1. Introduction to Basic Logical Thinking – Deduction, Induction

 

 

 

Abstract. In this lecture I will explain the basics of logical reasoning, including the function of premises, logical fallacies and the distinction between form and content. I will discuss deduction and induction and their relation to each other.

 

 

 

1       Introduction

Look at this statement “So, as we all know the earth is flat. It is because of our planets flatness that the earth cannot possibly revolve around the sun. So the sun must not only be smaller than the earth it must also move around this flat earth like a flash light over a map.”

This is not a logical argument. I’ll explain why later, but first what is logic?

 

Logic is the study of reasoning and its aim is to distinguish correct from incorrect reasoning by establishing the rules or patterns of successful arguments.

Logic is not concerned with the actual mental (or physical) process employed by a thinking entity when it is reasoning.

The investigation of the actual reasoning process falls more appropriately within the province of psychology, neurophysiology, etc.

It is important to have a background in logic if you are pursuing a career, seeking to further your education or even simply for complex everyday decision making.

For example if you were told that if you made a pair of wings you could jump from a tall structure and fly like a bird.

Would you do it?

You probably wouldn’t because this is considered a fallacy.

 

 

 

Logical Fallacies

The previous fallacy is called Affirming the consequent (this is also a predicate logic fallacy).

This fallacy occurs when one takes a true conditional statement and invalidly infers its converse even though the converse may not be true.

For example:

Birds can fly because they have wings

I made a pair of wings out of a couple of pieces of computer paper.

Therefore I can fly.

You may not know the particular name of the fallacy but you know that you can’t just make a pair of wings out of some sheets of paper and fly.

Logical Fallacies are common errors in reasoning that will undermine the logic of your argument.

They can be either

1.   illegitimate arguments or

2.   irrelevant points,

Logical Fallacies often identified because they lack evidence that supports their claim.

 

 

 

 

 

Inferring

But how does logical reasoning work?

Reasoning is a special mental activity called inferring.

This can also be called making (or performing) inferences.

To infer is to draw conclusions from premises.

Thus Logic is the study of making inferences.

In place of the word ‘premises’, you can also put: ‘data’, ‘information’, ‘facts’.

Examples of Inferences:

(1) You see smoke and infer that there is a fire.

(2) You count 19 persons in a group that originally had 20, and you infer that someone is missing.

Inferences are made on the basis of various sorts of things – data, facts, information, states of affairs.

In order to simplify the investigation of reasoning, logic treats all of these things in terms as a single sort of thing – statements.

The word ‘statement’ is intended to mean a declarative sentence.

An argument is a declarative sentence, which is to say a sentence that is capable of being true or false.

A statement is a sentence that is either true or false. Statements (or propositions) have a truth-value.

 

 

 

Statements

      The following are examples of statements.

◦     it is raining

◦     I am hungry

◦     2+2 = 4

Examples of sentences that are not statements

are you thirsty?

open the door, please

Logic correspondingly treats inferences in terms of collections of statements, which are called arguments.

An argument is a collection of statements, one of which is designated as the conclusion, and the remainder of which are designated as the premises.

The premises of an argument are intended to support (justify) the conclusion of the argument.

Indicator words help us identify the elements of an argument.

Conclusion indicators (such as “therefore,’” “so,” “it follows that”) alert you to the appearance of a conclusion, while premise indicators (such as “since,” “because,” “it follows from”) alert you to the appearance of a premise.

An explanation can sometimes be taken for an argument, and vice versa. Both arguments and explanations often use the same indicator words.

The difference is the conclusion is at issue in an argument.

Thus, even when an explanation involves indicator words, if there is nothing at issue, the passage does not become an argument:

“Much of Central Africa speaks French because that region was colonized by French speaking countries.”

Here, an explanation is offered for

- Much of Central Africa speaks French –

Which is an already accepted fact.

There is no intent to prove anything or settle some sort of issue.

