# A Concise Introduction to Logic

## Basic Concepts

### Arguments, Premises and Conclusions

- Logic
- The science of evaluating arguments.

- Argument
- A group of statements with a premise and conclusion.

- Statement
- A true or false sentence.

- Truth Values
- Truth and Falsity

The statements in an argument come in two flavors.

- Premises
- The statement of reasons or evidence.

- Conclusion
- The statement the premises support.

Inference is often used as an synonym with argument, and proposition with argument. They have more specific definitions.

- Inference
- The process of reasoning which the argument outlines.

- Proposition
- The content of the statement.

### Recognizing Arguments

Arguments must contain a factual and inferential claim.

- Factual Claim
- One or more statements claiming to present evidence.

- Inferential Claim
- A claim that the evidence implies something.

There are many types of writing that are not arguments.

- Noninferential Passages
- Where there is no claim that the premise supports the conclusion.

Noninferential passages include the following:

- Warnings
- Advice
- Statements of Belief and Opinion
- Loosely Associated Statements
- Reports

Expository passages can sometimes be arguments if they attempt to prove the topic sentence.

Illustrations give examples in an attempt to show something. They can sometimes be arguments if they purport to prove something.

Explanations are not arguments as the conclusion is usually obvious. They are made of two parts.

- Explanans
- The claim to shed light on.

- Explanandum
- The accepted definition.

Conditionals are statements not arguments. They may serves as the premise or conclusion. They are made up of two parts.

- if (antecedent) then (consequent)
- (consequent) if (antecedent)

Related are the concepts of sufficient and necessary conditions.

- Sufficient Condition
- P is all that’s needed for Q to happen.

- Necessary Condition
- Q cannot occur without P.

### Deduction and Induction

- Deductive Argument
- An argument where it is impossible for the conclusion to be false given true premises.

Examples of deductive arguments include the following:

- Mathematical Arguments
- Arguments from Definition
- Syllogisms

- Inductive Argument
- An argument where it is improbable that the conclusion is false given true premises.

Examples of inductive arguments include the following:

- Prediction
- Argument from Analogy
- Generalization
- Argument from Authority
- Argument based on Signs
- Casual Inference

Often times inductive arguments are thought to go from particular to general and deductive the other way around. This is often true, but not always.

- Particular Statement
- A statement that makes a claim about one or more members of a class.

- General Statement
- A statement that makes a claim about all members of a class.

### Validity, Truth, Soundness, Strength, Cogency

- Valid Deductive Argument
- Conclusion must be true given true premises.

- Invalid Deductive Argument
- Conclusion may be false given true premises.

- Sound Argument
- Valid argument with true premises.

A good deductive argument is a sound one.

- Strong Inductive Argument
- Improbable conclusion is false with true premises.

- Weak Inductive Argument
- Conclusion doesn’t follow probably from premises.

- Cogent Argument
- Strong argument and true premises.

- Total Evidence Requirement
- Premises don’t overlook crucial evidence that could undermine conclusion.

A good inductive argument is a cogent one.

Evaluating every argument involves two steps.

- Evaluate the link between the premise and conclusion (the inferential claim). This will show validity or strength.
- Evaluate the truth of the premises (the factual claim). This will show soundness or cogency.

### Argument Forms: Proving Invalidity

The validity of a deductive argument is determined by the argument’s form.

- Substitution Instance
- Uniforming replacing terms in an argument.

The following argument is valid.

- All A are B.
- All B are C.
- $\therefore $ All A are C.

The following argument is invalid.

- All A are B.
- All C are B.
- $\therefore $ All A are C.

The validity of a deductive argument can determined by forming a counterexample.

- Counterexample Method
- Constructs a substitution instance having true premises and a false conclusion.

The method can be described as such:

- Take an invalid argument and extract form.
- Construct a substitution instance with true premises and false conclusion.
- This proves the form invalid and subsequently the argument.

