LOGISTICS Truth Tables

Truth Value Assignments and Conditional Truth Tables

• Standard Truth Table Setup:

  • For two atomic propositions, the truth table is structured by assigning standard truth values across rows and columns.
  • Column 1 ($A$): Two truths followed by two falses ($T, T, F, F$).
  • Column 2 ($B$): Alternating truths and falses ($T, F, T, F$).

• Truth Conditions of the Conditional ($A \rightarrow B$):

  • Evaluating a conditional statement (e.g., "If I win the lottery, I'll give you a million dollars") involves identifying when a statement holds true versus when it can be proven false:
    • True Antecedent, True Consequent ($T \rightarrow T$): Yields True. Winning the lottery and providing the million dollars fulfills the condition. The speaker cannot be called a liar.
    • True Antecedent, False Consequent ($T \rightarrow F$): Yields False. Winning the lottery and failing to give the million dollars violates the commitment. This is the only scenario where the conditional statement is false.
    • False Antecedent, False Consequent ($F \rightarrow F$): Yields True. Not winning the lottery and not giving the million dollars does not break the promise. The condition promised was contingent on winning.
    • False Antecedent, True Consequent ($F \rightarrow T$): Yields True. Not winning the lottery but giving the million dollars anyway does not invalidate the original statement.

• Fundamental Rule of Conditionals:

  • A conditional statement is false if and only if the antecedent ("if" clause) is true and the consequent ("then" clause) is false.

Components and Mechanics of Conditional Statements

• Structural Terminology:

  • Antecedent: The initial clause of a conditional statement (represented by $A$ in $A \rightarrow B$). Derived from the Latin prefix ante- meaning "before."
  • Consequent: The secondary clause of a conditional statement (represented by $B$ in $A \rightarrow B$). Represents the outcome or consequence of the antecedent occurring.

• Linguistic Variations and Guarantee vs. Exclusivity:

  • The English expression "$A$ only if $B$" expresses a conditional connection, though it differs in natural phrasing from "If $A$, then $B$".
  • A conditional statement guarantees the consequent given the antecedent, but it does not claim the antecedent is the sole condition under which the consequent can occur.

• Mechanical Analogy (Car Ignition):

  • Consider the conditional: "If the car cranks, then it has gas" ($\text{Car Cranks} \rightarrow \text{Has Gas}$).
  • If the car cranks, it guarantees that the car has gas, a good battery, an ignition switch, spark plugs, and a starter.
  • The presence of gas alone does not guarantee the car will crank, but cranking guarantees the presence of gas.

Truth-Functional Connectors in Propositional Logic

• Propositional logic relies on four primary truth-functional connectors:
  1. Conjunction: $A \land B$ ("$A$ and $B$")
  2. Disjunction: $A \lor B$ ("$A$ or $B$")
  3. Negation: eg A ("not $A$")
  4. Conditional: $A \rightarrow B$ ("if $A$ then $B$")

The Eight Valid Rules of Inference

• Rules of inference are deductive argument forms that are universally valid by virtue of their logical structure.

• Memorizing these forms allows logical proofs to be constructed directly without relying on truth tables as a crutch.

• 1. Modus Ponens (Affirming the Antecedent):

  • Form:
    • Premise 1: $A \rightarrow B$
    • Premise 2: $A$
    • Conclusion: $B$
  • Explanation: Affirming that the antecedent ($A$) is true guarantees that the consequent ($B$) is also true.

• 2. Modus Tollens (Denying the Consequent / Negating the Consequent):

  • Form:
    • Premise 1: $A \rightarrow B$
    • Premise 2: eg B
    • Conclusion: eg A
  • Explanation: Denying or negating the consequent ( eg B) proves that the antecedent ( eg A) did not occur.

• 3. Hypothetical Syllogism (Chain Logic):

  • Form:
    • Premise 1: $A \rightarrow B$
    • Premise 2: $B \rightarrow C$
    • Conclusion: $A \rightarrow C$
  • Explanation: Links conditional statements in a continuous logical chain. This rule extends across arbitrarily long chains (e.g., $A \rightarrow Y$, $Y \rightarrow Z \implies A \rightarrow Z$).

• 4. Simplification:

  • Form:
    • Premise: $A \land B$
    • Conclusion: $A$
  • Explanation: If a conjunctive statement asserts that both atomic propositions exist together as true, either proposition can be isolated independently as true.

