Mathematical Logic: Propositions, Connectives, and Truth Tables

Foundations of Mathematical Logic and Propositional Calculus

• Role of Mathematical Logic:

  • Serves as an indispensable foundation for fostering critical and structured thinking.
  • Provides a formal framework to represent reasoning from everyday language and academic contexts in an exact symbolic notation, streamlining logical evaluation, systematic analysis, and formal validation.
  • Facilitates the development of structured reasoning schemes to apply in real-world contexts.

• Definition of a Proposition:

  • A proposition is defined as an expression of language that holds a truth value of either true or false, but cannot be both simultaneously.
  • It represents a declarative statement that unambiguously affirms or denies something (Copi et al., 2016).

• Examples of Valid Propositions:

  • "La Tierra gira alrededor del Sol" (The Earth revolves around the Sun): Qualifies as a proposition because it makes a clear declarative assertion; its truth value is true.
  • "5 es un número par" (5 is an even number): Qualifies as a proposition because it asserts a mathematical statement; its truth value is false (number $5$ is odd).

• Non-Examples of Propositions:

  • "¿Qué hora es?" (What time is it?): Does not qualify as a proposition because it is an interrogative sentence that neither affirms nor denies any fact.
  • "Estudia más" (Study more): Does not qualify as a proposition because it constitutes an imperative command or order rather than an affirmative or negative declarative statement.

• Propositional Variables and Symbolic Notation:

  • In symbolic logic, individual propositions are represented algebraically using lowercase propositional letters such as $p$, $q$, and subsequent letters.
  • Example assignment:
  • Let $p$ represent: Está lloviendo (It is raining).
  • Let $q$ represent: Hace frío (It is cold).
  • These propositional variables can be combined via logical connectives to construct complex arguments and structured logical expressions.

Core Logical Connectives

• Negation:

  • Symbol: Represented symbolically as $\neg$ or $\sim$.
  • Semantic Function: Inverts or denies the truth value of a given proposition, expressing its exact contradictory meaning.
  • Truth Property: For any proposition $p$, if $p$ is true ($V$), then $\neg p$ is false ($F$); conversely, if $p$ is false ($F$), then $\neg p$ is true ($V$).
  • Linguistic Translation: Expressed in natural language as "no" (not).
  • Example: If proposition $p$ is defined as "Hoy es lunes" (Today is Monday), the negated proposition $\neg p$ is "Hoy no es lunes" (Today is not Monday).

• Conjunction:

  • Symbol: Represented symbolically as $\land$.
  • Semantic Function: Connects two propositions by asserting that both constituent statements occur or hold true simultaneously.
  • Truth Rule: A conjunction is true if and only if both individual propositions are true. If either proposition (or both) is false, the entire conjunction evaluates to false.
  • Linguistic Translation: Read and written in mathematics and logic as "y" (and).
  • Example: Given $p$: Estudio matemáticas (I study mathematics) and $q$: Estudio física (I study physics), the conjunction $p \land q$ corresponds to: "Estudio matemáticas y estudio física" (I study mathematics and I study physics).

• Disjunction (Inclusive Disjunction):

  • Symbol: Represented symbolically as $\lor$.
  • Semantic Function: Joins two propositions such that it is sufficient for at least one of them to be true for the entire compound statement to be true.
  • Truth Rule: The disjunction evaluates to false solely when both component propositions are false. If at least one component is true (inclusive nature), the expression is true.
  • Linguistic Translation: Expressed in natural language and mathematical logic as "o" (or).
  • Example: Given $p$: Tengo tarea (I have homework) and $q$: Tengo examen (I have an exam), the disjunction $p \lor q$ corresponds to: "Tengo tarea o tengo examen" (I have homework or I have an exam).

• Conditional (Material Implication):

  • Symbol: Represented symbolically as $\rightarrow$.
  • Semantic Function: Establishes a relation of implication between an antecedent proposition and a consequent proposition (if the antecedent occurs, then the consequent follows).
  • Truth Rule: The conditional evaluates to false in exactly one scenario: when the antecedent (first proposition) is true and the consequent (second proposition) is false. In all other combinations, the conditional evaluates to true.
  • Linguistic Translation: Expressed in logic and language as "si… entonces" (if… then), where propositions replace the ellipses.
  • Example: Given $p$: Estudio (I study) and $q$: Apruebo el examen (I pass the exam), the conditional $p \rightarrow q$ translates to: "Si estudio, entonces apruebo el examen" (If I study, then I pass the exam).

