Reason and Responsibility

Philosophy encompasses three primary classical branches:

• Epistemology: The study of the nature of knowledge, exploring what knowledge is and what human beings are capable of knowing.
• Metaphysics: The study of the nature of reality and existence.
• Logic: The science of reasoning and argumentation, which forms the underlying methodological foundation for evaluating philosophical claims.
• Historical origins of deductive logic:
  • Ancient Greek philosopher Aristotle (pupil of Plato) is widely credited with inventing formal deductive logic.
  • Reasoning and argumentation existed prior to Aristotle, just as physical objects fell prior to Sir Isaac Newton's formulations of gravity; Aristotle was the first to systematically codify and outline the logical principles underlying correct argumentation.

Fundamental Terminology in Logic and Argumentation

• Logic:
  • Defined formally as the science of reasoning and argumentation.
  • Logic qualifies as a science because it consists of a set of systematic, consistent, and verifiable principles that govern valid reasoning and sound structure.
• Argument:
  • Defined formally as a connected series of statements intended to establish a truth or proposition.
  • An argument is distinct from an egocentric dispute, emotional conflict, or mere verbal disagreement between individuals.
  • Proper evaluation of an argument requires analyzing the structural relationship between its premises and its conclusion, rather than evaluating the conclusion in isolation.
  • Evaluating conclusions based on superficial traits (e.g., physical appearance or political persuasion of the arguer) is logically irrelevant.
  • The central focus of logical analysis is the inferential path taken from premises to conclusion (e.g., in theological proofs by historical figures with the title of Saint, the conclusion that God exists is known in advance; the logical substance lies entirely in the structure of the path used to arrive at that conclusion).
• Proposition:
  • A statement that possesses a truth value.
  • Truth Value: A fundamental property in logic denoting that a statement can be either true or false.
  • Examples of propositions:
  • "You are Cuban."
  • "Today is Thursday."
  • "Tallahassee is the capital of New York."
  • Note: The truth value of a proposition may be factually true (e.g., a contingently correct statement) or factually false (e.g., asserting Tallahassee as the capital of New York).
  • Non-propositional utterances (lacking truth value):
  • Interrogatives / Questions: Requests for information cannot be true or false.
  • Imperatives / Commands: Directives such as "Go get me a soda" cannot be true or false.

The Logical Concept of Validity

• Definitions of Validity:
  • Formulation 1: An argument form such that if the premises are true, the conclusion must be true (or cannot fail to be true / is necessarily true).
  • Formulation 2: An argument form where true premises guarantee the truth of the conclusion.
• Three Major Misconceptions Regarding Validity:
  1. Validity is not a synonym for factual truth:
  • An argument can consist entirely of factually false propositions and still remain logically valid.
  1. Validity is strictly a property of arguments:
  • Whole arguments are valid or invalid.
  • Individual claims, propositions, premises, and conclusions possess truth values (true or false); they can never be valid or invalid.
  1. Validity refers exclusively to the argument's structural form:
  • Validity depends on the symbolic and structural pattern connecting the premises to the conclusion, not on the concrete subject matter or factual correctness of the propositions.

Structure and Analysis of Argument Forms

• Anatomy of a Formal Argument:
  • Premises: The propositions located above the structural horizontal line; these provide the reasons or evidence.
  • Conclusion: The proposition located below the structural horizontal line; this represents the claim established by the premises.
• Form 1: Modus Ponens (Valid Form)
  • Symbolic Representation:

$\text{Premise 1: If } p\text{, then } q$

$\text{Premise 2: } p$

$\text{Conclusion: Therefore, } q$

• Concrete Example 1A (Factually True Statements):
  • Premise 1: If today is Thursday, then tomorrow is Friday.
  • Premise 2: It is Thursday.
  • Conclusion: Therefore, tomorrow is Friday.
  • Status: Valid. True premises guarantee a true conclusion.
• Concrete Example 1B (Factually False Statements):
  • Premise 1: If today is Tuesday, then tomorrow is Saturday.
  • Premise 2: Today is Tuesday.
  • Conclusion: Therefore, tomorrow is Saturday.
  • Status: Valid.
  • Explanation: Although Premise 1, Premise 2, and the Conclusion are all factually false according to standard calendars, the argument is structurally valid. If it were hypothetically true that Saturday immediately followed Tuesday and that today were Tuesday, the conclusion that tomorrow is Saturday could not fail to be true.
  • Form 2: Affirming the Consequent (Invalid Form)
• Definition of Invalidity:
  • An argument form such that even if the premises were true, the conclusion may still be false.
  • True premises do not guarantee a true conclusion in an invalid form.
• Symbolic Representation:

