Day 4: Deductive Reasoning and Logical Rules

Course Objectives

This lesson introduces the structure and rules of deductive reasoning. Students learn the form of syllogisms, how to use basic logical symbols, and how to apply inference rules.

After completing this lesson, students will be able to:

Understand the features and uses of deductive reasoning
Analyze the structure and validity of syllogisms
Use basic logical symbols to express propositions and arguments
Apply inference rules for valid reasoning
Use Venn diagrams to test syllogisms

Deductive Reasoning Basics

Deductive reasoning moves from general principles to specific cases. If the premises are true, the conclusion must also be true.

Characteristics

From general to specific
Necessity: true premises guarantee a true conclusion
High certainty
No new information: the conclusion is already contained in the premises

Example

All mammals have hearts. (General principle)
All dogs are mammals. (General principle)
Therefore, all dogs have hearts. (Specific case)

This example shows that if the premises are true, the conclusion necessarily follows.

Syllogisms

A syllogism is a basic form of deductive reasoning consisting of two premises and a conclusion. It involves three terms: the major term, the minor term, and the middle term.

Structure

Major premise: contains the major term and the middle term
Minor premise: contains the minor term and the middle term
Conclusion: contains the minor term and the major term

Where:

Major term: predicate of the conclusion
Middle term: appears in both premises but not in the conclusion
Minor term: subject of the conclusion

Example

All humans are mortal. (Major premise)
Socrates is human. (Minor premise)
Therefore, Socrates is mortal. (Conclusion)

In this example:

The major term is "mortal"
The middle term is "human"
The minor term is "Socrates"

Validity of Syllogisms

The validity of a syllogism depends on its form rather than its content. A valid form guarantees that if the premises are true, the conclusion must be true.

Some basic rules for syllogistic validity are:

The middle term must be distributed in at least one premise
If a term is distributed in the conclusion, it must be distributed in the corresponding premise
At least one premise must be affirmative
If one premise is negative, the conclusion must be negative
Two negative premises cannot yield a valid conclusion

Basic Logical Symbols

Logical symbols represent relations and operations in logic. They make logical expressions concise and precise.

Propositional Symbols

Yesterday we learned these basic symbols:

∧: conjunction (and)
∨: disjunction (or)
¬: negation (not)
→: conditional (if...then)
↔: biconditional (if and only if)

Predicate Symbols

Predicate logic also uses:

∀: universal quantifier, "for all"
∃: existential quantifier, "there exists"
P(x): predicate meaning "x has property P"

Expressing Propositions

For example, "All humans are mortal" can be written as:

∀x(H(x) → M(x))

Here H(x) means "x is human" and M(x) means "x is mortal".

"Some students like math" can be written as:

∃x(S(x) ∧ L(x,m))

where S(x) means "x is a student" and L(x,m) means "x likes math".

Basic Rules of Inference

Rules of inference allow us to derive conclusions from known premises. Some basic rules are:

Modus Ponens

If p → q is true and p is true, then q is true.

If it rains, the ground gets wet. (p → q)
It rains. (p)
Therefore, the ground is wet. (q)

Modus Tollens

If p → q is true and q is false, then p is false.

If it rains, the ground gets wet. (p → q)
The ground is not wet. (¬q)
Therefore, it didn't rain. (¬p)

Hypothetical Syllogism

If p → q and q → r are true, then p → r is true.

If it rains, the ground gets wet. (p → q)
If the ground is wet, shoes get dirty. (q → r)
Therefore, if it rains, shoes get dirty. (p → r)

Disjunctive Syllogism

If p ∨ q is true and ¬p is true, then q is true.

Either it rains or it snows. (p ∨ q)
It does not rain. (¬p)
Therefore, it snows. (q)

Addition

If p is true, then p ∨ q is true (for any q).

It rains. (p)
Therefore, either it rains or it snows. (p ∨ q)

Simplification

If p ∧ q is true, then p is true and q is true.

It is raining and windy. (p ∧ q)
Therefore, it is raining. (p)
Therefore, it is windy. (q)

Conjunction

If p is true and q is true, then p ∧ q is true.

It rains. (p)
It is windy. (q)
Therefore, it is raining and windy. (p ∧ q)

Validity Proof

A validity proof shows that an argument is logically sound. Here is a simple example:

Example

Prove the following argument is valid:

1. If it rains, the ground is wet. (p → q)
2. If the ground is wet, shoes get dirty. (q → r)
3. It rains. (p)
4. Therefore, shoes get dirty. (r)

Proof

p → q (premise 1)
q → r (premise 2)
p (premise 3)
q (from 1 and 3, modus ponens)
r (from 2 and 4, modus ponens)

Therefore, the conclusion r is valid.

Venn Diagrams

Venn diagrams use circles to represent sets. They help us analyze the validity of syllogisms.

Basic Form

The overlapping parts of circles show the intersections of sets.

Using Venn Diagrams

For example, consider this syllogism:

All humans are mortal. (Major premise)
Socrates is human. (Minor premise)
Therefore, Socrates is mortal. (Conclusion)

In the diagram:

One circle represents people
Another circle represents mortal things
Socrates is marked as a point

The major premise places the circle for people entirely inside the circle for mortal things.

The minor premise locates Socrates inside the circle of people.

Therefore, the diagram shows Socrates is also inside the circle of mortal things.

Course Summary

Today we learned:

The features and uses of deductive reasoning
The structure and validity of syllogisms
The use of basic logical symbols
Common inference rules such as modus ponens, modus tollens, and hypothetical syllogism
How to conduct a simple validity proof
How to use Venn diagrams to test syllogisms

These skills will help you better understand deductive reasoning and strengthen logical thinking.

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