 

 

 

Principle of Charity

We must remember to enlist the principle of charity and reconstruct arguments so that they give the benefit of the doubt to the person presenting the argument.

 

 

 

Form and Content

There is a distinction in logic between form and content.

There is a distinction in logic between arguments that are good in form and arguments that are good in content.

This distinction is best understood by way of an example or two.

(a1) all cats are dogs

all dogs are reptiles

therefore, all cats are reptiles

This argument is valid in form not in content.

The information is factually incorrect.

Remember that an argument is at its best when it is factual, valid, and sound.

An argument is factually correct if and only if all of its premises are true.

An argument is valid if and only if its conclusion follows from its premises.

An argument is sound if and only if it is both factually correct and valid.

 

 

 

Natural Deduction, validity and factuality

For an argument to be valid it must follow the rules of natural deduction. Another way to think of this caveat is that for an argument to be logical an argument must be correct in form.

The previous “cats are reptiles” argument is logical, even though it is not factual.

When we are assessing the truth of the premises, whether they are factual, we are doing argument analysis.

When both an argument is both logical and factual, we will consider it sound for our purposes.

The average person is not going to be familiar with all of the rules of natural deduction but here are a few;

1.  Modus Ponens or Law of Detachment (Modus Ponens = mode that affirms)

If A then B

A therefore B

In more accessible terms

If I wake up then I will eat that pancake

I woke up

I’m gonna eat that pancake.

·         Modus T Modus Tollens, (Denying the Consequent)

If A then B

not A therefore not B

In more accessible terms

If that toilet is dirty, then I going to fight somebody

The toilet is not messy

I’m not gonna fight somebody

There are a few more but they generally are not necessary unless you are going into a field that requires intensive study of formal logic, like philosophy of computer science, or mathematics, etc.

 

 

 

Two kinds of arguments

The two kinds of arguments that are utilized the most are deductive arguments and inductive arguments.

A deductive argument is one in which the conclusion is claimed to follow necessarily from the premises.

In other words, the premises are claimed to guarantee the conclusion, or it is impossible for the conclusion to be false if the premises are true.

An inductive argument is one in which the conclusion is claimed to follow with a degree of probability.

In other words, the premises make it likely for the conclusion to be true, or it is improbable that the conclusion is false if the premises are true.

 

Often you will come to a fact through induction and from there reason the consequences of adhering to that fact through deduction.

Both are equally important and neither makes sense without the other.

 

 

 

 

2       Deduction

A deductive argument is one in which the conclusion is claimed to follow necessarily from the premises.

In other words, the premises are claimed to guarantee the conclusion, or it is impossible for the conclusion to be false if the premises are true.

For example,

All jet aircraft in the air have jet engines. A Boeing 737 is flying in the air, thus it has a jet engine.

The scientific method uses deduction to test hypotheses and theories.

Deductive reasoning starts out with a general statement, or hypothesis, and examines the possibilities to reach a specific, logical conclusion.

In deductive inference, we make a prediction about the consequences of a theory.

“We predict what the observations should be if the theory were correct. We go from the general — the theory — to the specific — the observations”

In deductive inference, we make a prediction about the consequences of a theory.

 For example

1.         2+2=4

This is a deductive statement because our observations were draw from a theory we assumed was correct. It is not possible for the conclusion to be false if the premises are true.

 

 

Theory

 

                                                                                                                                

A theory in science is an explanation of a law.

A law describes a force, but makes no attempt to explain how the force works (like the gravitational force, or all living organisms arose in an evolutionary process).

A theory is an explanation of a natural phenomenon.

Einstein’s General Theory of Relativity explains how gravity works by describing gravity as the effect of curvature of four dimensional space-time. In deductive reasoning, our theory could be defined by us.

 

 

 

Deductive Syllogism

Deductive reasoning generally follows these steps.

First, there is a premise,

then a second premise,

and finally an inference (Conclusion).

A common form of deductive reasoning is the syllogism

A syllogism consists of two statements

1.         a major premise

2.         and a minor premise

From here we

3.         — reach a logical conclusion.