• Bicondicional (Material Equivalence):

  • Symbol: Represented symbolically as $\leftrightarrow$.
  • Semantic Function: Asserts that two propositions share identical truth values.
  • Truth Rule: The biconditional evaluates to true when both constituent propositions are simultaneously true or simultaneously false; it evaluates to false when their truth values differ.
  • Linguistic Translation: Expressed in logic as "si y solo si" (if and only if).
  • Example: Given $p$: Es cuadrado (It is a square) and $q$: Tiene cuatro lados iguales y ángulos rectos (It has four equal sides and right angles), the biconditional $p \leftrightarrow q$ states: "Es cuadrado si y solo si tiene cuatro lados iguales y ángulos rectos" (It is a square if and only if it has four equal sides and right angles).

Truth Tables for Basic Connectives

• Truth Value Combinations for Two Propositions:
  • Any given proposition $p$ can take one of two truth values: true ($V$) or false ($F$).
  • A second independent proposition $q$ can likewise take two truth values: true ($V$) or false ($F$).
  • Combining two independent binary propositions results in exactly $2^2 = 4$ unique truth configurations:

• Truth Table for Conjunction ($p \land q$):
  • Confirms that the conjunction is true exclusively when both inputs are true:

• Truth Table for Disjunction ($p \lor q$):
  • Confirms that the inclusive disjunction is true whenever at least one variable contains a truth value of $V$, and yields $F$ exclusively when both propositions are $F$:

Evaluation of Compound Propositions

• Arbitrary Expansion of Logical Systems:

  • Truth table methodology extends to accommodate arbitrarily many propositions and logical connectives to model complex arguments and structural problem-solving scenarios.

• Step-by-Step Evaluation Order for $\neg p \rightarrow q$:

  • Step 1 (Unary Negation): Determine the negation of the proposition $p$, denoted as $\neg p$. The resulting column reflects the exact opposite truth value of $p$.
  • Step 2 (Conditional Implication): Apply the conditional operator between the antecedent $\neg p$ and the consequent $q$, denoted as $\neg p \rightarrow q$.
  • Conditional Criterion: The expression evaluates to false solely when the antecedent ($\neg p$) is true ($V$) and the consequent ($q$) is false ($F$).

• Complete Truth Table for $\neg p \rightarrow q$:

• Row-by-Row Analysis:
  • For $p = V$ and $q = V$: Negation yields $\neg p = F$. The implication $F \rightarrow V$ yields $V$.
  • For $p = V$ and $q = F$: Negation yields $\neg p = F$. The implication $F \rightarrow F$ yields $V$.
  • For $p = F$ and $q = V$: Negation yields $\neg p = V$. The implication $V \rightarrow V$ yields $V$.
  • For $p = F$ and $q = F$: Negation yields $\neg p = V$. The implication $V \rightarrow F$ yields $F$.

Contextual and Real-World Applications of Truth Tables

• Application Example 1: Conditional Implication in Academic Performance:
  • Defined Propositions:
  • $p$: Juan estudia para el examen (Juan studies for the exam).
  • $q$: Juan aprueba el examen (Juan passes the exam).
  • Compound Statement ($p \rightarrow q$):
  • "Si, Juan estudia para el examen, entonces, Juan aprueba el examen" (If Juan studies for the exam, then Juan passes the exam).
  • Truth Table and Real-World Interpretations:

• Condition Analysis: Across all four logical cases of $p$ and $q$, the only scenario that fails the conditional statement (evaluates to false or violates the conditional rule) occurs when Juan studies for the exam and does not pass ($p = V$, $q = F$).

  • Application Example 2: Inclusive Disjunction in Extracurricular Activities:

• Defined Propositions:

  • $p$: Luis va al cine (Luis goes to the movies).
  • $q$: Luis va a entrenar (Luis goes to train).

• Compound Statement ($p \lor q$):

  • "Luis va al cine o va a entrenar" (Luis goes to the movies or goes to train).

• Truth Table and Real-World Interpretations:

• Condition Analysis: Across all possible case combinations for $p$ and $q$, the single outcome that fails the statement (evaluates to false or fails to satisfy the condition) is when Luis does not go to the movies and does not go to train ($p = F$, $q = F$).