Categorical and Propositional Logic
Categorical Deductive Logic
Overview of Categorical Logic
Focuses on asserting true facts about relationships between categories or concepts.
Categorical Proposition
Definition: A simple, direct proposition that asserts the relationship between two classes of objects or concepts, affirmatively, negatively, totally, or partially.
Basic Components of a Categorical Proposition
Subject:
The term that is being predicated in relation to another category or concept.
Example: In the proposition "All birds are animals," "birds" is the subject.
Predicate:
The term that predicates (defines or describes) the subject, either affirmatively or negatively.
Example: In the proposition "All birds are animals," "animals" is the predicate.
Forms of Categorical Propositions
Total of Four Possible Forms:
A: Universal Affirmative – "All S are P".
E: Universal Negative – "No S is P".
I: Particular Affirmative – "Some S are P".
O: Particular Negative – "Some S are not P".
Quantity and Quality of Categorical Propositions
Quantity: Refers to whether the proposition is universal (all) or particular (some) in context to S and P.
Quality: The affirmative or negative character of the proposition.
Quantifier: Indicates the extent of class inclusion or exclusion (terms like "all," "no," "some").
Copula: The verb that links S and P (forms of "to be" such as "are" or "are not").
Singular vs. Non-standard Quantifiers
Singular Terms: When the subject is unique and treated as universal (referring to a class with a single member).
Non-standard Quantifiers: Terms that present ambiguity regarding quantity, dependent on context (e.g., many, few, every, most).
Standard Quantifiers: Clear terms referring to subject classes: "some," "all," "none."
Existential Import
Definition: A proposition has existential import if its truth value presupposes and depends upon the real, independent existence of the things it refers to.
Usage: Important in distinguishing traditional and modern interpretations of categorical logic.
Square of Opposition
Definition: A graphical representation to highlight relationships between standard-form categorical propositions that differ in quality, quantity, or both.
Characteristics of the Modern Square of Opposition
Universal statements have a truth value, even if their subject class is empty.
Universal statements lack existential import: they can be framed as If-then statements.
Particular statements inherently have existential import.
Relationships in the Modern Square
The only definitive relationship between forms is contradictories:
Contradictories: A and O cannot both be true or false; E and I cannot both be true or false.
Traditional Square of Opposition
Definition: Illustrates logical relationships, compatibility or incompatibility, between A, E, I, and O.
Four Relationships Identified:
Contraries: A and E can’t both be true but may both be false.
Subcontraries: I and O can’t both be false but may both be true.
Subalterns: Relation between a universal proposition and its particular counterpart.
Valid to move from universal to particular with true statements.
Moving from particular to universal is valid only with false statements.
Contradictories: Two propositions never share the same truth value.
Venn Diagrams
Usage: Represent categorical propositions visually through interlocking circles, showing logical relationships in the square of opposition.
Representation:
“X” symbolizes a particular subject for propositions I and O.
Shaded areas indicate propositions for universal subjects A and E.
Categorical Syllogisms
Definition: A deductive argument composed of exactly two premises and a conclusion, all categorical propositions.
Standard Form Example:
1. All men are mortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.
Structure of Syllogisms
Terms in Syllogism: Three distinct categories, each occurring twice.
Major Term: The predicate in the conclusion.
Minor Term: The subject in the conclusion.
Middle Term: Links major and minor terms and only appears in premises.
Premises and Conclusion Forms
Major Premise: First premise, containing the major term.
Minor Premise: Second premise, housing the minor term.
Mood and Figure of Syllogisms
Mood: Represents the combination of the propositional forms of the three statements (e.g., AAA, OEO).
Figure: Determined by the position of the middle term in the premises, with four potential arrangements.
Examples of Syllogism**
AAA-1:
1. All the books in the Bible were written in ancient languages.
2. Job is a book in the Bible.
3. Job was written in an ancient language.
IAA-2:
1. Some things written in ancient languages are the books in the Bible.
2. Job is a book in the Bible.
3. Job is written in an ancient language.
Diagramming Categorical Syllogisms
Use three circles representing the three terms: M (Middle), S (Subject), P (Predicate).
Testing validity requires only illustrating the premises.
If the conclusion is illustrated by the premises, the syllogism is valid.
Propositional Logic
Definition: The logic of compound statements with building blocks as simple propositions (as opposed to categorical terms).
Truth Value: For propositions, truth values are always true or false, unlike categorical terms, which are not definitively true or false.
Expansion of Classical Logic
Propositional logic symbolizes logical forms of compound propositions and operations/functions using symbols.
Logical Operators and Compound Statements
Definition: Special symbols (connectives/operators) interconnecting ordinary language statements into compound statements.
Simple Statement: A statement that comprises only a subject and a predicate (e.g., "All butterflies are insects").
