Introduction to Logic (Phil 201): Comprehensive Course Notes

Foundational Concepts in Philosophy and Logic

Philosophy is the systematic inquiry into the principles and presuppositions of any field of study. Etymologically, the term derives from classical Greek word fragments:

• Philia / phile / philo: Love of, affinity for, or liking of.
• Sophia / sophos / sophist: Wisdom.

Thus, philosophy translates directly to the "love of wisdom."

The Core Branches of Philosophy

The discipline of philosophy is organized into major branches determined by the fundamental nature of the questions addressed within each domain. These divisions are not absolute; their boundaries overlap significantly.

1. Metaphysics (Ontology)

Metaphysics, or ontology, is the study of ultimate reality—investigating what is "really real." It examines the first principles of the natural order and the ultimate generalizations accessible to the human intellect. Ontology focuses specifically on identifying existent things and defining the relationships between their categories.

Central inquiries include:

• What types of entities exist? Do only concrete, particular objects exist, or do abstract, general entities exist as well?
• How is existence possible, and what governs the conditions of possibility and necessity for events?
• Identity and change across time: Is an individual the exact same person they were as an infant, as of yesterday, or as of a single moment ago?
• The ontological status of non-extended entities: If ideas lack physical dimensions, mass, shape, or color, in what manner do they exist? (An idea of the Empire State Building is neither physically larger nor smaller than an idea of a pocketbook; ideas lack spatial extension).
• The nature of space and time.
• The reality of abstract mathematical and geometric entities: In what manner do geometric figures exist? Are points, lines, and planes physically real, and what constitutes their substance?
• The ontological distinction between mind, soul, spirit, and physical matter: Do they share an identical fundamental substance?

2. Epistemology

Epistemology is the philosophical study of knowledge. It specifically examines the nature, origin, structure, methods, scope, and cognitive limits of human knowledge.

A primary epistemological question involves the degree of truth inherent to propositions—for example, evaluating the statement "The earth is round" and determining whether its truth-value depends strictly on the context and precision of the utterance.

3. Axiology (Value Theory)

Axiology investigates the nature, criteria, and metaphysical status of value. Core areas of investigation include:

• Nature of value: Whether value constitutes the fulfillment of subjective desire, psychological pleasure, personal preference, a behavioral disposition, or a broader human interest.
• Criteria of value: Whether values are governed by objective, universal standards or subjective metrics.
• Status of value: The relationship connecting values to empirical scientific facts, and the ultimate worth of human values.

Axiology is subdivided into two primary fields:

• Ethics: The study of values governing human conduct and moral dilemmas. It addresses: (1) the rightness and wrongness of actions, (2) the determination of things that are inherently good or desirable, and (3) the conditions under which human actions are blameworthy or praiseworthy.
• Aesthetics: The study of value in the arts, encompassing the inquiry into feelings, judgments, and standards of beauty. It explores judgments of sensory perception, taste, and emotional expression. Key questions address whether artistic creation is an intellectual or representational activity; what realistic representations in pop art signify; whether art depicts sensible or ideal objects; whether artistic merit is objective; why architectural and painterly forms frequently mirror mathematical principles; and the demarcation between art and physical reality.

4. Logic

Logic is the branch of philosophy that evaluates arguments. It is defined as the normative science that appraises reasoning by providing systematic methods for criticizing the arguments of others and constructing sound arguments of one's own.

Historical Development of Logic

Logic has evolved across classical antiquity, the medieval period, and modern mathematics:

• Aristotle (384–322 B.C.): Recognized as the father of formal logic. He developed syllogistic (categorical) logic and catalogued a substantial body of informal fallacies.
• Peter Abelard (1079–1142): The foremost logician of the Middle Ages. Abelard refined Aristotelian syllogistic logic and formulated a distinct theory of universals.
• William of Occam (c. 1285–1349): Oxford philosopher who advanced modal logic (the formal analysis of possibility, necessity, belief, and doubt) and contributed foundational concepts to metalanguage.
• Gottfried Wilhelm Leibniz (1646–1716): Introduced formal symbolic logic, establishing concepts later expanded by Augustus De Morgan, George Boole, William Stanley Jevons, and John Venn.
• John Stuart Mill (1806–1873): British philosopher who revitalized inductive logic, formalizing systematic methods for empirical inquiry.
• Gottlob Frege (1848–1925): Founded modern mathematical logic, formulating quantification theory and formal logic systems.
• Modern Technological Role: Logic provides the conceptual, mathematical, and algorithmic architecture necessary for designing digital computers and electronic circuitry.

