Introduction to Logic

1. Logic: The Art of Correct Reasoning

Logic is the study of the methods and principles used to distinguish correct from incorrect reasoning.

Why logic matters. When we need reliable judgments, reason is our best tool. Logic provides the objective criteria to evaluate reasoning, helping us tell the difference between sound arguments and faulty ones. It's not about how the mind thinks, but whether the conclusion truly follows from the premises.

Reasoning is key. All reasoning is thinking, but not all thinking is reasoning. Logic focuses specifically on the kind of thinking where inferences are made and conclusions are drawn from premises. It's the science of this process, ensuring correctness.

A practical skill. Studying logic helps us recognize our innate reasoning capacities and strengthen them through practice. It illuminates the principles of correct reasoning, making us more careful thinkers and more rational actors in all spheres of life.

2. Arguments: Building Blocks of Logic

In any argument we affirm one proposition on the basis of some other propositions.

Propositions are statements. Propositions are the fundamental units of reasoning; they are statements that assert something is or is not the case and are always either true or false. Sentences express propositions, but different sentences can express the same proposition.

Arguments link propositions. An argument is a group of propositions where one (the conclusion) is claimed to follow from the others (the premises), which provide support. Arguments are not disputes, but structured clusters of propositions exhibiting an inference.

Structure matters. The order of premises and conclusion doesn't determine argument quality, but identifying them is crucial. Indicator words like "therefore" or "because" can help, but context is often key. Arguments can be simple (one premise) or complex, sometimes with unstated premises (enthymemes).

3. Deduction vs. Induction: Certainty vs. Probability

A deductive argument makes the claim that its conclusion is supported by its premises conclusively. An inductive argument, in contrast, does not make such a claim.

Deduction guarantees. Deductive arguments claim their premises provide incontrovertible grounds for the conclusion. If this claim is correct, the argument is valid; if not, it's invalid. Validity is formal – if premises are true, the conclusion must be true.

• Validity applies only to arguments, not propositions.
• A valid argument with true premises is sound, guaranteeing a true conclusion.

Induction estimates. Inductive arguments claim premises give only some degree of probability to the conclusion. Their conclusions are never certain, so they are evaluated as better or worse, stronger or weaker, not valid or invalid.

• New information can strengthen or weaken an inductive argument.
• Inductive arguments are crucial for establishing matters of fact and causal connections.

Different goals. Deduction aims to clarify what is already known or implied by premises. Induction aims to discover new facts and establish probable truths about the world based on experience and evidence.

4. Language: The Double-Edged Sword of Logic

Good definitions are plainly very helpful in eliminating verbal disputes, but there are other uses of definition that are important in logic.

Language functions. Language serves multiple purposes: informative (asserting facts), expressive (conveying feelings), and directive (guiding actions). It also has ceremonial and performative uses. Logicians focus primarily on the informative function.

Emotive vs. neutral. Words carry both descriptive (literal) and emotive (attitudinal) meanings. Emotive language can be persuasive but also manipulative, especially in advertising or politics. Neutral language is preferred for unbiased reasoning.

Definitions clarify. Definitions explain the meaning of symbols (words, phrases).

• Stipulative: Assigns a new meaning (neither true nor false).
• Lexical: Reports existing meaning (can be true or false).
• Precising: Reduces ambiguity or vagueness.
• Theoretical: Encapsulates a theory.
• Persuasive: Influences attitudes using emotive language.

Meaning has layers. The meaning of a general term includes its extension (the objects it applies to) and its intension (the attributes shared by those objects). Intension determines extension, but not vice versa. Definitions by genus and difference explain intension by identifying a larger class and the distinguishing attribute.

5. Fallacies: Recognizing Errors in Reasoning

Logicians, however, commonly use the term “fallacy” more narrowly, to designate not just any error in reasoning, but typical errors—mistakes in reasoning that exhibit a pattern that can be identified and named.

What fallacies are. Fallacies are arguments that seem correct but contain a mistake in reasoning. They are typical patterns of error, often arising from language pitfalls (informal fallacies) or structural flaws (formal fallacies).