## Language: Meaning and Definition

### Varieties of Meaning

We can divide language by its purpose.

- Cognitive Meaning
- Terminology that conveys information.

- Emotive Meaning
- Terminology that expresses or evokes feelings.

Logic is concerned with the cognitive meaning, so we must distance ourself from the emotive meaning.

- Value Claim
- A claim something is good, bad, right, wrong important or not important.

Value claims require evidence and emotive language sometimes is used to obscure that certain claims are unfounded. This does not mean cognitive meanings are perfect.

- Vague Expression (Continuous)
- Where the meaning allows for many interpretations and is imprecise.

- Ambiguous Expression (Discrete)
- Where the meaning can have different clearly distinct meanings in a context.

These defective expressions can cause dispute.

- Verbal Disputes
- Disputes that arise over the meaning of language.

- Factual Disputes
- Disputes that arise over disagreement about facts.

### The Intension and Extension of Terms

- Term
- Words that may serve as the subject of a statement (e.g. proper names, common names, descriptive phrases).

There are two flavors of terms:

- Intensional Meaning
- Qualities or attributes the term connotes.

- Extensional Meaning
- Members of the class the term denotes.

Because connotations are subjective and may be different to different individuals, we use conventional connotation.

- Conventional Connotation
- The attributes commonly attributed by competent speakers.

- Empty Extension
- A class with no members that can have attributes.

The key connection between these ideas is that intension determines extension.

### Definitions and Their Purposes

- Definition
- Words that assign a meaning to some other words

- Definiendum
- The word to be defined.

- Definiens
- The words that do the defining.

There are different types of definitions:

- Stipulative Definition
- A coinage that assigns meaning to a term for the first time.

- Lexical Definition
- Reportr meaning already present in a language.

- Precising Definition
- Reduces vagueness in a term.

- Theoretical Definition
- Assigns meaning by suggesting a theory.

- Persuasive Definition
- Causes specific attitude toward definiendum.

### Definitional Techniques

There are different ways to product definitions.

- Extensional (Denotative) Definitions
- Assign meaning based on indicating members the definiendum denotes.

These come in three different types:

- Demonstrative (Ostensive) Definitions
- Very primitive: pointing or demonstrating physically.

- Enumerative Definitions
- Names members of the class.

- Definition by Subclass
- Names subclasses of the class.

Extensional definitions are mainly used for lexical and stipulative definitions, but cannot be used as a precise definition. Extension suggests intension but cannot determine it.

- Intension (Connotative) Definition
- Assigns meaning based on quality or attributes.

They come in four types:

- Synonymous Definition
- A single word synonym.

- Etymological Definition
- Assigns meaning by disclosing history of the word.

- Operational Definition
- Specifies experimental procedure to determine definition.

There if a final technique that can be particularly useful.

- Definition by Genus and Difference
- Assigns meaning by conveying the difference between a genus and species.

- Genus
- A relatively larger class.

- Species
- A relatively smaller class.

- Specific Difference
- An attribute that distinguishes species in genus.

Take the example: ice (species) means frozen (difference) water (genus).

### Criteria for Lexical Definitions

- Conforms to standards of grammar.
- Conveys essential meaning.
- Neither too broad nor too narrow.
- Avoids circularity.
- Isn’t negative where it could be positive.
- Avoids figurative, vague, obscure, and ambiguous language.
- Avoids persuasive terminology.
- Indicates context which definiens pertains.

## Informal Fallacies

### Fallacies in General

- Fallacy
- A defect in an argument based on something other than false premises.

- Formal Fallacy
- Identified by form of argument alone (applies only to deductive arguments).

- Informal Fallacy
- Only identified by the content of argument.

### Fallacies of Relevance

- Fallacies of Relevance
- Premises are logically irrelevant to conclusion.