The Nature and Structural Components of Arguments

Definition of an Argument

In formal logic, an argument is a group of statements wherein one or more statements (the premises) are claimed to provide supporting evidence or rational justification for believing another statement (the conclusion). Depending on how effectively the premises support the conclusion, an argument is classified as logically good or bad.

Premise 1: Man is mortal.
Premise 2: Professor Kitaw is man.
Conclusion: Professor Kitaw is mortal.

Components of an Argument

• Statement: A declarative sentence that possesses a definite truth value—it is either strictly true ($T$) or false ($F$).
• Proposition: The informational content or semantic meaning conveyed by a statement. In general logical analysis, "proposition" and "statement" are used interchangeably.
  • Example: "Cairo is in Ethiopia" is a statement possessing a truth value (specifically, False).
• Premises: The declarative statements setting forth the evidential grounds, data, or reasons for accepting the conclusion. An argument can contain multiple premises.
• Conclusion: The single statement supported or implied by the premises. An argument possesses one and only one conclusion.
• Inference: The cognitive and logical process by which a conclusion is derived from premises. The term is often used interchangeably with "argument."
• Immediate Inference: An argument consisting of exactly one premise and one conclusion.
  • Example:
  • Premise: All Japan cell phones are original apparatuses.
  • Conclusion: Therefore, it is false that no Japan cell phones are original apparatuses.

Indicator Words

Premises and conclusions are frequently identified by distinctive structural indicators:

Recognizing Arguments

A passage contains an argument if and only if it explicitly purports to prove something. To satisfy this condition, the passage must satisfy two necessary claims: a factual claim and an inferential claim.

1. Factual Claim vs. Inferential Claim

1. Factual Claim: A claim that at least one of the statements presents empirical evidence, grounds, or reasons. This claim pertains directly to the premises.
2. Inferential Claim: The claim that a conclusion follows logically from the asserted factual evidence. This claim governs the logical nexus connecting premises to conclusion.

Inferential claims can be expressed in two ways:

• Explicit Inferential Claims: Stated directly through premise or conclusion indicator words.
• Implicit Inferential Claims: Stated without indicator words, where the structural, inferential relationship between statements implies a derivation.
  • Example of an implicit inferential claim (Boikess and Edelson, cited in Hurley):     "The price reduction [seen with electronic calculator] is the result of a technological revolution. The calculator of 1960s used integrated electronic circuit that contained bout a dozed transistors or similar components on a single chip. Today, mass production chips, only a few millimeters square contain several thousand such components."

To identify whether a passage constitutes an argument, evaluate:

1. The presence of indicator words.
2. The existence of an inferential connection among statements.

2. Non-Argumentative Passages

Many texts lack an inferential claim and do not constitute arguments:

• Passages Lacking an Inferential Claim: Warnings, expressions of subjective belief or opinion, loosely associated statements, factual reports, expository passages, and illustrations.
• Conditional Statements: A conditional assertion formatted as "If [antecedent], then [consequent]" does not constitute an argument. The antecedent and consequent signify an asserted hypothetical or causal dependency rather than an inferential premise-to-conclusion relationship.
  • Example: "If Alemu studies hard, he will pass the exam." (Here, "Alemu studies hard" is the antecedent; "he will pass the exam" is the consequent).
  • Structural Rules for Conditionals:
  1. A single conditional statement is never an argument on its own.
  2. A conditional statement can serve as a premise, a conclusion, or both within an argument. Premise 1: If Alemu studies hard, he will pass the exam. Premise 2: Alemu really studies hard. Conclusion: Therefore, Alemu will pass the exam.        
  3. The inferential meaning of a conditional statement can be restructured into an argument.
• Explanations: Passages that elucidate why an acknowledged fact occurred, rather than attempting to prove that it occurred. Explanations consist of two components:
  • Explanandum: The accepted factual statement being explained.
  • Explanans: The statement or group of statements offering the causal or mechanical account.
  • Contrast with Arguments: Arguments seek to demonstrate the truth of a disputed conclusion via premises; explanations accept the explanandum as true and identify its causes.
  • Restructuring Explanations into Arguments:
  • Explanation: "The sky appears blue from the earth's surface because light rays from the sun are scattered by particles in the atmosphere."
  • Argument Form: "Light rays from the sun are scattered by particles in the atmosphere. Therefore, the sky appears blue from the earth's surface."
  • Interpretive Ambiguity: Some passages contain twofold indicators permitting dual interpretations as either explanations or arguments. For example: "Women become intoxicated by drinking a smaller amount of alcohol than men, because men metabolize part of the alcohol before it reaches the bloodstream, whereas women do not" (Hurley, p. 23).

Deductive vs. Inductive Arguments

Arguments are bifurcated into deductive and inductive classifications according to the absolute strength of their inferential claim.

Deductive Arguments

A deductive argument claims that the conclusion follows with absolute, inescapable necessity from the premises. Assuming the premises are true, it is impossible for the conclusion to be false.

Premise 1: All Ethiopians love unity.
Premise 2: ILS students are Ethiopians.
Conclusion: Therefore, ILS students love unity.

Inductive Arguments

An inductive argument claims that the conclusion follows with probability from the premises. Assuming the premises are true, it is improbable that the conclusion is false.

Premise 1: The vast majority of AU students are from Tigray region.
Premise 2: Engineering students are AU students.
Conclusion: Therefore, Engineering students are from Tigray region.

Deciding Between Deduction and Induction

Classifying an argument as deductive or inductive depends on three primary factors:

1. Special Indicator Terminology:
   • Deductive indicators: Necessarily, certainly, absolutely, definitely.
   • Inductive indicators: Probably, improbable, plausible, likely, unlikely, reasonable to conclude.
2. Actual Strength of the Inferential Claim: The objective connection between premises and conclusion, regardless of indicators.
3. Argument Form and Structural Category:

Note on Scientific Inferences: Scientific arguments can be deductive or inductive depending entirely on their logical structure and whether their conclusions are claimed with necessity or probability.

Rejection of the Historical View: The traditional view that deduction moves strictly "from the general to the particular" and induction moves "from the particular to the general" is rejected by modern logicians as technically inaccurate.

Evaluating Arguments

Evaluating any argument requires resolving two distinct questions:

1. The Inferential Claim: Do the premises logically support the conclusion?
2. The Factual Claim: Are all the premises objectively true?

Evaluating Deductive Arguments

Deductive arguments are appraised across three criteria: Validity, Truthfulness, and Soundness.

Validity vs. Invalidity

Validity is governed entirely by structural form—the strict logical relationship connecting premises to conclusion—not by factual truth.

• Valid Deductive Argument: An argument constructed such that if the premises are assumed true, it is mathematically impossible for the conclusion to be false. The conclusion follows with absolute necessity.
• Invalid Deductive Argument: An argument constructed such that even if the premises are assumed true, it remains possible for the conclusion to be false. The inferential link fails to achieve absolute necessity.

There is no middle ground between valid and invalid arguments.

Premise 1: All cats are animals. (True)
Premise 2: Dogs are animals. (True)
Conclusion: Therefore, dogs are cats. (False)
Evaluation: Invalid.

Truth-Values and Validity

The factual truth or falsity of an argument's statements is irrelevant to validity, with one single exception: an argument with true premises and a false conclusion is always invalid.