Types of informal fallacies:

• Relevance: Premises are irrelevant to the conclusion (e.g., Appeal to Emotion, Red Herring, Ad Hominem).
• Defective Induction: Premises are relevant but too weak (e.g., Appeal to Inappropriate Authority, Hasty Generalization).
• Presumption: Premises assume too much (e.g., Begging the Question, Complex Question).
• Ambiguity: Meaning shifts within the argument (e.g., Equivocation, Composition, Division).

Why they deceive. Informal fallacies often exploit psychological biases or linguistic imprecision, making irrelevant or weak premises seem convincing. Detecting them requires careful analysis of the argument's content and context.

Avoiding pitfalls. Understanding common fallacy patterns helps us identify flawed reasoning in others' arguments and avoid making such mistakes in our own. It's a crucial skill for critical thinking.

6. Classical Logic: The Power of Categorical Syllogisms

A categorical syllogism is a deductive argument consisting of three categorical propositions that together contain exactly three terms, each of which occurs in exactly two of the constituent propositions.

Categorical propositions. Classical (Aristotelian) logic focuses on arguments about classes of objects, expressed in four standard forms:

• A (Universal Affirmative): All S is P
• E (Universal Negative): No S is P
• I (Particular Affirmative): Some S is P
• O (Particular Negative): Some S is not P

Syllogism structure. A categorical syllogism has two premises and a conclusion, with three terms: major (predicate of conclusion), minor (subject of conclusion), and middle (in both premises). Standard form places the major premise first, then the minor, then the conclusion.

Mood and figure. The mood is the sequence of proposition types (e.g., AAA, EIO). The figure is the position of the middle term in the premises (four possible figures). Mood and figure together define the syllogism's form.

Validity is formal. A syllogism's validity depends solely on its form, not its content. Only 15 of the 256 possible forms are valid in the modern (Boolean) interpretation. Techniques like Venn diagrams or applying syllogistic rules (e.g., undistributed middle, illicit process) test validity.

7. Modern Logic: Symbols Unlock Deeper Structure

Symbols greatly facilitate our thinking about arguments.

Beyond syllogisms. Many valid deductive arguments cannot be analyzed by classical syllogistics because their validity depends on the internal structure of propositions or on logical connections beyond the A, E, I, O forms. Modern symbolic logic provides tools for this.

Symbolic language. An artificial language with precise symbols avoids the ambiguities of natural language. Key truth-functional connectives are:

• Conjunction (and): •
• Negation (not): ~
• Disjunction (or): ∨ (inclusive)
• Material Implication (if-then): ⊃
• Material Equivalence (if and only if): ≡

Truth tables. The truth value of a truth-functional compound statement is determined solely by the truth values of its components. Truth tables display all possible truth value combinations, defining connectives and testing argument forms for validity.

Argument forms. Validity is a formal property. An argument form is valid if it has no substitution instance with true premises and a false conclusion. Truth tables provide a mechanical test for this.

8. Quantification: Analyzing Internal Propositional Structure

His discovery of quantification has been called the deepest single technical advance ever made in logic.

Analyzing non-compound statements. Traditional logic struggled with arguments whose validity depended on the internal structure of non-compound statements (like "Socrates is human" or "All humans are mortal"). Quantification theory provides the tools to analyze these.

Individuals and predicates. Singular propositions assert an attribute of an individual (e.g., "Socrates is mortal").

• Individual constants (a, b, c...) denote individuals.
• Capital letters (A, B, C...) symbolize predicates (attributes).
• Propositional functions (Ax, Bx) contain variables and become statements upon substitution (Aa, Ba).

Quantifiers generalize. Propositional functions can become general propositions through quantification:

• Universal quantifier (x): "(x)Mx" means "Everything is mortal."
• Existential quantifier (∃x): "(∃x)Bx" means "Something is beautiful."

