Matpic
mathematics, in pictures
Concept

Negation

The logical operation that turns a statement into its opposite, true exactly when the original is false. Applied to a quantified sentence it swaps every for some and some for every, and pushes itself inward until it lands on the property at the end.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Also named here as predicate logic — the same set of essays touches all of them, so they are one junction rather than several.

j is one more than i, counting round — as a grid, with both quantifier readings. A grid of marks for a relation, with the row and column facts the two quantifier orders ask about.

Every row, or one column

For every person there is someone who loves them, and there is someone who loves everyone, are the same six words in a different order. Draw the relation as a grid and they become two obviously different questions — one about rows, one about columns.

logic · Quantifiers Six sentences from two quantifiers, and which imply which. A diagram of the six sentences that can be built from two quantifiers and a relation, arranged from strongest to weakest with arrows for implication, each labelled with how many of the 512 relations on three points satisfy it.

Six sentences from two quantifiers

One relation, two variables, 'for every' and 'there is': there are eight ways to arrange them and six different sentences come out. Which of them imply which is a small, complete diagram, found by checking all 512 relations on three points — and the diagram crosses over in the middle, which is where every confusion about the order of quantifiers lives.

logic · Quantifiers

Named alongside it

The objects these essays reach for when they reach for this one.

Predicate logicQuantifierQuantifier orderRelationCounterexampleDualityExhaustive searchUniformityWitness

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