These include:

- Appeal to Force
- Appeal to Pity
- Appeal to the People
- Argument against the Person
- Accident
- Straw Man
- Missing the Point
- Red Herring

### Fallacies of Weak Induction

- Fallacies of Weak Induction
- The connection between premises and conclusion are not strong enough to support.

These include:

- Appeal to Unqualified Authority
- Argument to Ignorance
- Hasty Generalization
- False Cause
- Slippery Slope
- Weak Analogy

### Fallacies of Presumption, Ambiguity, and Grammatical Analogy

- Fallacies of Presumption
- Premises presume what’s purported to prove.

These include:

- Begging the Question
- Leading Question
- Complex Question
- False Dichotomy
- Suppessing Evidence

- Fallacies of Ambiguity
- The conclusion relies on a linguistic ambiguity.

These include:

- Equivocation
- Amphiboly

- Fallacies of Grammatical Analogy
- A defective argument appears good by being similar to a non-fallacious argument

These include:

- Composition
- Division

### Fallacies of Ordinary Language

There are a couple reasons for informal fallacies.

- One’s intent may be to mislead.
- Mental carelessness and unchecked emotion.
- Unexamined presuppositions in one’s world view.

## Categorical Propositions

### The Components of Categorical Propositions

- Categorical Proposition
- A proposition that relates two classes (subject, predicate).

- Standard Form Categorical Proposition
- Relates the subject and predicate completely clearly.

- All S are P.
- Some S are P.
- No S are P.
- Some S aren’t P.

- Quantifiers
- Specify how much subject is included in predicate (e.g. all, some).

- Copula
- Links the subject and predicate (e.g. are, aren’t)

### Quality, Quantity, and Distribution

- Quality
- Either affirmative or negative.

- Quantity
- Either universal or particular.

- Distribution
- If proposition makes assertion about all members of a class.

Universals distribute subjects, negatives distribute predicates. See here for more details.

### Venn Diagrams and the Modern Square of Opposition

- Existential Import
- A proposition which claims the existence of subject term.

- Aristotellian Viewpoint
- Universal propositions about existing things have existential import.

- Boolean Viewpoint
- Universal propositions don’t have existential import.

Both recoginize particular propositions recoginize existence.

- Modern Square of Opposition (Boolean)
- Shows mutually contradicting pairs of propositions.

- Contadictory Relation
- Proposition value given by necessity because opposition is known.

Just by logic alone, if we know an A proposition is true, we can infer that O is not true.

- Logically Undetermined Truth Value
- When logic alone cannot determine the truth value of a proposition.

- Immediate Inferences
- Arguments with only one premise.

In immediate inferences, if the venn diagrams of the premise and conclusion are the same, the argument is valid.

- Unconditionally Valid
- Valid regardless of whether terms are existing things.

- Existential Fallacy (Boolean)
- Argument is invalid because the premise lacks existential import.

All Boolean inferences are unconditionally valid.

### Conversion, Obversion, and Contraposition

Different operations can, if used correctly, yield logically equivalent statements.

- Conversion
- Switches the subject and predicate. This is only logically equivalent for E and I propositions.

- Logically Equivalent Statements
- Statements that necessarily have the same truth value.

- Illicit Conversion
- Fallacy of conversion of A and O propositions.

- Obversion
- Changes quality without changing quantity and replaces the predicate with its complement. This is logically equivalent for all propositions.

- Class Complement
- Everything outside the class.

- Contraposition
- Switch subject and predicate and replace both with complements. This is logically equivalent for A and O.

- Illicit Contraposition
- Fallacy of contrapositions for E and I props.

### The Traditional Square of Opposition

- Traditional Square of Opposition (Aristotellian)
- Shows logically necessary relationships between propositions.

- Contradictory
- Statements have opposite truth values.

- Contrary
- At least one is false (not both true).

- Subcontrary
- At least one is true (not both false).

- Subalternation
- True from universal to particular. False from particular to universal.

There are a number of fallacies to consider under the Aristotellian perspective.

