2.1 Atomic and Molecular Propositions

Propositions are the building blocks of logical reasoning. Atomic propositions are simple statements, while molecular propositions combine them using logical connectives. Understanding these types helps us analyze complex arguments and statements.

Propositional variables and truth values are key concepts in logic. Variables represent any proposition, allowing us to study logical structures. Truth values (true or false) determine the validity of statements and arguments in different scenarios.

Propositions

Types of Propositions

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• Propositions are declarative sentences that are either true or false, but not both
• Atomic propositions are the simplest type of proposition that cannot be broken down into smaller propositions
  • Consist of a single statement that is either true or false (The sky is blue)
  • Represented by a single letter variable, usually lowercase letters like $p$, $q$, or $r$
• Molecular propositions are more complex propositions formed by combining one or more atomic propositions using logical connectives
  • Created using logical operators such as conjunction (and), disjunction (or), negation (not), implication (if-then), and equivalence (if and only if)
  • Can be broken down into smaller, constituent atomic propositions ($p \land q$ is a molecular proposition composed of atomic propositions $p$ and $q$)

Representing Propositions

• Propositions can be represented using symbols to abstract away the content and focus on the logical structure
  • Allows for analyzing the relationships between propositions without getting caught up in the specific meaning of the sentences
• Atomic propositions are typically represented by single lowercase letters like $p$, $q$, $r$, etc.
  • The choice of letter is arbitrary and does not affect the logical structure ($p$: It is raining, $q$: The grass is wet)
• Molecular propositions are represented by combining the symbols for atomic propositions with logical connectives
  • Conjunction: $p \land q$ (It is raining and the grass is wet)
  • Disjunction: $p \lor q$ (It is raining or the grass is wet)
  • Negation: $\lnot p$ (It is not raining)

Truth and Variables

Truth Values

• Every proposition has a truth value, which is either true (T) or false (F)
  • The truth value indicates whether the proposition accurately describes reality or not
  • Atomic propositions have a single truth value determined by the state of affairs they describe ($p$: The sky is blue - true)
• The truth values of molecular propositions are determined by the truth values of their constituent atomic propositions and the logical connectives used
  • Truth tables are used to systematically represent all possible combinations of truth values for the atomic propositions and the resulting truth value of the molecular proposition
  • For example, the truth table for conjunction ($\land$) shows that $p \land q$ is only true when both $p$ and $q$ are true

Propositional Variables

• Propositional variables are symbols (usually lowercase letters) that represent arbitrary propositions
  • They serve as placeholders for any proposition, allowing for the study of logical forms without referring to specific content
  • For example, $p$ could represent "It is raining" in one context and "The sky is blue" in another
• Propositional variables are essential for constructing and analyzing logical arguments and proofs
  • They enable the formulation of general logical principles and rules that apply to any proposition, regardless of its content
  • For instance, the law of excluded middle states that for any proposition $p$, either $p$ or $\lnot p$ must be true: $p \lor \lnot p$ is always true
• Truth assignments are functions that assign truth values (T or F) to propositional variables
  • They represent different possible scenarios or states of affairs under which the propositions may be evaluated
  • For example, if $p$ represents "It is raining," a truth assignment of $p = T$ would indicate a scenario where it is indeed raining