Logical Reasoning

In today’s lesson we will look at:

• what we mean by logical reasoning

• different types of logical reasoning:

– deductive reasoning

– inductive reasoning

– abductive reasoning

Logical Reasoning• Wouldn’t it be good if you were having an argument

with someone, and you could just say, “Let’s work it out!” like you would with a Maths question?

• That’s been the goal of philosophers and mathematicians for thousands of years – to develop a system of thinking that is objective, like arithmetic.

• Unfortunately there is no “argument calculator”, but there are systems of logical reasoning that we can use to help us to decide what makes sense.

• You have previously looked at Boolean logic; today we will look at three other methods of logical reasoning.

Deductive Reasoning• Deductive reasoning relies on the idea of true

statements having a consequence.

• If you are given two statements that are true, then you can deduce that a third statement is also true.

• This is the basis of the syllogism, devised by Aristotle.

• For example:

1. No reptiles have fur

2. All snakes are reptiles

Conclusion: No snakes have fur

More Examples

Another example:

1. All humans are mortal

2. All Greeks are human

Conclusion: All Greeks are mortal

What about this one?

3. Some dogs are dangerous

4. Some dangerous things are volcanoes

Conclusion: Some dogs are volcanoes?

No - at least one of the statements must be universal (i.e. they apply to all “all” or “no” things)

Further Examples

Another example:

1. Some big dogs like cheese

2. All dogs that like cheese are friendly

Conclusion: Some friendly dogs are big

What about this one?

3. No snakes have wheels

4. Some snakes are green

Conclusion: Some green things have no wheels

Inductive Reasoning• Inductive reasoning is really the opposite of this – it’s

about generalising things from an observation, i.e.

1. Almost all A are B

2. C is A

Conclusion: C is almost certainly B

• For example:

1. Almost all birds can fly

2. An eagle is a bird

Conclusion: An eagle can almost certainly fly

• Can you think of an example where that doesn’t work?

More Examples• Another example:

1. Most plane flights do not crash

2. This is a plane flight

Conclusion: It is almost certain not to crash

• i.e. the plane not crashing is the outcome that best matches what you’ve previously observed

• Inductive reasoning is also about spotting patterns:

Further Examples

What is the next pattern in the sequence?

What’s happening in this sequence?

Abductive Reasoning• Abductive reasoning is about finding the most likely

explanation for what has happened. It’s what doctors use to diagnose illnesses and mechanics use to find faults.

• Imagine that you look out of the window and see that your lawn is wet - what are the possible causes?

– it’s just been raining

– it might be dew

– someone might have watered it with a hose

– aliens with water pistols landed in your garden

• Based on your experience, which is most likely?

• What about if you haven’t got a hose and the sky is grey? What about if it’s summer and the sky is blue?

Base Rate Fallacy• We need to be careful when using this type of logic.

• A city of 1 million people has 100 criminals. CCTV and face-recognition software sounds an alarm when it spots a criminal, but:

i. If it scans a criminal it only sounds the alarm 99% of the time

ii. If it scans a non-criminal it sounds the alarm 1% of the time

• The alarm sounds – what is the probability that the person is a criminal?

a) 99 per cent

b) less than 1 per cent

c) 98 per cent

Prosecutor’s Fallacy• A man is on trial for murder. The defendant shares the

same rare blood type as the perpetrator and just 1% of the population of 300 million.

• Ignoring all other evidence, what is the probability that the man is guilty based on just the blood type match?

a) 10 per cent

b) 99 per cent

c) 0.00003333 per cent

• Why? Because 1% of 300 million is quite a lot – there are 3 million people whose blood type matched that of the murderer.