- Illicit Subcontrary
- Illicit Contrary
- Illicit Subalternation

- Existential Fallacy
- When subcontrary, contrary or subalternation are used on nonexistent subjects.

- Conditionally Valid
- When the validity is unknown because the subject’s existence is unknown.

## Categorical Syllogisms

### Standard Form, Mood, and Figure

- Syllogism
- A deductive argument with two premises and one conclusion.

- Categorical Syllogism
- A deductive argument with 3 categorical propositions, capable of being put into standard form.

- Major Term
- Predicate of the conclusion.

- Minor Term
- Subject of the conclusion.

- Middle Term
- The term in the permises but not in the conclusion.

The standard-form categorical syllogism has a few conditions:

- All three statements are standard-form categorical propositions.
- The two occurrences of each term are identical.
- Each term is used in the same sense.
- The major premise is first, minor premise second, conclusion last.

- Mood
- The letter of the premises and conclusion types (e.g. AOE).

Each argument form has a different validity.

### Rules and Fallacies

If any of the following fallacies are commited, the syllogism is invalid.

- Fallacy of the Undistributed Middle
- When the middle term is not distributed at least once.

- Illicit Major or Minor Fallacy
- A term is undistributed in the premise, but distributed in the conclusion.

- Fallacy of Exclusive Premises
- When both premises are negative.

There are two other fallacies that forbid exactly what they say:

- Affirmative Conclusion from a Negative Premise
- Negative Conclusion from Affirmative Premises

The existential fallacy is valid under the Aristotellian viewpoint as long as the critical term denotes at least one thing.

- Existential Fallacy
- Where both premises are universal and the conclusion is particular.

### Enthymemes

- Enthymeme
- An argument expressible as a categorical syllogism but missing a premise or conclusion.

These are common in ordinary spoken word and frequently other parts of the syllogism are implied.

### Sorites

- Sorites
- Chain of categorical syllogisms where the intermediate conclusions are left out.

## Propositional Logic

- Propositional Logic
- Fundamental elements are whole statements.

### Symbols and Translation

- $\neg A$
- Negation

- $A\wedge B$
- Conjunction

- $A\vee B$
- Disjunction

- $A\Rightarrow B$
- Material Implication

- $A\iff B$
- Material Equivalence

- Main Operator
- Has the scope of all other statements.

- Well Formed Formulas
- Syntactically correct arrangements of symbols.

### Truth Functions

- Truth Function
- Compound propositions whose truth values is completely determined by components.

- Statement Variables
- Lowercase and can stand for any statement.

- Statement Form
- Statements variables and operators such that uniform substitution resulsts in a statements.

Definitions are given by truth tables, which show all possible truth values for operators.

### Truth Tables for Propositions

Statements can be classified for their truth value from form alone.

- Logically True (Tautologous)
- True regardless of the component truth values.

- Logically False (Self-contradictory)
- False regardless of the component truth values.

- Contigent Statement
- Varies depending on the truth value of the components.

Statements can also be compared.

- Logically Equivalent
- Same truth value when the variables take on the same value.

- Contradictory Statements
- Always opposite truth values.

- Consistent
- At least one situation where both are true simultaneously.

- Inconsistent
- No situation for which both are true simultaneously.

### Truth Tables for Arguments

Arguments can be tested for validity using truth tables. If there is a line with all true premises and a false conclusion, the argument is invalid. If this is not the case, it is valid.

- Corresponding Conditional
- A conditional with the conjunction of the argument’s premises as the antecedent and the conclusions as the consequent.

A valid argument’s corresponding conditional must be a tautology.

### Indirect Truth Tables

There is a very quick way to determine validity.

- Assume the argument is invalid.
- If there is a contradiction, then it is valid.
- If not, then it’s invalid.

When doing these, work backwards.

There is also a method for testing consistency.

