Advanced Conditional Propositions Notes

Advanced Conditional Propositions

Deeper Look at Conditional Propositions

  • Purpose & Explanation
    • This lesson offers an in-depth exploration of advanced conditional propositions, contrasting expressions in English with symbolic representations.
    • Symbols aid in understanding logical relationships and implications, indicating when different expressions can reflect the same logical meaning.

Negative Conditions

  • Definition: Negative conditions often rely on the use of “unless” or negate the antecedent.
  • Examples:
    • "Unless you come through the door, you can’t get into the room."
    • "If you don’t come through the door, you can’t get into the room."
  • Logical Equivalence:
    • Both sentences symbolize as (~D → ~R) where:
    • D = "you come through the door"
    • R = "you can get into the room"

Contraposition

  • Definition: Contraposition involves switching the antecedent and consequent of a conditional proposition, maintaining logical equivalence.
  • Example: From “if you don’t come through the door, you can’t get into the room,” we derive:
    • Original: (~D → ~R)
    • Contrapositive: (R → D)
  • General Rule:
    • Definition of Contraposition: (p → q) ≡ (~q → ~p)

Types of Conditions

  1. Sufficient Conditions

    • Definition: A condition is sufficient when its occurrence is enough to guarantee the consequent.
    • Example: "If you are good (G), then I’ll give you a lollipop (L)" → G → L.
    • Raising expectations: If the child is good, she feels entitled to the lollipop.

  2. Necessary Conditions

    • Definition: A necessary condition must be met for the consequent to possibly occur.
    • Example: "A high school diploma (D) is required for being accepted to a university (U)" → Symbolized as (D → U).
    • Alternative expressions:
      • Negative Statement: "If you are not a high school graduate (~D), then you will not be accepted (~U)" → (~D → ~U).
      • Positive Statement: "If you are accepted (U), then you must have a high school diploma (D)" → (U → D).

  3. Necessary and Sufficient Conditions

    • Combination: When conditions act both as necessary and sufficient, often expressed as “if and only if.”
    • Example: "If and only if you are good, you will get a lollipop" → (G → L) ∙ (L → G).
    • Alternative Expressions: (i) Bi-conditional: (p ↔ q); (ii) Logical equivalence: (p ≡ q).

“Only If” Propositions

  • Understanding Usage: The phrase “only if” can denote both necessary and sufficient conditions or merely necessary conditions.
  • Examples:
    • "Only if you are good will I give you a lollipop" interpreted as a necessary condition.
    • Ambiguity arises: "If you train hard, only then will you achieve a specific goal" might represent only a necessary component, not a guarantee.

Combined Propositions

  • Recognizes that propositions can intertwine multiple hypothetical expressions.
  • Example: "If Gloria learns English well (G), then she will get a good job (J) and make lots of money (M)" → Symbolized as [G → (J ∙ M)].

Problem Set Exercises

  • Tasks provided to create personal symbols and symbolize conditional propositions for practice, enhancing understanding of logical expressions.
    • Example task 1: "Joe goes to the game if and only if Jamal plays quarterback; and Andrew kicks field goals."
    • Incorporate practical applications based on ordinary language expressions involving conditional logic.