Chapter 3: Logical Reasoning

3.1 Statements

Definition of a Statement

A statement is a sentence that is either true or false, but not both. Questions, commands, exclamations, and algebraic expressions with unknown variables are non-statements.

Truth Values of Statements

Every statement has a truth value: either True (T) or False (F).

• Example Statement: "8 is an even number." (True)
• Example Non-Statement: "Please turn off the lights." (Command / No truth value)

Negation of a Statement

The negation of a statement $p$ is denoted as $\sim p$ (read as "not $p$"). It reverses the truth value of the original statement by inserting the word "not" or "no".

Determining Truth Values of Compound Statements

Compound statements combine two statements using "AND" or "OR":

• "p AND q": True ONLY when BOTH $p$ and $q$ are true.
• "p OR q": True if AT LEAST ONE of $p$ or $q$ is true.

Quantifiers: "All" and "Some"

• "All": Refers to every single object or element in a specified set.
• "Some": Refers to at least one (or a part of) element in a specified set.

3.2 Implications

Forming Implications

An implication is a statement written in the form "If $p$, then $q$".

• $p$ is called the antecedent (hypothesis).
• $q$ is called the consequent (conclusion).

Implication in Terms of "If and only if"

A statement "$p$ if and only if $q$" consists of two separate implications combined:

1. If $p$, then $q$
2. If $q$, then $p$

Converse, Inverse, and Contrapositive

Given the primary implication: "If $p$, then $q$":

• Converse: If $q$, then $p$
• Inverse: If $\sim p$, then $\sim q$
• Contrapositive: If $\sim q$, then $\sim p$

Note on Logical Equivalence: An implication and its contrapositive always share the same truth value. Similarly, the converse and inverse share the same truth value.

3.3 Arguments

Definition and Types of Arguments

An argument is a process of making a conclusion based on a set of premises.

1. Deductive Arguments

A process of deriving a specific conclusion from general premises. Deductive arguments are assessed for validity and soundness.

Form I (Direct Deduction)

Premise 1: All $A$ are $B$.
Premise 2: $C$ is $A$.
Conclusion: $C$ is $B$.

Form II (Modus Ponens)

Premise 1: If $p$, then $q$.
Premise 2: $p$ is true.
Conclusion: $q$ is true.

Form III (Modus Tollens)

Premise 1: If $p$, then $q$.
Premise 2: Not $q$ is true ($\sim q$).
Conclusion: Not $p$ is true ($\sim p$).

2. Inductive Arguments

A process of deriving a general conclusion based on specific observations or cases. Inductive arguments are assessed for cogency and whether they are strong or weak.