Chapter Review

Logical Deductions

Logical Deductions

Deductive Reasoning Foundations

Deduction draws conclusions that necessarily follow from given premises, moving from general rules to specific conclusions. If premises are true and reasoning is valid, the conclusion is guaranteed.

Key Points

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    Deduction moves general → specific; induction moves specific → general
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    A valid deductive conclusion is guaranteed by its premises — no room for probability
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    Truth preservation: true premises + valid form → true conclusion
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    Every deductive argument has premises (given info) and a conclusion (what follows)
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    'Must be true' or 'necessarily follows' signals a deductive conclusion is required

Categorical Syllogisms

A categorical syllogism has exactly two premises and one conclusion, linking three categories through quantifiers (all, no, some). The middle term bridges the other two categories but does not appear in the conclusion.

Key Points

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    Four quantifier forms: All A are B, No A are B, Some A are B, Some A are not B
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    Major premise = general rule; minor premise = specific case; conclusion = what follows
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    Middle term appears in both premises but not the conclusion
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    Standard valid form: All A are B, All B are C → All A are C
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    Standard valid form: All A are B, No B are C → No A are C
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    Do not use outside knowledge — only premises determine validity

Venn Diagram Verification

Venn diagrams visually test syllogism validity by representing categories as overlapping circles. Shading marks empty regions (universal statements); crosses mark existing members (particular statements).

Key Points

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    Shading = empty region (used for 'All A are B' or 'No A are B')
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    Cross mark = at least one member exists (used for 'Some A are B')
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    Diagram the premises first, then check if the conclusion is already represented
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    'All A are B' does NOT mean 'All B are A' — the converse is a common trap
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    'Some A are B' does NOT imply 'Some A are not B'

Distributed Terms and Validity Rules

A term is distributed when the statement makes a claim about every member of that category. Two quick validity rules: the middle term must be distributed in at least one premise, and any term distributed in the conclusion must also be distributed in its premise.

Key Points

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    'All A are B': subject A distributed, predicate B not distributed
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    'No A are B': both A and B distributed
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    'Some A are B': neither A nor B distributed
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    'Some A are not B': predicate B distributed, subject A not distributed
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    Middle term must be distributed in at least one premise (otherwise syllogism is invalid)
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    Term distributed in conclusion must be distributed in its corresponding premise

Binary Relations and Their Properties

A binary relation connects two elements. The three key properties — transitivity, symmetry, and reflexivity — determine what new relations can be deduced from given ones.

Key Points

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    Transitive: if aRb and bRc, then aRc — enables chaining (e.g., 'greater than', 'ancestor of')
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    Symmetric: if aRb, then bRa — works both directions (e.g., 'equals', 'sibling of')
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    Antisymmetric: if aRb and bRa, then a = b (e.g., ≤, ⊆)
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    Reflexive: aRa for every element (e.g., 'equals', 'same age as')
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    Equivalence relation = reflexive + symmetric + transitive (maximum deductive power)
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    Not all relations are transitive — 'friend of' is not transitive
Formula

Deriving New Relations

To predict new relations from given ones, identify the relation type, determine its properties, then chain transitive relations and reverse symmetric ones. Write relations as ordered pairs to spot chains.

Key Points

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    Step 1: Identify the relation type (ordering, equality, spatial, etc.)
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    Step 2: Determine properties (transitive? symmetric? reflexive?)
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    Step 3: Chain transitive relations to connect distant elements
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    Step 4: Reverse symmetric relations to fill missing pairs
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    For n elements in a strict ordering, total ordered pairs = n(n−1)/2
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    Always check deduced relations for contradictions with given ones
Formula

Linear Ordering Problems

Linear ordering problems arrange elements along one dimension (height, rank, position) using direct, relative, and negative clues. The strategy is to place fully determined elements first, then narrow down partial constraints.

Key Points

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    Direct clues fix a position ('A is the tallest' → A at top)
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    Relative clues constrain range ('D is between B and E' → D in interval)
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    Negative clues eliminate arrangements ('F is not next to G')
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    Start with the most restrictive clues — fixed positions and complete chains
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    'A is taller than B' does NOT mean A is immediately above B
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    For 'which must be true' questions, test each option against all valid arrangements

Grouping and Classification Problems

Grouping problems assign elements to categories using inclusion rules (must be together), exclusion rules (cannot be together), and capacity constraints. Start with forced assignments and treat paired elements as a single unit.

Key Points

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    Inclusion rules force assignments ('A must be in Group 1')
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    Exclusion rules forbid arrangements ('A and B cannot be in same group')
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    Capacity constraints limit group size ('each group has exactly 3 members')
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    Treat 'must be together' pairs as one combined element to simplify
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    'If A in Group 1 then B in Group 2' is NOT the same as its converse
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    For 'must be true' questions, try to construct a valid arrangement violating the option

Structure Mapping

Structure mapping transfers the relational pattern of one arrangement to a new set of elements via a bijection (one-to-one correspondence). Every relation in the original must have a matching relation in the new structure.

Key Points

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    Extract all relations from the original, not just the obvious ones
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    Abstract into rules: replace specific elements with Element 1, Element 2, etc.
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    Apply the abstract pattern to new elements via bijection
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    The new structure must satisfy every constraint from the original
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    Do not introduce extra relations that do not exist in the original
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    Verify: every element in old set maps to exactly one in new set and vice versa

Combined Deduction Problems

The hardest questions combine ordering, grouping, and relation deductions simultaneously. Cross-dimensional constraints (linking order position to group membership) are especially powerful and should be exploited early.

Key Points

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    Start with the most restrictive constraint across all dimensions
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    Cross-dimensional clues ('tallest person is in Group A') connect multiple systems
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    Draw a grid with elements vs. positions/groups; mark cells as must/cannot/possibly
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    Systematic elimination: remove any arrangement violating even one constraint
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    Focus on 'must be true' and 'cannot be' deductions first for definitive answers
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    When time is limited, prioritize constraints that give binary (yes/no) information

Formulas

Equivalence Relation

A relation is an equivalence relation if and only if it is reflexive, symmetric, and transitive

Strict Ordering Pair Count

Number of ordered pairs in a strict (complete) ordering of n elements

Distribution Rules

Which terms are distributed in each syllogistic form: A = universal affirmative, E = universal negative, I = particular affirmative, O = particular negative

Syllogism Validity Conditions

Two necessary conditions for a categorical syllogism to be valid