MATH 1105: Logical Reasoning in Mathematics

This course provides an in-depth exploration of logical reasoning, specifically focusing on mathematical propositions, their types, and the logical operations that can be applied to them. Understanding the components of logical statements is crucial for the study of mathematics and its applications. The fundamental aspects explored here include propositions, logical connectives, and arguments, as well as methods to validate these logical constructs.

Understanding Propositions and Statements

In logic, a statement is defined as a verbal assertion that is either true or false, though it cannot be both simultaneously. This characteristic is termed the truth value of the statement. For instance:

Example Statements: 1. Pampanga is in Region III. 2. –7 – 9 = 16. 3. Do your homework. 4. x = -4 is a solution of x² = 16. 5. Where are you going?

Some statements can be composite, meaning they consist of two or more statements combined by various logical connectives. The types of logical connectives used include:

1. Conjunction

The conjunction of two statements is formed by combining them with the word "and". This composite statement is represented as p ∧ q. The truth table for conjunction shows that this statement is true only if both individual statements are true. For example:

If p is "It is raining" and q is "The sun is shining", then p ∧ q translates to "It is raining, and the sun is shining".

The truth value of p ∧ q can be summarized in the following truth table:

p	q	p ∧ q
T	T	T
T	F	F
F	T	F
F	F	F

2. Disjunction

Disjunction of two statements is formed using the word "or". This is denoted as p ∨ q and is true if at least one of the statements is true. For example:

If p is "He enrolled in Math I" and q is "He dropped Math II", then p ∨ q translates to "He enrolled in Math I or he dropped Math II".

The truth values of p ∨ q are expressed as follows:

p	q	p ∨ q
T	T	T
T	F	T
F	T	T
F	F	F

3. Negation

Negation involves a statement's inversion, usually expressed as "It is false that..." or by inserting "not". This is symbolically represented as ~p. For instance:

If p is "Pampanga is located in Region III", then ~p is "It is false that Pampanga is located in Region III".

The truth value operates as follows:

If p is true, then ~p is false, and vice versa.

4. Conditional Statements

Conditional statements take the form of "if p then q" and are denoted as p → q. A conditional statement is considered true unless the first part is true while the second part is false. For example:

If Pampanga is in Region III, then 2 + 2 = 4.

5. Biconditional Statements

Statements of the form "p if and only if q" are referred to as biconditional statements, represented as p ↔ q. The truth of p ↔ q is established when both statements share the same truth value.

Logical Equivalence

Two propositions are said to be logically equivalent when their truth tables yield identical results. This equivalence is denoted as P ≡ Q. An example of logical equivalence includes proving the relationship:

(p → q) ∧ (q → p) ≡ (p ↔ q)

Logical Reasoning: Arguments

In symbolic logic, an argument consists of premises that support a conclusion, typically formatted as:

P1, P2, ..., Pn / ∴ Q

Arguments can be validated through three methodologies: direct truth table evaluation, shortened truth table methods, and graphical representation via diagrams. The validity of an argument can be assessed based on the tautological nature of the compound proposition formed by the premises leading to the conclusion.

Method 1: Truth Table

Theorem: An argument P1, P2, ..., Pn / ∴ Q is valid if (P1 ∧ P2 ∧ ... ∧ Pn) → Q is a tautology.

Method 2: Shortened Truth Table

To prove invalidity, it suffices to find a combination of truth values where the premises are true but the conclusion is false. This approach avoids constructing the full truth table.

Application of Venn Diagrams

Venn diagrams provide a visual method to represent logical relationships among various sets. For example, let’s analyze the argument:

Babies are irrational. Nobody is despised who can manage a crocodile. Irrational people are despised. Therefore, babies cannot manage a crocodile.

In this situation:

The set representing babies is a subset of irrational individuals.
Irrational persons are included within the parameter of despised individuals.
The disjointed nature of despised persons and people who can manage crocodiles supports the conclusion drawn that babies cannot manage crocodiles.

Algebra of Logic

This section outlines essential rules of logical replacement and inference, providing a systematic approach for establishing valid arguments. Key rules to note include:

Rules of Replacement

Idempotence: P ≡ (P ∨ P), P ≡ (P ∧ P)
Commutativity: (P ∨ Q) ≡ (Q ∨ P), (P ∧ Q) ≡ (Q ∧ P)
Associativity: (P ∨ Q) ∨ R ≡ P ∨ (Q ∨ R), (P ∧ Q) ∧ R ≡ P ∧ (Q ∧ R)
De Morgan’s Laws: ~(P ∨ Q) ≡ (~P ∧ ~Q), ~(P ∧ Q) ≡ (~P ∨ ~Q)
Material Implication: (P → Q) ≡ (~P ∨ Q)

Rules of Inference

Addition: If P is true, then P ∨ Q holds.
Simplification: If P ∧ Q is true, then both P and Q must be true.
Modus Ponens: Given P → Q and P, you can conclude Q.
Hypothetical Syllogism: If P → Q and Q → R, then P → R.

Through the application of these principles and methodologies, students will learn to analyze logical arguments rigorously, enhancing their mathematical reasoning and problem-solving capabilities. Understanding these logical frameworks is pivotal to mastering the quantitative disciplines.

Logical Reasoning and Propositions in Mathematics

Logical Reasoning and Propositions in Mathematics