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:

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.

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:

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

Examples of inductive arguments include the following:

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.

  1. Evaluate the link between the premise and conclusion (the inferential claim). This will show validity or strength.
  2. 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.

The following argument is invalid.

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:

  1. Take an invalid argument and extract form.
  2. Construct a substitution instance with true premises and false conclusion.
  3. 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
Report 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

  1. Conforms to standards of grammar.
  2. Conveys essential meaning.
  3. Neither too broad nor too narrow.
  4. Avoids circularity.
  5. Isn’t negative where it could be positive.
  6. Avoids figurative, vague, obscure, and ambiguous language.
  7. Avoids persuasive terminology.
  8. 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:

Fallacies of Weak Induction

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

These include:

Fallacies of Presumption, Ambiguity, and Grammatical Analogy

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

These include:

Fallacies of Ambiguity
The conclusion relies on a linguistic ambiguity.

These include:

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

These include:

Fallacies of Ordinary Language

There are a couple reasons for informal fallacies.

  1. One’s intent may be to mislead.
  2. Mental carelessness and unchecked emotion.
  3. 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.
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.

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:

  1. All three statements are standard-form categorical propositions.
  2. The two occurrences of each term are identical.
  3. Each term is used in the same sense.
  4. 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:

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

¬A
Negation
AB
Conjunction
AB
Disjunction
AB
Material Implication
AB
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.

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

When doing these, work backwards.

There is also a method for testing consistency.

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

Argument Forms and Fallacies

There are a variety of different valid argument forms:

There are also several invalid forms:

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:

Rules of Replacement

Rules of Replacement
Pairs of logically equivalent statement forms.
(Metalogical)
The symbol for logical equivalence.
Axiom of Replacement
Logically equivalent statements may replace on another.

The following are the Rules of Replacement.

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.

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

Indirect Proof

Indirect Proof
Assumes the negation to proof the statement.
  1. Use the symbol AIP and indentation to denote assumption.
  2. Demonstrate a contradiction.
  3. 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.

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:

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:

  1. Negations are deleted.
  2. 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:

  1. Relevance of similarities
  2. Number of similarities.
  3. Nature or degree of dissimilarity.
  4. Number of primary analogues.
  5. Diversity amongst primary analogues (more is stronger).
  6. 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:

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:

  1. Occurrence of a problem
  2. Formulating a hypothesis
  3. Drawing implications from the hypothesis
  4. 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.

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.

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.