Can True and False be represented without quantifies?

Question

From http://en.wikipedia.org/wiki/First-order_logic#Logical_symbols

Sometimes the truth constants T, Vpq, or ⊤, for "true" and F, Opq, or ⊥, for "false" are included. Without any such logical operators of valence 0, these two constants can only be expressed using quantifiers.

• How can True and False be represented without quantifies?

• Can True and False be represented without quantifies, e.g. $\tilde{p} ∨ p$ and $\tilde{p} ∧ p$? Thanks.

Answer 1

The bolded sentence you quote from Wikipedia is wrong -- true and false can indeed be represented as $\neg p\lor p$ and $\neg p\land p$ for any well-formed-formula $p$. In some logics, such as intuitionistic logic, $\neg p\lor p$ may not be unambiguously true, but then at least $p\to p$ will be.

The only way to make the sentence true would be to somehow prevent there from being any wff $p$ you could use in these constructions in the first place -- such as if

• The language we're working in contains no propositional letters (nullary predicates), and no individual constants (nullary function letters), and
• For some reason we require that everything we write down is a sentence (without free variables).

In which case, expressing $\bot$ and $\top$ requires quantifiers for the trivial syntactic reason that one needs quantifiers to be allowed to say anything at all in the first place.