Week 3 - Mini-lecture 3: Deductive reasoning

Introduction

This mini lecture is considered the most difficult in the unit due to the introduction of new concepts related to logic and reasoning, which may be unfamiliar to those without a background in philosophy.
The concepts resemble algebraic equations with letters and require understanding logical statements called syllogisms.
It is crucial to avoid misunderstanding logical statements, as this can lead to losing marks on the exam. Full attention is required, especially for those without prior knowledge in philosophy and logic.
Learning these concepts may require rote learning to grasp the premises, but with time, they will become clear.
Deductive reasoning will be covered in this mini lecture, while other types of reasoning will be discussed in mini lecture four.

Basic Terms of Reasoning

Reasoning involves drawing inferences and conclusions about the world.
In cognitive psychology research, reasoning focuses on mental processes that lead to valid conclusions.
Judgement, on the other hand, studies the processes that lead to errors.
Three types of reasoning: deductive, inductive, and abductive. Deductive reasoning will be the focus.

Deductive Reasoning

Deductive reasoning is a process of drawing conclusions from premises by constructing valid arguments.
It involves using formal rules of proof to reason about syllogisms, which are logical statements.

Properties of Deductive Reasoning

Conclusions can be drawn with certainty; they are either certainly true or certainly false.
It involves applying a general rule to a specific case to draw a specific conclusion.
Deductive reasoning does not add any new knowledge; it utilizes general rules applied to specific cases.

Validity vs. Truth

The key question in deductive reasoning is whether a conclusion follows logically from its premises.
A conclusion is valid if it follows logically from its premises.
A valid conclusion is different from a true conclusion.
A conclusion is valid if it follows logically from its premises; if the premises are also true, then the conclusion is true.
The premises may not always be accurate.

Examples

Example of a valid and true conclusion:
- Premise one: All humans are mortal.
- Premise two: Socrates is a human.
- Conclusion: Therefore, Socrates is mortal.
Example of a valid but untrue conclusion:
- Premise one: All humans have tails.
- Premise two: Socrates is human.
- Conclusion: Therefore, Socrates has a tail.

Language and Venn Diagrams

Syllogisms involve terms like "all," "none," or "some."
Venn diagrams can help explain how reasoning works or doesn't work.
Example:
- Premise one: All humans are mortal. (All humans are within the circle of mortality).
- Premise two: Socrates is a human. (Socrates is within the circle of humans).
- Conclusion: Therefore, Socrates is mortal (Socrates also falls within the circle of mortality).

Snake Example

Premise one: Some snakes are venomous.
Premise two: Some Australian animals are snakes.
Invalid Conclusion: Therefore, some Australian snakes are venomous.
Explanation: It's possible that some of the Australian animals that are snakes are not venomous.

Conditional Syllogisms

A conditional syllogism is a deductive reasoning problem with a major premise in the form of "if P, then Q."
- Major premise: If Daniel gets a puppy, then he will be happy.
  - P (antecedent): Daniel gets a puppy.
  - Q (consequent): He will be happy.
- The minor premise affirms or denies either the antecedent or the consequent.
- The conclusion relates to the portion of the major premise that is not mentioned in the minor premise.

Modus Ponens

Modus ponens syllogisms, or affirming the antecedent, make the syllogism valid.
- If P, then Q. P is the case. Therefore, Q is the case.
- If Daniel gets a puppy, then he will be happy. Daniel got a puppy. Therefore, Daniel is happy.

Affirming the Consequent

Affirming the consequent does not logically follow and is invalid.
- If P, then Q. Q is the case. Therefore, P is the case (Invalid).
- If Daniel gets a puppy, then he will be happy. Daniel is happy. Therefore, Daniel got a puppy (Invalid).
- Just because Daniel is happy doesn't necessarily mean that he got a puppy.

