importance that it is worth thinking about for a few seconds before continuing to read. Check your thoughts on how this might be done by answering the question below.

Value: 10

Given an argument what is the best way to decide the argument's inference type?

The answer really should not be all that surprising. After all we already have a two-step method of checking whether any proposition p does not imply another proposition q, and validity is defined in terms implication. Hence we can extend the two step method we used to check to see if a proposition p implies another proposition q to check to see if an argument is valid or not.

The two step method for determining whether an argument is not valid.

To determine whether an argument is valid or not, many times it is easier to check to see if it is not valid. An argument is not valid only if both of the conditions [a] and [b] are true:

[a] It is possible for the argument to have all true premises, or in other words it could have been or could be the case that all of the premises are true.

[b] Under the possibility above, the argument can have a false conclusion.

An argument is not valid only if both of these conditions hold. The failure of one or both of these conditions means the argument is valid.

Think about conditions [a] and [b] as check boxes. To be able to place checks in both boxes means the argument's inference is not valid. If one can not place a check mark in both boxes, then the argument's inference is valid.

Let's see how this works by considering some exercises.

Exercise 2.2.2 Determine whether the following arguments have valid inferences:

(a)

If John makes the free throw, then the U of A will win the game.

Hence the U of A won the game.

(b)

If John makes the free throw, then the U of A will win the game.

John did not make the free throw

Hence the U of A lost the game.

(c)

If John makes the free throw, then the U of A will win the game.

John did not make the free throw

Hence the Thursday follows Wednesday in week-day ordering.

Solution:

For each argument we check first to see if we can think of some possible way both premises can be true. If so, we have checked condition [a]. If not, we stop and declare the argument to be valid (make sure you understand why we make this declaration, if you need to, re-read the previous paragraphs).

Second we check to see if under the possibility where all premises are true (condition [a]), it is possible to have a false conclusion. If so, then we have checked conditions [a] and [b] and hence conclude the inference is not valid. Otherwise we declare the argument's inference valid.

Now apply the above criteria and check your answers by completing the quiz below:

### Test your understanding: valid inferences

Value: 5

Which of the following arguments have valid inferences?