Top and bottom (⊤, ⊥)

How to read it

Stanford’s CS103 course notes put it in two lines: “The symbol ⊤ is a value that is always true. The symbol ⊥ is value that is always false.” They count the two as connectives, though, as the notes say, they don’t connect anything. In proofs ⊥ stands for a contradiction. forall x: Calgary reads it as something like “contradiction!” or “but that’s absurd!”, writes it on the line where a sentence and its negation have both appeared, and treats it as a sentence letter that is always false. A proof by contradiction can then be written (¬p → ⊥) → p (CS103): if assuming not-p leads to a contradiction, p. Unicode names the characters DOWN TACK ⊤ (U+22A4), “top”, and UP TACK ⊥ (U+22A5), “base, bottom”.

Also searched as: ⊤, ⊥, contradiction symbol, down tack, falsum, tautology symbol, upside down T, up tack, verum.

⊥ in logic and ⊥ in geometry

The same upside-down T means perpendicular in geometry: OpenStax’s Contemporary Mathematics writes two lines meeting at 90° as l₁ ⊥ l₂. Unicode keeps the two apart. Logic’s bottom is ⊥ UP TACK (U+22A5); the relation “perpendicular to” has its own character, ⟂ PERPENDICULAR (U+27C2), “orthogonal to”, which its chart notes is typeset with extra space on each side, as a relation between two things is. On engineering drawings the same shape is the perpendicularity tolerance. Standing alone as a statement, ⊥ is the logic’s “false”; between the names of two lines, it is “perpendicular to”.

Look-alikes

⊤ is easy to read as a capital T. Both are the turnstile ⊢ turned on its side.