Modus Tollens

Modus tollens syllogisms, where we deny the consequence, make the syllogism valid.
- If P, then Q. Q is not the case. Therefore, P is not the case (Valid).
- If Daniel gets a puppy, then he will be happy. Daniel is not happy. Therefore, Daniel did not get a puppy (Valid).

Denying the Antecedent

Denying the antecedent is invalid.
- If P, then Q. P is not the case. Therefore, Q is not the case (Invalid).
- If Daniel gets a puppy, then he will be happy. Daniel did not get a puppy. Therefore, Daniel is not happy (Invalid).
- There could be many other reasons why Daniel is happy or unhappy.
It requires rote learning to know the differences between them.

Truth Table

This table shows how people can get logic wrong if they haven't been trained in logic.

Statement	Description	Correct Judgement	Percentage of Participants who Judge Logic as Valid
If P, then Q
P therefore Q	Affirming the Antecedent	Valid	97%
Not Q therefore Not P	Denying the Consequent	Valid	60%
If Q therefore P	Affirming the Consequent	Invalid	Less than Half
If not P therefore not Q	Denying the Antecedent	Invalid	Less than Half

Wason Selection Task

Study by Wason in the 60s.
Aims to understand how syllogisms can be used as a tool for reasoning.
The strongest way to test a conditional rule is to seek out disconfirming evidence.
We should try to test a rule by both affirming the antecedent and denying the consequent.

License Selection Tasks

Participants were presented with cards with letters on one side and numbers on the other side.
They were told to test the rule: "If a card has a vowel on one side, then it has an even number on the other side."

Results

46% of participants chose A.
33% chose only A.
4% of participants (the correct answer) chose A and 7.
The major premise: If there is a vowel on one side, then there is an even number on the other side.

Explanation

The minor premise that might produce disconfirming evidence involves either affirming the antecedent or denying the consequence. Those are the only two valid ways of phrasing that.
You need to confirm this: If there's a vowel on one side, then there is an even number on the other side. So you would flip that and say if there is an even number to test it, and what we would have to do, and this is what might trip you up, is you have to deny the consequent for that to be a logical statement to then say, therefore not P. So you actually need to flip this to an odd number because if you were going to affirm the consequent and flipped over the even number, it doesn't logically follow that, therefore, P. But if you denied the consequent and said therefore not Q or not Q, so then an odd number and you flip that, you could then say, therefore not P.
So if that number had the vowel, you flip that, not Q, and it didn't have a vowel. You could then find that just confirming evidence or confirming evidence, depending on what the… Because it doesn't mean that this is… This is where the truth and the validity goes apart. It doesn't mean that that's going to follow the rule, but that's how you test the rule. So if P and Q is it P is the case, therefore Q is the case. So that's why you would do A because you're testing that P is true. You've got your fair value. You flip it and you see what's on the other side.
If P, then Q and say Q is not the case. So you flip the odd number because then you could be saying, therefore P is not the case. So those are two. Those are the two valid ones. So those minor premises that are associated with the invalid conclusions, which we know is if not P or if yes Q.
Confirming this one over the nine, this one, you won't produce a valid result from those ones. So they might be true and follow the rule, but you're not actually confirming because they're based on invalid conclusions that you would draw, and that is invalid even if it's true, if you're not actually testing it by flipping these two around.
So that's why it's not four because you're confirming because that's an even number. You're confirming the consequent, which doesn't logically follow that, therefore P. And for this one too, if you're saying if not P, it doesn't logically follow, then if not Q because you're saying this, therefore that doesn't automatically mean not that, therefore not that. And you know, we could bring up those Venn diagrams again and show you why. But that's how it goes when we're talking about the conditional syllogisms.

Key Concepts for the Exam

Components of a syllogism: antecedent, consequent, and conclusion.
Structure of a syllogism.
Difference between validity and truth.
Definition of a conditional syllogism.
Valid and invalid kinds of conditional syllogisms.
Affirming the antecedent and denying the consequent are valid.
Denying the antecedent and affirming the consequent are invalid.