- Assume consistent statements (that they’re all true).
- If there is a contradiction then it’s inconsistent.
- If not, then it’s consistent.

### Argument Forms and Fallacies

There are a variety of different valid argument forms:

- Disjunctive Syllogism
- Pure Hypothetical Syllogism
- Modus Ponens (Asserting Mode)
- Modus Tollens (Denying Mode)
- Constructive Dilemma
- Destructive Dilemma

There are also several invalid forms:

- Affirming the Consequent
- Denying the Antecedent

One can refute these invalid forms with some techniques.

- Grasping by the Horns
- Prove conjunction premise false by proving either conjunct false.

- Escaping by the Horns
- Prove disjunctive premise false.

A substitution instance of a valid form always leads to a valid argument. However, a substitution instance of an invalid form may be valid.

## Natural Deduction in Propositional Logic

- Natural Deduction
- The conclusion is derived from the premise by discrete steps.

- Rules of Inference
- Rules that logical proofs depend on.

### Rules of Implication

- Rules of Implication
- Basic arguments whose premises imply their conclusion.

These rules are:

- Modus Ponens
- Modus Tollens
- Pure Hypothetical Syllogism
- Disjunctive Syllogism
- Constructive Dilemma
- Simplification
- Conjunction
- Addition

### Rules of Replacement

- Rules of Replacement
- Pairs of logically equivalent statement forms.

- $\equiv $ (Metalogical)
- The symbol for logical equivalence.

- Axiom of Replacement
- Logically equivalent statements may replace on another.

The following are the Rules of Replacement.

- De Morgan’s Rule
- Commutativity
- Associativity
- Distribution
- Double Negation
- Transposition
- Material Implication
- Material Equivalence
- Exportation
- Tautology

Rules of implication apply to whole lines, but rules of replacement may occur within lines.

### Conditional Proof

- Conditional Proof
- A method for deriving conditional statements.

This is a good method if the conclusion is a conditional.

- Use the symbol ACP and indentation to denote an assumption.
- Assume the antecedent of the conclusion.
- Derive the truth value for consequent.
- Discharge the hypothetical statements as a conditional (use the symbol CP).

### Indirect Proof

- Indirect Proof
- Assumes the negation to proof the statement.

- Use the symbol AIP and indentation to denote assumption.
- Demonstrate a contradiction.
- Discharge the negation of the assumption (use the symbol IP).

## Predicate Logic

- Predicate Logic
- The combination of syllogistic logic and propositional logic. The fundamental component is the predicate.

### Symbols and Translation

- Predicate Symbol
- Uppercase letters used to indicate class designation.

- Individual Constants
- Lowercase letters used to indicate names of individuals.

- Singular Statement
- Makes an assertion about a specific individual.

So for example, “Socrates is mortal” could become $Ms$.

- Statement Function
- An expression after the quantifier is removed.

Statement functions have free variables, statements have bound variables. There are symbols for quantifiers.

- Universal Quantifier ($\forall $)
- Existential Quantifier ($\exists $)

### Using the Rules of Inference

- Instantial Letter
- Letter representing an instance that is applied.

- Instantiation
- Deletes quantifiers and replaces bound variables of a quantifier an instantial letter.

- Generalization
- Introduces a quantifier.

There are different types of instantiation and generalization:

- Universal Instantiation
- Universal Generalization
- Existential Instantiation
- Existential Generalization

Take note that there are some exceptions in these.

### Change of Quantifier Rule

We can translate between quantifiers using a change of quantifier rule (tagged with CQ). It is valid to change the quantifier if:

- Negations are deleted.
- Negations not present are inserted.

### Proving Invalidity

- Counterexample Method
- Prove wrong by using substitution instances.

- Finite Universe Method
- Restrict the universe and evaluate using truth tables.

### Relational Predicates and Overlapping Quantifiers

- Monadic Predicate
- Assigns one attribute to an individual.

