Introduction

Logic is the part of Philosophy that studies reasoning. More specifically, Logic studies:

• How to define the concept of logical consequence.

• The forms of valid reasoning.

• The methods that distinguish valid from invalid reasoning.

• Logical fallacies: invalid reasoning that seems valid.

Aristotle (4th century BC) was the first philosopher to study reasoning in detail, finding forms of valid reasoning. Since then, Logic has developed as one of the most important formal sciences.

Logic is a key tool to all three scientific methods:

• Deduction uses proofs (demonstrations) as its main method of acquiring new knowledge. Valid proofs are proofs that follow valid rules of reasoning, that is logical reasoning.

• Induction analizes large amounts of data in order to extract general conclusions. Logic provides methods for extracting valid conclusions, the empirical sciences provide the initial data.

• Hypotesis plus deduction employs deduction for extracting conclusions from initial hypothesis, so logic is again employed for the advance of knowledge.

Proving validity

Valid arguments are the subject matter of Logic. The validity of an argument only depends on its form: the logical form of the premises and of the conclusion. The content of an argument is of no interest to Logic, only its logical form.

Arguments are expressed in natural language (in English, or Spanish or in any other language spoken by human beings). Premises and conclusion are sentences of a natural language. We need to know the logical form of natural language sentences in order to study the validity of an argument. So the tasks ahead are two:

1. From natural language sentences to their logical forms.

2. Study the logical form of an argument in order to decide its validity. 

In this section we will restrict these two tasks to a very concrete (and simple) type of deductive arguments: the arguments first analized by Aristotle in the IVth century B.C. Since then, Logic has advanced tremendously and more powerful (and complex) tecniques have been developed for analising more complex arguments. 

Logical form

Let's consider the following arguments:

Are you convinced by these arguments? Do you think the conclusion is necessarily true if the premises are true. Although the content is different, premises and conclusion have the same logical form in both cases.

Let's now consider these other arguments:

Are you convinced by these arguments? Do you think the conclusion is necessarily true if the premises are true. Again, although the content is different, premises and conclusion have the same logical form in both cases.

In order to understand (and to prove to yourself and to others) that the first pair of arguments are valid while the second pair are not, we need to understand first which are the logical forms behind them. Since these are very simple arguments, their logical form is equally simple, we just need some concepts.

Basic concepts

These are concepts we will not define; they are simple enough to be understood by themselves:

• The concept of element or object: any entity that is singular and different from other entities.

• The concept of set: any grouping of elements forms a set.

• The concept of membership of an element in a set: any element may or may not belong to a given set.

Derived concepts

Defined in terms of the basic conceps:

• The empty set: a set without elements.

• The intersection set: given two sets, the intersection set is a new set that includes as its members all the elements that are members of the two initial sets. When there are no common elements, the intersection set is the empty set.

• The difference set: given two sets, the difference of the first one minus the second set is a new set including only the elements of the first set that are not members of the second. If all the elements of the first set are members of the second, then the difference from the first to the second is the empty set.

• Equality between two sets: two sets are equal when they have the same elements as members.

Some notation

In order to study logical forms, it is most useful to write them, and for that we need some notational conventions:

• Elements: a, b, c, ...

• Sets: A, B, C, ...

• Membership: ∈

• Not a member: ∉

• Empty set: Ø

• Intersection: ∩

• Difference: −

• Equality: =

• Inequality: ≠

In addition to all the previous symbols, we will use the symbol ∴ for indicating the conclusion formula in an argument. The symbol ∴ separates the premises from the conclusion.

With all these concepts and a way to write them, we can write the logical form of the previous arguments:

In the above formulas, we have applied the following conventions (they are not essential, other symbols could be used, but is convenient to state them just for clear communication):

• A for the set of persons in the first argument and for the set of targaryans in the second.

• B for the set of mortals in the first argument and for the set of dragon lovers in the second.

• a for Socrates and also for Daenerys.

The formulas are the same although the sentences are different. This is an expected result since the sentences have the same logical form although their content is different (they talk about different things).

We can do the same exercise with the second pair of arguments:

This time the same symbols are used to represent different contents:

• A for the set of elephants in the first argument and for the set of rock stars in the second.

• B for the set of trunk owners in the first argument and for the set of people loved by fans in the second.

• a for Dumbo and also for Quevedo.

Again, the formulas are the same since they have the same logical form but different content.

Just looking at the logical forms of the first pair and the second pairs of arguments we can spot the differences and maybe even understand why the first pair are valid arguments and the second pair are  invalid. Arguments having the first logical form are always valid, arguments of the second form are always invalid.

Graphical demonstration of validity and invalidity

There is a simple way of demonstrating the validity or invalidity of an argument such as the previous ones: we just need to draw the logical forms of the premises:

1. First, draw the two premises in a diagram.

2. Second, without drawing the conclusion, inspect the diagram and look if the conclusion is already drawn. If it is, the argument is valid: the conclusion follows from the premises. But if the conclusion is not drawn, then the argument is invalid.

Let's look at the graphical representations of two of our previous arguments:

Extending our logic

The arguments we are going to analyse are not as simple as the ones already seen. They may contain four different types of sentences, so we need to use all our symbols and also to extend our graphical technique for proving the validity (or invalidity) of a wider variety of arguments.

Once we able to translate several kinds of sentences from natural language to formulas, we should be able to demonstrate the validity or invalidity of more complex arguments. For that, we need to extend our graphical technique accordingly.

Let's consider the following argument:

1. No comet has a metallic core.

2. Some asteroids have a metallic core.

3. Therefore some asteroids are not comets.

Formalization

• Comets : C

• To have a metallic core: M

• Asteroids: A 

C ∩ M = Ø,  A ∩ M ≠ Ø  ∴  A − C ≠ Ø

Is this a valid argument? Let's make a drawing:

In the third, final diagram, with both premises already drawn, we can see that the conclusion A − C ≠ Ø is already drawn, since there is an element in the zone A − C. So, if the premises are true, the conclusion is necessarily true; we have proven the validity of the argument.

Logical consequence, validity and correctness

We can now provide general definitions of these three important logical concepts.

• An argument is valid when there is a relation of logical consequence between its premises and its conclusion. Thus, arguments are valid or invalid depending on whether there is or there isn't logical consequence.

• There is logical consequence (symbolized ∴ ) between premises and conclusion when the truth of the premises implies necessarily the truth of the conclusion. In other words, it is impossible that the premises are true and the conclusion is false.

It is very important to understand that the relation of logical consequence is a conditional relation: if all the premises are true, then the conclusion must be true. But if just one of the premises is false, then the conclusion may be true or false: the relation is broken.

Because of this conditional nature of the logical consequence, a valid argument does not guarantees the truth of the conclusion. It only guarantees the truth of the conclusion provided that the premises are true. The truth of the conclusion is conditional to the truth of the premises.

If validity does not guarantees the truth of the conclusion, then we need another concept for expressing that:

• An argument is correct when it is valid and its premises are true. Correcteness requires two ingredients:

  • one is provided by Logic: validity

  • the other is provided by science, common sense or any other source of true information. 

Examples:

1. All unicorns are magical. Socrates was a unicorn. Therefore Socrates was magical is a valid argument: if the premises were true, the conclusion had to be true. However, it is not correct since the premises are false.

2. There are red frogs. Therefore there are frogs that are not red is an invalid argument: there is no logical consequence from the premise to the conclusion; it is possible for the premise to be true and the conclusion false. Although both the premise and the conclusion are true on Earth now, it is possible (maybe in the future or in the past) that the only frogs that exist are red, making true the premise but false the conclusion.