Logical Reasoning Quantity Math

Practice flashcards covering logical reasoning quantity terms, their ranges, negations, single-statement inferences, and combinations involving overlaps.

Few A are B

Most A are not B ($A \xrightarrow{most} \sim B$)

A few

Equivalent to "some"

Almost all A are B

Most A are B ($A \xrightarrow{most} B$) & Some A are not B ($A \xleftrightarrow{some} \sim B$)

Determining "must be true" in quantity math

Use the quantity term's weakest possibility only (e.g., some = $1$)

Determining "could be true" in quantity math

Use the quantity term's full range of possibilities (e.g., some = $1-100$)

Inference from All A are B ($A \to B$)

Some B are A ($B \xleftrightarrow{some} A$)

Inference from Most A are B ($A \xrightarrow{most} B$)

Some B are A ($B \xleftrightarrow{some} A$)

Inference from Not all A are B ($\sim(A \to B)$)

Some A are not B ($A \xleftrightarrow{some} \sim B$)

Inference from No A are B ($A \to \sim B$)

No B are A ($B \to \sim A$)

Quantity Overlap Rule

If you add up the weakest interpretations of the quantity terms and the sum exceeds $100$, they must overlap, which results in an inference.

Overlap: All ($100$) + Most ($51$)

$151 =$ Overlap of $51 =$ Most

Overlap: Most ($51$) + Most ($51$)

$102 =$ Overlap of $2 =$ Some

Overlap: Most ($51$) + Some ($1$)

$52 =$ No overlap = No inference

Combined Inference: All A are B & All A are C

All A are B & C (A \to B \text{ & } C)

Combined Inference: All A are B & Most A are C

Most A are B & C (A \xrightarrow{most} B \text{ & } C)

Combined Inference: All A are B & Some A are C

Some A are B & C (A \xleftrightarrow{some} B \text{ & } C)

Combined Inference: Most A are B & Most A are C

Some A are B & C (A \xleftrightarrow{some} B \text{ & } C)

Combined Inference: Most A are B & Some A are C

Nothing! This is a trap!

SOME (Range and Negation)

Range: $1\%-100\%$. Negation: NO ($0\%$).

MOST (Range and Negation)

Range: $51\%-100\%$. Negation: NOT MOST ($0\%-50\%$).

ALL (Range and Negation)

Range: $100\%$. Negation: NOT ALL ($0\%-99\%$).

NO (Range and Negation)

Range: $0\%$. Negation: SOME ($1\%-100\%$).

Synonyms for SOME

a few, a significant number, certain, many, several, frequently, occasionally, often, sometimes.

Synonyms for MOST

majority, preponderance, generally, routinely, typically, usually.

Synonyms for ALL

any, each, every, always, invariably, universally, without exception.

Synonyms for NO

none, never.