5.2 Logical Statements

Logical statements form the backbone of philosophical reasoning. They help us analyze arguments, identify flaws, and build stronger cases. Understanding necessary and sufficient conditions, counterexamples, and conditional statements is crucial for clear thinking.

Validity in logic isn't about truth, but about structure. A valid argument guarantees that if the premises are true, the conclusion must be true. This chapter equips you with tools to evaluate arguments, spot fallacies, and construct sound reasoning.

Logical Statements

Necessary vs sufficient conditions

Top images from around the web for Necessary vs sufficient conditions

• 

  Categorical Logic – Critical Thinking View original

  Is this image relevant?

• 

  Categorical Logic – Critical Thinking View original

  Is this image relevant?

• 

  Logical reasoning - Wikipedia View original

  Is this image relevant?

• 

  Categorical Logic – Critical Thinking View original

  Is this image relevant?

• 

  Categorical Logic – Critical Thinking View original

  Is this image relevant?

1 of 3

Top images from around the web for Necessary vs sufficient conditions

• 

  Categorical Logic – Critical Thinking View original

  Is this image relevant?

• 

  Categorical Logic – Critical Thinking View original

  Is this image relevant?

• 

  Logical reasoning - Wikipedia View original

  Is this image relevant?

• 

  Categorical Logic – Critical Thinking View original

  Is this image relevant?

• 

  Categorical Logic – Critical Thinking View original

  Is this image relevant?

1 of 3

• Necessary condition must be met for a statement to be true, if not met the statement is false (being a mammal is necessary for being a dog)
  • If an animal is not a mammal it cannot be a dog, but being a mammal does not guarantee it is a dog (cat, whale)
• Sufficient condition guarantees the truth of a statement if met, the statement must be true (being a dog is sufficient for being a mammal)
  • If an animal is a dog it must be a mammal, but not all mammals are dogs (human, elephant)
• If A is a necessary condition for B, then B is a sufficient condition for A (being a dog is necessary for being a golden retriever, being a golden retriever is sufficient for being a dog)

Counterexamples for universal claims

• Universal claims assert something is true for all cases using words like "all," "every," "always," or "never" (all birds can fly)
• Counterexamples are specific instances that disprove a universal claim, must be relevant and clearly demonstrate the falsity of the claim (penguins, ostriches)
• Effective counterexamples clearly and directly contradict the universal claim, are specific and well-defined, not exceptions to the rule but demonstrate the rule is false
• Steps to construct a counterexample:
  1. Identify the universal claim being made
  2. Consider possible exceptions or cases that do not fit the claim
  3. Select a specific, well-defined example that clearly contradicts the claim
  4. Present the counterexample and explain how it disproves the universal claim

Validity of conditional statements

• Conditional statements are in the form "If P, then Q" (P → Q), P is the antecedent and Q is the consequent (if it is raining, then the ground is wet)
• A conditional statement is valid if whenever the antecedent (P) is true, the consequent (Q) must also be true, counterexamples can prove a conditional statement is invalid
• Counterexample types for conditional statements:
  • P is true but Q is false (P ∧ ¬Q), shows the consequent does not necessarily follow from the antecedent (a dishonest politician)
  • Q is true but P is false (¬P ∧ Q), does not disprove the conditional statement but shows Q can be true without P being true (ground wet from sprinkler, not rain)
• Evaluating the validity of conditional statements involves considering possible counterexamples where P is true but Q is false, if found the statement is invalid, if none can be found it is considered valid
• Logical connectives, such as "and," "or," and "not," are used to combine or modify simple statements in conditional statements

Types of Reasoning and Logical Structures

• Deductive reasoning: A form of logical argument where the conclusion necessarily follows from the premises if they are true
• Inductive reasoning: A method of drawing general conclusions from specific observations or examples
• Syllogism: A form of deductive reasoning consisting of a major premise, a minor premise, and a conclusion
• Truth tables: Used to determine the truth value of complex statements involving logical connectives
• Fallacy: An error in reasoning that undermines the logic of an argument