- Relational Predicate
- Establishes a connection between objects.

Overlapping quantifiers can occur and order only matters if the quantifiers are different.

## Analogy and Legal and Moral Reasoning

- Argument from Analogy
- Inductive, rests on similarity between comparisons

- Primary Analogue
- Analogue mentioned in the premise.

- Secondary Analogue
- Analogue mentioned in the conclusion.

Evaluating:

- Relevance of similarities
- Number of similarities.
- Nature or degree of dissimilarity.
- Number of primary analogues.
- Diversity amongst primary analogues (more is stronger).
- Specificity of conclusion.

- Legal Reasoning
- Rests on precedent and analogues with other legal arguments.

## Causality and Mill’s Methods

- Sufficient Condition
- Condition which is enough to produce a phenomenon.

- Necessary Condition
- All neccessary conditions must be present for a phenomenon.

- Necessary and Sufficient Condition
- Both of these at the same time.

If we try to prevent something, we search for the necessary condition. If we want to cause something, we look for a sufficient one. The conjunction of all necessaries is a sufficient condition.

Mill’s Methods can help use identify causal connections between events:

- Method of Agreement
- Method of Difference
- Joint Method of Argreement and Difference
- Method of Residues
- Method Concomitant Variation

## Hypothetical/Scientific Reasoning

### The Hypothetical Method

- Hypothesis
- Conjectures about the explanation for a phenomenon when direct evidence is not available.

- Hypothetical Reasoning
- The process used to produce hypotheses.

Hypothetical reasoning in used in scientific and philosophical inquiry. The method involves four steps:

- Occurrence of a problem
- Formulating a hypothesis
- Drawing implications from the hypothesis
- Testing the implications

A hypyothesis is not derived from the evidence, but is used to explain the underlying pattern. It directs the search for evidence, allowing us to ignore what is irrelevant and what is relevant. Discovering one of the implications of a hypothesis does not prove it (fallacy of affirming the consequent), but gives it more credibility.

### The Proof of Hypotheses

- Empirical Hypotheses (Provable)
- Concern the production of some thing or the occurrence of some event that can be observed.

- Theoretical Hypotheses (Confirmed to Degrees)
- Concern how something should be conceptualized.

Empirical hypotheses can be proven when the hypothesized thing/event is observed. Theoretical hypotheses cannot be proven, but only confirmed to varying degrees. If many implications are shown incorrect, it may be disproved.

### The Tentative Acceptance of Hypotheses

Our acceptance of a hypothesis should depend on a few criteria.

- Adequacy
- The extent to which a hypothesis fits the facts it intends to explain. More accurate predictions yield a more adequate hypothesis.

- Internal Coherence
- The extent to which the component ideas of a hypothesis are rationally interconnected.

- External Consistency
- When a hypothesis does not disagree with other well-confirmed hypotheses.

- Fruitfulness
- The extent to which a hypothesis suggests new ideas for future analysis and confirmation.

## Science and Superstition

- Science
- Hypotheses which fit the criteria of a good hypothesis.

- Superstition
- Hypotheses which do not fit the criteria of a good hypothesis.

### Evidentiary Support

A good hypothesis should be constructed according to a few principles.

- Precise Language
- Falsifiability
- Ockham’s Razor
- Lead to true predictions
- Avoids ad hoc modification
- Leads to new insight

### Objectivity

Superstitions exist to satisfy emotional need. Often they are mysterious (and people are fascinated by the mysterious). They are typically supported by sloppy thinking or due to perception which has been affected.

Our perception is often deceieved by different causes.

- The Placebo Effect
- Pareidolia
- Perceptual Set
- Autokinetic effect
- Hallucinations (hypnagogic, hypnopompic, collective)
- Confabulation

### Integrity

Evidence especially in superstition is often faked. Problems of coherence and external consistency are often ignored. The superstitions ignore these problems, but scientists work until these problems are solved in their own hypotheses.