Lecture Notes: Symbolizing Arguments in Logic

Introduction

• Practice in translating statements into logical notation
• Distinguish between atomic and compound statements
• Assign sentence letters to atomic statements
• Focus on logical structure, revealed by statement operators

Main Goals

• Analyze and construct arguments
• Focus primarily on deductive arguments
  • A sound deductive argument guarantees a true conclusion
  • Soundness requires true premises and a valid relationship between premises and conclusion

Methods for Checking Validity

• Counterexample method
  • Create a scenario where premises are true but the conclusion is false
  • Limitations include the challenge of conceiving all possible counterexamples
• Direct checking for validity requires symbolizing the argument

Symbolizing Arguments

• Break compound statements into atomic statements and operators
• Assign uppercase English letters to atomic statements
  • Arbitrary letter choice, but avoid confusion by assigning different letters to different atomic statements
• Use logical connectives (symbols) for conjunction, negation, disjunction, conditional, and biconditional

Example Analysis

1. Statement: "If Sheila doesn't answer the phone, then she's either at work or she's mad at me."

   • Identify Compound Statement: Presence of 'if', 'or' indicates a compound statement
   • Main Operator: Conditional (indicated by 'if-then')
   • Breakdown:
     • Antecedent: Sheila doesn't answer the phone (negation)
     • Consequent: She's at work or she's mad at me (disjunction)
   • Symbolization:
     • Assign letters: P = Sheila does answer the phone, W = She's at work, M = She's mad at me
     • Translated statement: ¬P → (W ∨ M)

2. Statement: "Ann donated money to charity and she encouraged Gabe to, but he didn't."

   • Identify Compound Statement: Presence of 'and', 'but', 'didn't'
   • Main Operator: Conjunction (indicated by comma and 'but')
   • Breakdown:
     • Left Conjunct: Ann donated money to charity and encouraged Gabe to donate money
     • Right Conjunct: Not Gabe donated money
   • Symbolization:
     • Assign letters: A = Ann donated money, E = Ann encouraged Gabe, G = Gabe donated money
     • Translated statement: (A ∧ E) ∧ ¬G

3. Statement: "If you invite neither Mia nor Vincent to the party, then someone else will win the dance contest."

   • Identify Compound Statement: Presence of 'if-then', 'neither-nor'
   • Main Operator: Conditional
   • Breakdown:
     • Antecedent: Neither Mia nor Vincent invited (negated disjunction or conjunction of negations)
     • Consequent: Someone else wins the dance contest
   • Symbolization:
     • Assign letters: M = Invite Mia, V = Invite Vincent, S = Someone else wins
     • Translated statement: ¬(M ∨ V) → S

General Advice

• Practice is crucial for mastering symbolization
• Engage in discussions and seek help if needed
• Treat learning logic like learning a skill; it requires consistent practice and application

Conclusion

• Next steps involve practicing symbolization and applying it to determine argument validity
• Stay active in learning and practice regularly
• Attend office hours and participate in discussions if clarification is needed