Logical Deductions
Deduction and Logical Necessity
Deduction is the process of drawing a conclusion that necessarily follows from given premises. If the premises are true and the reasoning is valid, the conclusion must be true — there is no room for doubt. Deductive reasoning moves from general rules to specific conclusions, unlike inductive reasoning which moves from specific observations to general patterns.
General-to-specific movement: Deduction starts with established rules or facts and applies them to particular cases. If all humans are mortal and Socrates is human, then Socrates is mortal.
Certainty: A correct deductive conclusion is guaranteed by its premises. This distinguishes deduction from inductive reasoning, which only produces probable conclusions.
Truth preservation: In a valid deduction, truth flows from premises to conclusion. If the premises are true, the conclusion cannot be false.
Every deductive argument consists of premises (the given information) and a conclusion (what necessarily follows). The logical bridge between them is the form of the argument. If the form is valid and the premises are true, the conclusion is inescapable.
Major premise: A general statement or rule — e.g., 'All mammals have lungs.'
Minor premise: A specific statement about a particular case — e.g., 'A whale is a mammal.'
Conclusion: The statement that follows necessarily — e.g., 'A whale has lungs.'
Deductive vs Inductive Reasoning
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Deductive: All birds have feathers. A robin is a bird. → A robin has feathers. (Certain)
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Inductive: Every robin I have seen has feathers. → All robins have feathers. (Probable)
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Deductive conclusions are guaranteed; inductive conclusions are probable at best.
Syllogistic Reasoning
A categorical syllogism is a deductive argument with exactly two premises and one conclusion, where each statement describes a relationship between categories using quantifiers such as 'all', 'no', or 'some'. Syllogistic reasoning is the backbone of formal deduction.
Standard form: Every syllogism has a major premise, a minor premise, and a conclusion. Each statement links two categories (terms) through a quantifier.
Quantifiers: 'All A are B' (universal affirmative), 'No A are B' (universal negative), 'Some A are B' (particular affirmative), 'Some A are not B' (particular negative).
Middle term: The category that appears in both premises but not in the conclusion. It acts as the bridge between the other two categories.
Four Standard Syllogistic Forms
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All A are B. All B are C. → All A are C.
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All A are B. No B are C. → No A are C.
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Some A are B. All B are C. → Some A are C.
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No A are B. All C are B. → No A are C.
Venn diagrams provide a visual method for testing whether a syllogism is valid. Each category is represented as a circle; shading shows empty regions and crosses show that at least one member exists. If the diagram of the premises forces the conclusion to be true, the syllogism is valid.
Shading (empty regions): Used for universal statements ('All A are B' means the part of A outside B is empty).
Cross marks (existing members): Used for particular statements ('Some A are B' means the overlapping region has at least one member).
Testing validity: Diagram the premises first, then check whether the conclusion is already represented in the diagram. If it is, the argument is valid.
A term in a syllogism is distributed if the statement makes a claim about every member of that category. Recognising distributed terms helps quickly identify invalid syllogisms without drawing diagrams.
'All A are B': The subject A is distributed (the statement covers every A). The predicate B is not distributed.
'No A are B': Both A and B are distributed — the statement makes a claim about every member of both categories.
'Some A are B': Neither A nor B is distributed — the statement only claims something about some members of each category.
'Some A are not B': The predicate B is distributed but the subject A is not.
Distribution Quick Reference
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Universal affirmative (All A are B) → Subject distributed, Predicate not
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Universal negative (No A are B) → Both distributed
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Particular affirmative (Some A are B) → Neither distributed
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Particular negative (Some A are not B) → Predicate distributed, Subject not
Relations and Predicting New Relations
A binary relation connects two elements and describes how they are related. Given a set of known relations, deductive logic allows us to predict new relations that necessarily follow. This is the first core learning outcome: using stated relations to infer conclusions.