Below is an example of a categorical syllogism.

For example, the premise

“Every A is B”

could be followed by another premise,

“This C is A.”

Those statements would lead to the conclusion

“This C is B.”

Syllogisms are a great way to test deductive reasoning to make sure the argument is valid.

For example,

“All men are mortal.

Peter is a man.

Therefore, Peter is mortal.”

“All men are mortal” is a categorical term.

 

 

 

Categorical syllogism, terms and propositions

A categorical syllogism is an argument containing three categorical propositions: two premises and one conclusion.

All M are P.

All S are M.

All S are P.

A categorical term is something that will be categorized, like ‘hornet’ and ‘cat’.

It is usually a collective statement such as ‘all hornets’ or ‘some hornets’.

A categorical proposition is simply a statement about the relationship between categories.

It states whether one category or categorical term is fully contained with another, or is partially contained within another or is completely separate.

A hornet is an insect

Some hornets are friendly

No hornet is a cat

Propositions may have quality: either affirmative or negative.

They may also have quantity: such as ‘a’, ‘some’, ‘most’ or ‘all’. The ‘all’ quantity is also described as being universal and other quantities particular.

The first term in the proposition is the subject.

The second term is the predicate.

Some hornets (subject) are friendly (predicate)

A categorical term is said to be distributed if the categorical proposition that contains it says something about all members of that categorical term.

It is undistributed if the categorical proposition that contains it and it does not say something about all members of that categorical term.

There are four types of categorical proposition, each of which is given a vowel letter A, E, I and O.

A way of remembering these is:

Affirmative universal,

nEgative universal,

affIrmative particular and

nOgative particular.

 

 

 

Validity and soundness in deductive arguments

For deductive reasoning to be valid, the hypothesis must be correct. 

It is assumed that the premises, “All men are mortal” and “Peter is a man” are true.

Therefore, the conclusion is logical and true.

In deductive reasoning, if something is true of a class of things in general.

It is also true for all members of that class.

We judge deductive reasoning in terms of validity.

A deductive argument is valid when it is impossible for the conclusion to be false.

It is impossible for the conclusion to be false if the premises are true.

An invalid argument is one in which it is possible for the conclusion to be false, if the premises are true.

A sound argument is valid, and its premises are actually true.

All invalid arguments are, by definition, unsound.

A convenient test of validity is the counterexample method.

A counterexample to a statement is evidence that shows the statement is false, and it concerns truth value analysis.

It shows the possibility that premises assumed to be true do not make the conclusion necessarily true.

A single counterexample to a deductive argument is enough to show that an argument is invalid.

 

 

 

Deductive Fallacy

It’s possible to come to a logical conclusion even if the generalization is not true.

If the generalization is wrong, the conclusion may be logical, but it may also be untrue.

For example, the argument,

“All hairy men are grandfathers.

Harold is hairy.

Therefore, Harold is a grandfather,”

is logically valid but it is untrue because the original statement is false.

 

 

 

3       Induction

Earlier I explained that inductive reasoning tend to lead to facts and when we reason from said facts we tend to use deduction.

An inductive argument is one in which the conclusion is claimed to follow with a degree of probability.

In other words, the premises make it likely for the conclusion to be true, or it is improbable that the conclusion is false if the premises are true.

Inductive reasoning differs from deductive reasoning because it makes broad generalizations from specific observations.

We often use inductive reasoning when we need to make decisions based in evidence.

For example,

There is a large bowl of stew in your refrigerator.

It’s been there a few days.

Using deductive reasoning you might say

“My rule is to never eat food that’s been in the fridge past two days.”

But that’s not going to help you determine whether the food is safe to eat.

To do that you must think about times you’ve eaten food that is a few days old, consider the smell, the taste.

You must reason from specific experiences and not one rule or principle.

Inductive reasoning occurs when conclusions are drawn from some information.

When we evaluate inductive arguments, we use the following concepts: strong, weak, cogent, and uncogent.