• 5. Conjunction:

  • Form:
    • Premise 1: $A$
    • Premise 2: $B$
    • Conclusion: $A \land B$
  • Explanation: The inverse of simplification. If two atomic propositions are known to exist independently as true, they can be joined into a conjunctive statement.

• 6. Addition:

  • Form:
    • Premise: $A$
    • Conclusion: $A \lor B$
  • Explanation: If $A$ is known to be true, appending any arbitrary proposition $B$ via disjunction ($\lor$) creates a statement that is unconditionally true.
  • Common Confusion: Students frequently mistake addition for conjunction because it appends elements; however, addition strictly uses disjunction ($\lor$), providing a logical "out."

• 7. Disjunctive Syllogism:

  • Form:
    • Premise 1: $A \lor B$
    • Premise 2: eg A
    • Conclusion: $B$
  • Explanation: Given a choice between two options where one option is eliminated, the remaining option must be true.

• 8. Constructive Dilemma:

  • Form:
    • Premise 1: $A \lor B$
    • Premise 2: $A \rightarrow C$
    • Premise 3: $B \rightarrow D$
    • Conclusion: $C \lor D$
  • Explanation: Combines disjunctive options with conditional consequences to establish a disjunctive result.
  • Example: A burglar is in either the attic ($A$) or the basement ($B$). If in the attic ($A$), they face an attempted murder charge ($C$). If in the basement ($B$), they face an aggravated assault charge ($D$). Attempted murder carries a maximum life sentence, while aggravated assault carries a maximum 20-year sentence. Consequently, the defendant faces either a life sentence or a 20-year sentence ($C \lor D$).

Formal Logical Fallacies

• Invalid Reversals:
  • Reversing valid inference forms produces formal logical fallacies:
    • Fallacy of Affirming the Consequent: $A \rightarrow B$, $B \implies A$ (Invalid).
    • Fallacy of Denying the Antecedent: $A \rightarrow B$, eg A \implies eg B (Invalid).

Step-by-Step Proof Applications and Derivations

• Proof Strategy Example 1: Extracting Negation via Modus Tollens:

  • To isolate eg A from $A \rightarrow B$ and eg B, apply Modus Tollens (Denying the Consequent).
  • To extract eg eg B from a conjunction like $\neg \neg B \land D$, apply Simplification.

• Proof Strategy Example 2: Hypothetical Syllogism and Conjunction Steps:

  • Premises given:
    1. $A \rightarrow B$
    2. $B \rightarrow C$
    3. $A \land \neg D$
  • Step 1: Connect $A \rightarrow B$ and $B \rightarrow C$ using Hypothetical Syllogism to derive $A \rightarrow C$.
  • Step 2: Isolate $A$ from $A \land \neg D$ using Simplification.
  • Step 3: Apply Modus Ponens using $A \rightarrow C$ and $A$ to isolate $C$.
  • Step 4: Isolate eg D from $A \land \neg D$ using Simplification.
  • Step 5: Combine $C$ and eg D using Conjunction to achieve the final conclusion: $C \land \neg D$

Questions & Discussion

• Question: Can "I win the lottery only if you get the money" be read as meaning that you only get the money if the lottery is won, making it false if money is given without winning?

  • Response: Moving the clauses alters the formal statement structure. Reading "$A$ only if $B$" in English can sound unnatural, but in formal logic, providing money without winning the lottery does not make the original conditional false. The only scenario that invalidates the statement is winning the lottery and failing to give the money.

• Question: Is denying the consequent doing the same thing as affirming the antecedent?

  • Response: No. Affirming the antecedent uses Modus Ponens ($A \rightarrow B$, $A \implies B$), whereas denying the consequent uses Modus Tollens ($A \rightarrow B$, eg B \implies eg A). Both are valid logical forms, but they operate in opposite directions.\n\n* **Question**: In Exercise Step 4, how do we isolate C$when$C does not exist inside an "and" statement?\n * **Response**: First derive A \rightarrow C$via Hypothetical Syllogism from$A \rightarrow B$and$B \rightarrow C$. Then extract$A$from$A \land eg D$via Simplification. Finally, apply Modus Ponens to$A \rightarrow C$and$A$to derive$C$$.