Compound Statements
Connect two or more propositions based on the functionality of connectives.
Truth Value Determinants:
Truth values of simple propositions.
The way the propositions connect.
Logical Operators
Categories of Operators:
Negation: Denies the statement, denoted by a tilde (~).
Example: It contradicts the original statement (both can’t be true simultaneously).
Conjunction: Indicates all conjuncts must be true, symbolized by a dot (p•q).
Example: All conjuncts must hold true for the compound statement to be true.
Disjunction: Indicates at least one disjunct must be true, represented by a wedge (p˅q).
Example: “p or q” true if at least one is true.
Conditional: Shows a specific relation (antecedent to consequent), symbolized by a horseshoe (p ⊃ q).
Example: Conditions stated (e.g., if the dog barks, then the cat runs).
Biconditional: Indicates mutual relationship (either both true or both false), represented by a triple bar (p ≡ q).
Complex Statements and Structure
Well-formed Formulas: Utilize proper separators to avoid ambiguity.
Example: The correct punctuations and parentheses are critical for clarity.
Truth Functions
Definition: The truth value of any compound statement determined by:
Truth values of individual propositions.
Functional relationship of logical operators.
Truth Tables
Definition: Arrangements showing truth values for truth-functional propositions, detailing every case.
Process: For truth tables, the number of lines doubles with each added component.
Order of Operations
Process: The main operator remains outside of parentheses, controlling the truth value of the compound statement. Final operation is always executed last.
Contingent vs. Non-Contingent Statements
Definition:
Contingent statements are neither necessarily true nor false; scenarios must be sketched for all possible combinations of truth values.
Non-contingent statements do not rely on components' truth values.
Types of Non-Contingent Statements
Tautology: Always necessarily true.
Example: “Either the butterfly is an insect or it is not an insect.”
Self-Contradiction: Always necessarily false.
Example: “The Vatican is located within Rome and is not located within Rome.”
Logical Equivalence
Definition: Different truth-functional statements that possess identical truth tables, meaning they share truth values in all scenarios.
Examples:
Propositions “All butterflies are insects” vs. “All non-insects are non-butterflies”.
“I will neither paint the house nor cut the grass” equivalent to “I will not paint the house and I will not cut the grass”.
Contradictory Statements
Defined by opposite truth values on all lines of respective truth tables.
Consistent Statements: At least one line where both statements are true.
Inconsistent Statements: No lines where the statements can be simultaneously true.
Truth Tables for Arguments
Validity Check: If, when analyzing a truth table, all premises are true but the conclusion is false, the argument is invalid. In other scenarios, the argument is valid.
Indirect Truth Tables for Arguments
Definition: An abbreviated method of testing argument validity by determining if any scenario can allow the conclusion to be false while premises are true.
Deductive Argument Forms / Implication Rules
Modus Ponens
Definition: Valid argument form affirming the antecedent, leading to consequent affirmation.
Example:
If p, then q.
p.
Therefore, q.
Affirming the Consequent
Definition: Invalid fallacy affirming the consequent, leading back to the antecedent.
Example:
If the dog barks, then the cat runs.
The cat ran.
Therefore, the dog barked.
Modus Tollens
Definition: A valid argument form that denies the consequent to deny the antecedent.
Example:
If p, then q.
~q.
Therefore, ~p.
Denying the Antecedent
Definition: Invalid fallacy denying the antecedent to deny the consequent.
Example:
If I don't water the plant, it will die.
I watered the plant.
Therefore, it will not die.
Argument Forms
Identifying Argument Forms in Examples:
A: Affirming the Consequent
B: Modus Tollens
C: Modus Ponens
D: Denying the Antecedent
Valid Argument Check
Determining validity based on provided premises and conditions. Each conclusion drawn or denied based on whether the premises hold true.
Pure Hypothetical Syllogism
Definition: A syllogism where all premises and conclusion are purely hypothetical propositions.
Example:
If p, then q.
If q, then r.
Therefore, if p, then r.
Hypothetical Propositions
Assert that the antecedent's truth guarantees the consequent's truth; truth of one does not necessitate the truth of the other.
Conditional Propositions
Similar to A form categorical propositions; converse is not equivalent to original wording.
Biconditional Definition
Necessary conditional relationship showing that both components imply each other's truth.
Non-Standard Expressions of Hypothetical Statements
Various translations may express antecedents, but swapping them can change the meaning.
Disjunctive Syllogisms
Definition: If at least one disjunct must hold true, affirming one concludes the other must be false.
Example:
1. Suzie will choose coffee or tea.
2. Suzie did not choose tea.
3. Therefore, Suzie must have chosen coffee.
Disjunctive Operations
Definition: Inclusive vs. exclusive, validity hinges on the disjunction’s nature.
Affirmation must adhere to the broader sense unless explicitly stated otherwise.