$\text{Premise 1: If } p\text{, then } q$

$\text{Premise 2: } q$

$\text{Conclusion: Therefore, } p$

• Concrete Example 2A:
  • Premise 1: If I am human, then I am a rational animal.
  • Premise 2: I am a rational animal.
  • Conclusion: Therefore, I am human.
  • Status: Invalid.
• Counterexample Method (Demonstrating Invalidity via Substitution):
  • To prove a form invalid, substitute terms to construct an argument with demonstrably true premises and a false conclusion.
  • Premise 1: If I am in Buffalo, then I am in New York State. (True: Buffalo is a city in NY State; being in Buffalo is a sufficient condition for being in NY State).
  • Premise 2: I am in New York State. (True).
  • Conclusion: Therefore, I am in Buffalo. (False: One can be located in New York State outside the city limits or county of Buffalo).
  • Status: Invalid. The presence of true premises yielding a false conclusion proves that this structural form does not guarantee truth.
• Mechanics of Conditionals ($\text{If } p\text{, then } q$):
  • Antecedent: The proposition following "if" ($p$).
  • Consequent: The proposition following "then" ($q$).
  • A conditional expresses a relationship of sufficiency where the truth of the antecedent ($p$) guarantees the truth of the consequent ($q$).
  • Reversing this relationship (affirming the consequent $q$ to infer antecedent $p$) fails because the consequent can be satisfied through alternative conditions (e.g., being in New York State without being in Buffalo, or being a mammal without being human).
  • Form 3: Denying the Antecedent (Invalid Form)
• Symbolic Representation:

$\text{Premise 1: If } p\text{, then } q$

$\text{Premise 2: Not } p$

$\text{Conclusion: Therefore, not } q$

• Concrete Example 3A:
  • Premise 1: If today is Thursday, then we have Intro to Philosophy.
  • Premise 2: It is not Thursday.
  • Conclusion: Therefore, we do not have Intro to Philosophy.
  • Status: Invalid.
• Counterexample Method (Demonstrating Invalidity):
  • Premise 1: If I have a dog, then I am a mammal. (True: All dogs are mammals).
  • Premise 2: I am not a dog. (True: Subject is human).
  • Conclusion: Therefore, I am not a mammal. (False: Humans are biological mammals containing functional mammary glands).
  • Status: Invalid. Negating the antecedent does not preclude the consequent from occurring via other means.
  • Form 4: Modus Tollens (Valid Form)
• Symbolic Representation:

$\text{Premise 1: If } p\text{, then } q$

$\text{Premise 2: Not } q$

$\text{Conclusion: Therefore, not } p$

• Concrete Example 4A:
  • Premise 1: If Lassie is a snake, then Lassie is a reptile.
  • Premise 2: Lassie is not a reptile.
  • Conclusion: Therefore, Lassie is not a snake.
  • Status: Valid.
• Geographic Analogy (Spatial Containment):
  • Premise 1: If located in Tallahassee, then located in Florida. (Tallahassee is wholly contained within Florida).
  • Premise 2: Not located in Florida.
  • Conclusion: Therefore, not located in Tallahassee.
  • Explanation: If a set ($p$) is entirely contained within a larger set ($q$), being outside the larger set ($\text{Not } q$) guarantees being outside the contained set ($\text{Not } p$).

Summary Matrix of Truth Values and Validity

• Permutations of validity and truth components in formal deductive logic:
  • Valid Arguments can possess:
  • True Premises and a True Conclusion.
  • False Premises and a True Conclusion.
  • False Premises and a False Conclusion.
  • Forbidden Combination: A Valid argument can NEVER have all True Premises and a False Conclusion.
  • Invalid Arguments can possess:
  • True Premises and a True Conclusion.
  • True Premises and a False Conclusion.
  • False Premises and a True Conclusion.
  • False Premises and a False Conclusion.