Notation: A relation R between elements $a$ and $b$ is written $aRb$. For example, if 'taller than' is the relation, and Ali is taller than Bilal, we write Ali $R$ Bilal.
Relation set: The complete set of given relations forms the starting information. New relations are derived by applying logical rules to this set.
Implicit relations: Some relations are not stated directly but can be deduced. If $aRb$ and $bRc$, and the relation is transitive, then $aRc$ is an implicit relation.
Relations have structural properties that determine what new relations can be deduced. The three key properties are transitivity, symmetry, and reflexivity. Recognising these properties in a given relation tells you exactly what deductions are valid.
Transitive: If $aRb$ and $bRc$, then $aRc$. Examples: 'greater than', 'ancestor of', 'to the left of'. Transitivity enables chaining — the most powerful tool for generating new relations.
Symmetric: If $aRb$, then $bRa$. Examples: 'equals', 'is a sibling of', 'is married to'. Symmetry means the relation works in both directions.
Antisymmetric: If $aRb$ and $bRa$, then $a = b$. Examples: 'less than or equal to', 'is a subset of'. Distinct elements cannot relate to each other in both directions.
Reflexive: $aRa$ for every element. Examples: 'equals', 'is the same age as'. Every element relates to itself.
Property-Based Deduction Rules
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Transitive relation + chain of pairs → deduce new endpoint-to-endpoint relation
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Symmetric relation + given pair → deduce the reverse pair
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Equivalence relation (reflexive + symmetric + transitive) → deduce all three types of new relations
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Antisymmetric relation + both directions → deduce that elements are identical
To predict new relations from given ones, first identify the type of relation and its properties, then apply the appropriate deduction rules. The key strategy is to chain transitive relations, reverse symmetric ones, and combine both when dealing with equivalence relations.
Step 1 — Identify the relation: What type of relationship connects the elements? (ordering, equality, family, spatial, etc.)
Step 2 — Determine properties: Is the relation transitive? Symmetric? Reflexive? The properties determine which deductions are valid.
Step 3 — Apply rules: Chain transitive relations to connect distant elements. Reverse symmetric relations to fill in missing pairs.
Step 4 — Check for contradictions: If a deduced relation contradicts another given relation, re-examine the assumptions.
Ordering and Ranking Problems
Linear ordering problems give you a set of clues about the relative position of elements along a single dimension (height, rank, age, position in a row). The goal is to arrange all elements in a complete ordering using deductive logic. Each clue constrains the possible arrangements.
Direct clues: 'A is taller than B' places A above B. 'C is the shortest' fixes C at the bottom.
Relative clues: 'D is somewhere between B and E' places D in the interval between B and E, without specifying the exact position.
Negative clues: 'F is not next to G' eliminates the adjacent arrangement for that pair.
The systematic approach to ordering problems is to first place elements that are fully determined, then use the remaining clues to narrow down the positions of partially constrained elements. Draw a line or grid and fill in positions as clues are applied.
Step 1 — Identify fixed positions: Look for clues that pin an element to a specific spot (first, last, tallest, shortest). Place these first.
Step 2 — Build chains: Use transitive 'greater than' or 'less than' clues to create chains. If A > B, B > C, and C > D, then A > B > C > D is a complete chain.
Step 3 — Apply constraints: Use 'between', 'adjacent', and 'not next to' clues to eliminate remaining possibilities.
Step 4 — Check consistency: Verify that every clue is satisfied by the final arrangement. If not, re-examine the deduction.
Ordering Clue Types and Their Meanings
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'A is taller than B' → A is above B (not necessarily adjacent)
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'A is the tallest' → A is at the very top
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'A is immediately above B' → A is directly above B with no element between them
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'A is somewhere between B and C' → B, A, C or C, A, B (order of B and C matters)
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'A is not next to B' → A and B cannot be adjacent
Grouping and Classification Problems
Grouping problems provide rules about which elements can or must be placed together in categories. The goal is to determine which groupings are possible, necessary, or impossible based on the given constraints. Grouping is closely related to set membership — each element belongs to one or more groups.
