Logical reasoning (Arguments)

Introduction

There are different ways of thinking.  People in everyday life often do not think critically, and they think or decide things based on: 

1. traditions or customs,
2. emotions, 
3. religious beliefs,
4. popular opinions, or
5. ideology, etc.

Critical thinking is an approach to think clearly and rationally about what to do or what to believe mostly about the physical world.  Critical thinking can include different thinking methods or approaches.  A crucial part of critical thinking is to identify, construct, and evaluate arguments.

It should be noted that there are mental processes other than thinking and reasoning.  These processes include sensation, perception, memory, emotion, creativity, interaction with the body etc. 

What is an argument?

In everyday life, people often use "argument" to mean a quarrel between people. But in logic and critical thinking, an argument is a list of statements, one of which is the conclusion and the others are the premises or assumptions of the argument.

For present purposes, we shall take a statement to be any declarative sentence. A declarative sentence is a complete and grammatical sentence that makes a claim. So here are some examples of statements in English :

Snow is white.

The moon is made of green cheese.

Everyone is here.

As you can see, statements can be true or false, and they can be simple or complex. But they must be grammatical and complete sentences.

To give an argument is to provide a set of premises as reasons for accepting the conclusion. To give an argument is not necessarily to attack or criticise someone. Arguments can also be used to support other people's viewpoints.

Here is an example of an argument:

If you want to find a good job, you should work hard. You do want to find a good job. So you should work hard.

The first two sentences here are the premises of the argument, and the last sentence is the conclusion. To give this argument is to offer the premises as reasons for accepting the conclusion. [1]

The standard format

Presenting arguments in the standard format [2]

When it comes to the analysis and evaluation of an argument, it is often useful to label the premises and the conclusion, and display them on separate lines with the conclusion at the bottom :

Premise 1:  If you want to find a good job, you should work hard. 
Premise 2:  You do want to find a good job.
Conclusion:  So you should work hard.

Let us call this style of presenting an argument a presentation in the standard format. Please rewrite the following two arguments using the standard format:

1. We should not inflict unnecessary pain on cows and pigs. After all, we should not inflict unnecessary pain on any animal with consciousness, and cows and pigs are animals with consciousness.
2. If this liquid is acidic, the litmus paper would have turned red. But it hasn't, so the liquid is not acidic.

See answers here. 

Types of arguments

Deductive argument 

Deductive argument offers the strongest support: the premises ensure the conclusion, meaning that it is impossible for the conclusion to be false if all the premises are true. Such an argument is called a valid argument, for example: 

All men are mortal.

Socrates is a man. 

Therefore, Socrates is mortal. 

For valid arguments, it is not important whether the premises are actually true.  However, if they were true, the conclusion could not be false. Deductive reasoning plays a central role in formal logic and mathematics.

Non-deductive argument [3]

For non-deductive arguments (or logical reasoning), the premises make their conclusion rationally convincing without ensuring its truth. This is often understood in terms of probability: the premises make it more likely that the conclusion is true and strong inferences make it very likely. Some uncertainty remains because the conclusion introduces new information not already found in the premises. Non-deductive reasoning plays a central role in everyday life and in most sciences. 

Often-discussed types are:

inductive, 

abductive, and 

analogical reasoning. 

 

Inductive argument is a form of generalisation that infers a universal law from a pattern found in many individual cases. It can be used to conclude that "all ravens are black" based on many individual observations of black ravens. 

Abductive argument, also known as "inference to the best explanation", starts from an observation and reasons to the fact explaining this observation. An example is a doctor who examines the symptoms of their patient to make a diagnosis of the underlying cause. 

Analogical argument compares two similar systems. It observes that one of them has a feature and concludes that the other one also has this feature. [4]  The structure or form may be generalised like so:

P and Q are similar in respect to properties a, b, and c.

P has been observed to have further property x. 

Therefore, Q probably has property x also.

 

Analysis of deductive arguments

If an argument is valid, and all the premises are true (then by definition the conclusion is necessarily true) then the argument is called a sound argument.

Validity

An argument is valid if and only if there is no logically possible situation where all the premises are true and the conclusion is false at the same time.

Validity and Truth

Note the following situations:

The premises and the conclusion of an invalid argument can all be true.

Adam loves Beth. 
Beth loves Cathy. 
So Adam loves Cathy.

This argument is not valid, for it is possible that the premises are true and yet the conclusion is false. Perhaps Adam loves Beth but does not want Beth to love anyone else. So Adam actually hates Cathy. The mere possibility of such a situation is enough to show that the argument is not valid. 

The premises and the conclusion of a valid argument can all be false.

All pigs can fly. 

Anything that can fly can swim. 

So all pigs can swim.

Although the two premises of this argument are false, this is actually a valid argument. To evaluate its validity, ask yourself whether it is possible to come up with a situation where all the premises are true and the conclusion is false. (We are not asking whether there is a situation where the premises and the conclusion are all true.) Of course, the answer is 'no'. If pigs can indeed fly, and if anything that can fly can also swim, then it must be the case that all pigs can swim.

A valid argument with false premises can still have a true conclusion.

All pigs are purple in colour. 
Anything that is purple is an animal. 
So all pigs are animals.

 

Some exercises

Some exercises on Validity (we can consider the Soundness of the arguments as well):  https://philosophy.hku.hk/think/arg/valid1.php

Some exercises on Analogical arguments: https://philosophy.hku.hk/think/arg/analogy.php

Some exercises on good arguments: https://philosophy.hku.hk/think/arg/goodarg.php

References:

1. [A01] What is an argument?  Critical Thinking Web, Maintained by Joe Lau, Philosophy Department, University of Hong Kong.  https://philosophy.hku.hk/think/arg/arg.php
2. [A02] The standard format.  Critical Thinking Web, Maintained by Joe Lau, Philosophy Department, University of Hong Kong.  https://philosophy.hku.hk/think/arg/standard.php
3. Logical reasoning, Wikipedia. https://en.wikipedia.org/wiki/Logical_reasoning
4. Argument from analogy, Wikipedia. https://en.wikipedia.org/wiki/Argument_from_analogy#:~:text=Argument%20from%20analogy%20is%20a,the%20world%20and%20make%20decisions.