 

 

 

Inductive Inference

“In inductive inference,

1.         we go from the specific to the general by making many observations.

2.         After making these observations we discern a pattern,

3.         we then make a generalization, and infer an explanation or a theory,” 

A strong inductive argument is one such that if the premises are assumed to be true, then the conclusion is probably true.

If the premises are assumed to be true, then it is improbable that the conclusion is false.

A weak inductive argument is one such that if the premises are assumed to be true, then the conclusion is not probably true.

An inductive argument is cogent when the argument is strong and the premises are true.

An inductive argument is uncogent if either or both of the following conditions hold: the argument is weak, or the argument has at least one false premise.

“In science, there is a constant interplay between inductive inference (based on observations) and deductive inference (based on theory), until we get closer and closer to the ‘truth,’ which we can only approach but not ascertain with complete certainty.” 

The following is an example of inductive logic,

1.         “The coin I pulled from the bag is a penny.

2.         That coin is a penny.

3.         A third coin from the bag is a penny.

4.         Therefore, all the coins in the bag are pennies.”

Induction makes broad generalizations from specific observations.

1.         If you smoke, you may get cancer

2.         If you play the lottery you might win the jack pot

3.         You see large amounts of water hitting your window, it might be raining.

In each of these statements we make broad generalizations from specific observations.

If you smoke, you may get cancer.

But you might not.

If you play the lottery you might win the jack pot.

But you probably won’t.

You see large amounts of water hitting your window, it might be raining. It might not be raining, it could be a broken sprinkler.

You don’t know for sure.

 This doesn’t mean that Induction isn’t valuable.

We need induction for scientific analysis, and we use induction every day.

Ask you self how many times have you decided based on past experience? 

With induction we know that after seeing enough people who have smoked cigarettes getting cancer, we know that smoking increases the likelihood of getting cancer. 

Or maybe you take a path to get to the cafeteria every day for the past week and it hasn’t been blocked, but you know that one day it could be.

You know that the route isn’t going to be free forever, but we do know, with deductive logic that 2+2 will always equal 4.

 

 

 

Inductive Fallacy

Even if all of the premises are true in a statement, inductive reasoning allows for the conclusion to be false:

1.         “Harold is a grandfather.

2.         Harold is bald.

3.         Therefore, all grandfathers are bald.”

The conclusion does not follow logically from the statements.

•           Inductive reasoning has its place in the scientific method. Scientists use it to form hypotheses and theories.

•           Deductive reasoning allows them to apply the theories to specific situations.

 

 

 

Zetetic method

Let’s look at the false flat earth theory inductively. In 1864 a great man named Samuel Birley Rowbotham wrote a book called Zetetic Astronomy: The Earth not a Globe. According to the Zetetic method sensory observations reign supreme.

It’s like extreme empiricism.

This empiricism is so extreme is would make George Berkeley, the man who coined the popular term esse est percipi (To be is to be perceived) be like calm down you are taking your sensory experiences way too seriously.

According to Berkeley reality only exists when we perceive it.

When we taste, hear, see or smell it.

When we are not experiencing a thing, like your house, God is watching it so it still exists.

Reality exists as long and someone is watching it! Anyway according to the Zetetic method our perception that the earth is flat must lead to the deduction that the earth is, in fact, flatter than a pancake.

When you walk around, our look out from a high place does the earth look round?

No, so taking all of your experiences of looking and walking on it would you conclude that the earth is round?

Bear in mind that the surface of the earth being flat doesn’t necessarily mean that it can’t have high points, like mountains and tall buildings.

 

 

 

4       Conclusion

We should rely on experts to help us determine if an argument is factual. And the experts tend to agree that the earth is round and it revolves around the sun.

While my flat earth arguments were valid they were not factual.

I have tended to notice student falling into the lack of factuality pretty often. So please utilize expert knowledge when making arguments. Assume nothing.

I would like you to cite every argument you make with a reliable source.