Inclusion rules: 'All A are in Group 1' or 'If X is in Group 2, then Y must also be in Group 2' — these force certain assignments.
Exclusion rules: 'No A can be in Group 3' or 'X and Y cannot be in the same group' — these forbid certain arrangements.
Capacity constraints: 'Each group has exactly 3 members' — these limit how many elements each group can hold.
Solving grouping problems requires systematically applying inclusion and exclusion rules to eliminate impossible arrangements. The most effective strategy is to start with the most restrictive rules and work toward the less restrictive ones.
Step 1 — List all elements and groups: Write down every element and every available group or category.
Step 2 — Apply forced assignments: If a rule says 'A must be in Group 1', place A there immediately. These are free points.
Step 3 — Apply exclusion rules: If 'A and B cannot be in the same group', note this constraint. Once one is placed, the other's options narrow.
Step 4 — Fill remaining slots: Use capacity constraints and remaining rules to assign the leftover elements.
Grouping Rule Types
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Must-be-together: 'A and B are always in the same group' — treat them as a paired unit
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Cannot-be-together: 'A and B are never in the same group' — if one is placed, the other is excluded from that group
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Conditional: 'If A is in Group 1, then C must be in Group 2' — only activates when the condition is met
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Mutual exclusion: 'Each group contains exactly one of {X, Y, Z}' — distribute these elements across separate groups
Structure Mapping and Developing New Structures
Structure mapping means using the relationships in one known structure to build a new structure that follows the same pattern. This is the second core learning outcome: given an existing arrangement of elements and their relations, develop a new arrangement that preserves the same logical structure with different elements.
Pattern transfer: Identify the relational pattern in the given structure (e.g., 'each element is related to the next by rule R'), then apply the same pattern to a new set of elements.
Constraint preservation: The new structure must satisfy the same type of constraints as the original. If the original had a transitive ordering, the new one must also be transitively consistent.
One-to-one correspondence: Each element in the original structure maps to exactly one element in the new structure, and vice versa. This is called a bijection.
To develop a new structure from a given one, first extract all the relations that define the original, then construct the new structure by establishing the same relations among new elements. The key is to ensure every relation in the original has a corresponding relation in the new structure.
Step 1 — Analyse the given structure: List every relation between every pair of elements. Who is above whom? Which elements are grouped together? What are the constraints?
Step 2 — Abstract the pattern: Express the structure as a set of rules rather than specific elements. Instead of 'Ali is taller than Bilal', write 'Element 1 is taller than Element 2'.
Step 3 — Apply to new elements: Substitute the new elements into the abstract pattern. If Element 1 is now Sara and Element 2 is now Usman, then Sara is taller than Usman.
Step 4 — Verify consistency: Check that the new structure satisfies all the original constraints. If the original was a strict ordering, the new one must also be a strict ordering with no ties or contradictions.
Structure Mapping Checklist
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Have all relations from the original been transferred?
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Does every element in the new set correspond to exactly one element in the original?
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Are transitive chains preserved — if A > B > C in the original, does the new structure have the same chain?
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Are grouping and exclusion rules respected in the new structure?
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Is the new structure free of contradictions?
The most challenging questions combine ordering, grouping, and relation deductions in a single problem. These require you to apply multiple deduction techniques simultaneously: order elements along a dimension, assign them to groups, and track binary relations between them — all while respecting every constraint.
Combined approach: Start with the most restrictive constraint across all dimensions. A fixed position in ordering or a forced group assignment gives the most initial information.
Cross-dimensional constraints: Some clues link dimensions — e.g., 'the tallest person is in Group A'. These are especially powerful because they connect ordering information to grouping information.
Systematic elimination: When multiple arrangements are possible, eliminate options that violate any constraint. The remaining options are the valid solutions.