2.7 Logical Arguments

Arguments come in two main flavors: inductive and deductive. Inductive arguments use evidence to draw probable conclusions, while deductive arguments use logic to reach certain conclusions. Understanding these types helps us evaluate the strength and validity of arguments.

Conditional arguments, using "if-then" statements, follow specific laws like Modus Ponens and Modus Tollens. These laws help us analyze and construct valid arguments. Truth tables and Venn diagrams are handy tools for verifying the validity of arguments visually.

Types of Arguments and Their Evaluation

Inductive vs deductive arguments

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• Inductive arguments draw conclusions based on patterns, observations, or evidence
  • Conclusions are probable, not certain since they rely on empirical evidence (scientific studies, surveys, historical data)
  • Strength is evaluated by the likelihood of the conclusion being true given the premises
    • Strong inductive argument: Premises provide significant support for the conclusion (Most swans observed are white, therefore the next swan observed will likely be white)
    • Weak inductive argument: Premises provide little support for the conclusion (Some birds can fly, therefore penguins can fly)
• Deductive arguments draw conclusions based on logical reasoning from premises
  • Conclusions are certain if the premises are true and the argument is valid since they rely on logical structure (syllogisms, proofs)
  • Validity is evaluated by the structure of the argument, not the truth of the premises
    • Valid deductive argument: Conclusion necessarily follows from the premises (All men are mortal. Socrates is a man. Therefore, Socrates is mortal)
    • Invalid deductive argument: Conclusion does not necessarily follow from the premises (All cats are mammals. All dogs are mammals. Therefore, all cats are dogs)
  • A counterexample can be used to disprove a deductive argument by showing a case where the premises are true but the conclusion is false

Analyzing Conditional Arguments

Laws of conditional arguments

• Conditional arguments involve an "if-then" statement (conditional premise)
  • Antecedent: The "if" part of the conditional premise ($p$ in $p \rightarrow q$)
  • Consequent: The "then" part of the conditional premise ($q$ in $p \rightarrow q$)
• Law of Detachment (Modus Ponens) states if the conditional premise and the antecedent are true, then the consequent must be true
  • Example: If it is raining ($p$), then the ground is wet ($q$). It is raining ($p$ is true). Therefore, the ground is wet ($q$ is true).
• Law of Denying the Consequent (Modus Tollens) states if the conditional premise is true and the consequent is false, then the antecedent must be false
  • Example: If the switch is on ($p$), then the light is on ($q$). The light is not on ($q$ is false). Therefore, the switch is not on ($p$ is false).

Verifying Validity Using Truth Tables and Venn Diagrams

Validity verification techniques

• Truth tables list all possible combinations of truth values for the premises and conclusion

  • An argument is valid if the conclusion is true whenever all the premises are true
  • Example: If it is sunny ($p$), then I will go to the beach ($q$). It is sunny ($p$ is true). Therefore, I will go to the beach ($q$ is true). | $p$ | $q$ | $p \rightarrow q$ | $p$ | $q$ | |:---:|:---:|:-----------------:|:---:|:---:| | T | T | T | T | T | | T | F | F | T | F | | F | T | T | F | T | | F | F | T | F | F |

• Venn diagrams represent sets and their relationships using overlapping circles

  • An argument is valid if the conclusion set is a subset of the intersection of the premise sets
  • Example: All squares ($S$) are rectangles ($R$). All rectangles ($R$) are quadrilaterals ($Q$). Therefore, all squares ($S$) are quadrilaterals ($Q$). [A Venn diagram with three circles labeled $S$ inside $R$ inside $Q$, demonstrating the subset relationships.]

Elements of Logical Arguments

• Premise: A statement or proposition used as a basis for reasoning or argument
• Conclusion: The logical result or consequence derived from the premises
• Logical connectives: Words or symbols used to join simple statements into compound statements (e.g., "and", "or", "if-then")
• Propositional logic: A branch of logic that deals with propositions and their relationships, using symbols and connectives to represent and analyze arguments
• Fallacy: An error in reasoning that undermines the logic of an argument