• False Premises and False Conclusion Yielding a Valid Form:

  Premise 1: All fruits are apples. (False)
  Premise 2: Potato is a fruit. (False)
  Conclusion: Therefore, potato is an apple. (False)
  Evaluation: Valid.
  ```

![Concentric Euler diagram illustrating class containment for fruits, apples, and potato](https://assets.knowt.com/pdf-flow-prod/6724d9f5-c078-4ec9-82b0-8bc3da669f66-figures/16.png)

In this valid syllogistic structure, the class "Potato" is entirely subsumed within the class "Fruits," which is itself entirely subsumed within the class "Apples." The formal conclusion "potato is an apple" follows necessarily from the premises, demonstrating that validity depends strictly on logical form rather than empirical accuracy.

- **True Premises and True Conclusion Yielding an Invalid Form:**

Premise 1: All soft drinks are non-alcoholics. (True) Premise 2: Coca-Cola is non-alcoholic. (True) Conclusion: Therefore, Coca-Cola is soft drink. (True) Evaluation: Invalid.   ```

Although all three statements are empirically true, the structure is invalid. As diagrammed, both "Soft drink" and "Coca-Cola" fall inside the broader category of "Non-alcoholic," but that broad inclusion does not prove that Coca-Cola belongs inside the specific subset of soft drinks. The conclusion does not follow with logical necessity.

Soundness vs. Unsoundness

Soundness combines factual truth with formal validity:

• Sound Deductive Argument: An argument that is valid and has entirely true premises. A sound argument necessarily delivers a true conclusion.
• Unsound Deductive Argument: An argument that is either invalid, contains at least one false premise, or both. All invalid deductive arguments are automatically unsound.

Summary Table: Deductive Logic Combinations

Note on Table: The question mark ($?$) indicates that the specific outcome depends on the argument's structural form. An argument possessing true premises and a false conclusion can never be valid.

Evaluating Inductive Arguments

Inductive arguments are evaluated along three distinct axes: Strength, Truthfulness, and Cogency.

Strength vs. Weakness

Unlike the absolute dichotomy of validity, inductive strength is evaluated on a probabilistic continuum:

• Strong Inductive Argument: An argument where, if the premises are assumed to be true, it is highly probable that the conclusion is true.
  • Example: Premise: Famine had been occurring in every decade for the last several consecutive decades in Ethiopia. (True) Conclusion: Therefore, probably there will be famine in Ethiopia in the next decade. (Probably True) Evaluation: Strong.     
• Weak Inductive Argument: An argument where, even if the premises are assumed to be true, the conclusion does not follow with high probability.
  • Example: Premise 1: The dreams that Weyzero Debre repeatedly dreams sometimes occur directly. (True) Premise 2: Weyzero Debre dreamt that there will be a government change in Ethiopia. (True) Conclusion: Therefore, probably there will be a government change in Ethiopia. (Probably False) Evaluation: Weak (The conclusion has low probability because her dreams occur directly only occasionally).     

Factors determining inductive strength include sample size, representativeness, and the credibility of cited authorities or witnesses.

Truth-Values and Inductive Strength

Empirical truth or falsity does not determine whether an argument is inductively strong or weak:

• False Premises Yielding a Strong Inductive Argument:

  Premise: All previous Soccer world cups were won by the Ethiopian team. (False)
  Conclusion: Therefore, probably the next world cup will be won by the Ethiopian team. (Probably False)
  Evaluation: Strong (If the premise were assumed true, the conclusion would follow with high probability).
  ```
- **True Premises Yielding a Weak Inductive Argument:**

Premise: A few Ethiopian leaders were warriors. (True) Conclusion: Therefore, probably the next Ethiopian leader will be a man. (Probably True) Evaluation: Weak (The premise provides weak inferential support for the conclusion, even though both statements are empirically true).   ```

Cogency vs. Uncogency

• Cogent Inductive Argument: An argument that is inductively strong and possesses all true premises. If an argument satisfies both conditions, its conclusion is probably true.
• Uncogent Inductive Argument: An argument that is inductively weak, contains one or more false premises, or fails on both counts.

Summary Table: Inductive Logic Combinations

Note on Table: The question mark ($?$) indicates that inductive strength depends on how strongly the premises support the conclusion. Any argument with at least one false premise is automatically uncogent.