Traditional forms revisited. A, E, I, O propositions are translated using quantifiers and propositional functions:

• A: (x)(Φx ⊃ Ψx)
• E: (x)(Φx ⊃ ~Ψx)
• I: (∃x)(Φx - Ψx)
• O: (∃x)(Φx - ~Ψx)

Rules for proof. Four quantification rules (Universal Instantiation, Universal Generalization, Existential Instantiation, Existential Generalization) are added to the rules of inference, enabling formal proofs for arguments involving general propositions.

9. Causal Reasoning: Uncovering Connections in the World

Our ability to control our environment, to live successfully and to achieve our purposes, depends critically on our knowledge of causal connections.

Understanding "cause". The word "cause" has multiple meanings:

• Necessary condition: Circumstance without which the event cannot occur (e.g., oxygen for fire).
• Sufficient condition: Circumstance in whose presence the event must occur (e.g., being in a temperature range + oxygen for combustion).
• Critical factor: The one circumstance that made the difference in a specific case.
• Necessary and sufficient condition: The conjunction of all necessary conditions.

Causal laws. Every assertion of causation implies a general causal law: similar causes produce similar effects. These laws are discovered empirically, not deductively.

Mill's Methods. Five patterns of inductive inference help identify causal connections:

• Agreement: Common circumstance among cases where phenomenon occurs.
• Difference: Circumstance present when phenomenon occurs, absent when it doesn't.
• Joint Method: Combines agreement and difference.
• Residues: Subtracting known effects to find cause of remainder.
• Concomitant Variation: Phenomena varying together.

Limitations. These methods are powerful for testing hypotheses but are not foolproof rules for discovery or proof. They rely on prior analysis of relevant factors and observed correlations, which may be incomplete or misleading.

10. Scientific Method: Hypothesis, Testing, and Explanation

The aim of science is to discover general truths (chiefly in the form of causal connections like those discussed in the preceding chapter) with which the facts we encounter can be explained.

Explanation in science. Scientific explanations are theoretical accounts (hypotheses or theories) from which the facts to be explained can be logically inferred. They must be relevant, general, and empirically verifiable.

Stages of inquiry:

1. Identify the problem (a puzzling fact).
2. Devise preliminary hypotheses (tentative explanations).
3. Collect additional facts (guided by hypotheses).
4. Formulate refined explanatory hypothesis (accounts for all facts).
5. Deduce further consequences (predictions).
6. Test consequences (empirical verification/falsification).
7. Apply the theory (practical use).

Evaluating hypotheses. Competing scientific explanations are judged by:

• Compatibility with established theories.
• Predictive power (testable predictions).
• Simplicity (though this can be complex).

Hypotheses are provisional. Scientific explanations are held provisionally, subject to revision or rejection based on new evidence. Falsifiability is a key aspect of testability.

Classification as hypothesis. Even descriptive activities like classification involve hypotheses about which attributes are most important for revealing causal connections and suggesting fruitful laws.

11. Probability: Quantifying Uncertainty

Probability is the central evaluative concept in all inductive logic.

Probability measures likelihood. Inductive conclusions are probable, not certain. Probability quantifies this likelihood, expressed as a number between 0 (impossibility) and 1 (certainty).

Conceptions of probability:

• A priori: Based on the ratio of favorable outcomes to equipossible outcomes (e.g., coin flip).
• Relative frequency: Based on the observed frequency of an attribute within a reference class (e.g., mortality rates).

Calculus of probability. This mathematical branch computes probabilities of complex events from component probabilities using theorems:

• Product Theorem: For joint occurrences (multiply probabilities).
  • Independent events: P(a and b) = P(a) × P(b)
  • Dependent events: P(a and b) = P(a) × P(b if a)
• Addition Theorem: For alternative occurrences (add probabilities).
  • Mutually exclusive events: P(a or b) = P(a) + P(b)
  • Not mutually exclusive: Use indirect methods (e.g., 1 - P(neither)).

Expectation value. In decisions involving risk, expectation value helps evaluate choices. It's the sum of possible returns multiplied by their probabilities. Rational decisions often involve maximizing expectation value, balancing risk and return.

Last updated: May 5, 2025