Irish Logical Thinking flashcards

32 cards. Read the term and the definition here. The deck at the top of this page is optional.

Contrapositive

The valid logical transformation of 'If P then Q' into 'If not Q then not P'. It is logically equivalent to the original conditional, unlike the converse.

Converse error

Concluding 'If P then Q, therefore if Q then P'. The converse is not logically equivalent to the original conditional and yields invalid deductions.

Necessary condition

A condition that must hold for a claim to be true. 'A only if B' makes B necessary for A, meaning not-B guarantees not-A.

Sufficient condition

A condition whose presence guarantees the outcome. In 'if A then B', A is sufficient for B, though B may have other sufficient causes.

Negation of 'all'

The statement 'Some A are not B'. It is the only correct logical negation of 'All A are B'; 'No A are B' is far stronger and not equivalent.

Valid syllogism

A deductive form where true premises guarantee a true conclusion, such as All A are B, All B are C, therefore All A are C.

Inclusive versus exclusive or

Inclusive 'or' allows both alternatives to hold; exclusive 'or' permits exactly one. Wording such as 'but not both' signals exclusive disjunction.

Modus tollens

The valid argument form: If P then Q; not Q; therefore not P. It is the practical application of the contrapositive in deduction.

Percentage change formula

Change expressed as (new value minus old value) divided by the old value, multiplied by 100. The denominator is always the original value.

Successive percentage changes

Changes compound multiplicatively, not additively. A rise of 20% then a fall of 20% gives 1.2 times 0.8, a net 4% decrease.

Harmonic mean speed

For equal distances travelled at speeds x and y, average speed is 2xy divided by (x plus y). It is always below the arithmetic mean.

Compound interest formula

Final amount equals principal multiplied by (1 plus rate) raised to the number of periods. Growth is exponential rather than linear.

Work rate addition

Rates add when workers combine. If A finishes in a hours and B in b hours, their joint time is ab divided by (a plus b) hours.

Reverse a percentage increase

To recover the original value after a rise of r percent, divide by (1 plus r/100). Subtracting r percent from the new value is incorrect.

Weighted average

The sum of each value multiplied by its weight, divided by the total of the weights. It equals the simple mean only when weights are equal.

Percentage points versus percent

A move from 4% to 6% is 2 percentage points but a 50% relative increase. Confusing the two is a standard numerical trap.

Compound annual growth rate

The constant yearly rate taking an initial value to a final value over n years: (final divided by initial) to the power 1/n, minus 1.

Ratio scaling

A ratio a to b represents ak to bk for a common multiplier k. Given a total, solve for k first, then compute each part.

Chirality in rotation

Rotation preserves the clockwise order of vertices; reflection reverses it. A mirrored figure cannot be produced by rotation alone.

Alternating series rule

A sequence where odd and even positions follow separate rules. Analyse positions 1, 3, 5 and 2, 4, 6 independently.

Overlay cancellation

A matrix rule where identical overlapping lines in two frames cancel while differing lines persist in the third frame.

Symmetry axis count

The number of lines dividing a shape into mirror halves. In matrices, axis counts often form a progression across a row.

Euler's polyhedron formula

For any convex polyhedron, vertices minus edges plus faces equals 2. It links counts when one quantity is unknown.

Cube net opposite faces

Faces separated by exactly one face in a straight line on a net become opposite faces when folded and never share an edge.

Pigeonhole principle

Placing more items than categories guarantees at least one category holds two or more items. It proves existence without construction.

Invariant

A quantity unchanged by every permitted operation. It proves a target state is unreachable when the invariant differs.

Parity argument

Tracking whether a quantity is odd or even. If every move flips parity, reachability depends on the parity of the step count.

Working backwards

Solving from the target state or from the answer options toward the given conditions, which prunes branching search trees quickly.

Base-rate neglect

Ignoring how rare a condition is when interpreting a test result. Even an accurate test yields many false positives for rare conditions.

Simpson's paradox

An aggregate trend that reverses within every subgroup. Group-level data must be checked before drawing a combined conclusion.

Survivorship bias

Drawing conclusions only from visible successes while ignoring failures that dropped out of the sample, which distorts the inferred rule.

Gambler's fallacy

Believing independent trials must balance out. Past outcomes do not alter the probability of the next independent event.