2

Taking $\lnot$ and $\land$ as undefined notions, I have seen the following definitions

$a\lor b\textit{ for }\lnot(\lnot a\land\lnot b)$

$a\implies b\textit{ for }\lnot(a\land\lnot b)$

$a{\iff}b\textit{ for }(a\implies b)\land(b\implies a)$

How is for different from $\iff$? Why isn't it taken as a third undefined connective?

egreg
  • 238,574
jjb
  • 31
  • 1
    No I think it's a way to differentiate the $\Leftrightarrow$ used in the definition and the one that would have been used to explain you the logical concept! – Revolucion for Monica Apr 04 '16 at 19:40
  • Why isn't it considered an undefined connective, along with ${\lnot}$ and ${\land}$, though? – jjb Apr 04 '16 at 19:46
  • You are right "${\iff}$" is a "full fledged" connective with truth table \begin{matrix}a&b&c\0&0&1\0&1&0\1&0&0\1&1&1\end{matrix}. It is not contradictory with what @Marine1 said. It is at a different level. If you go later beyond boolean logic, you will understand these nuances. – Jean Marie Apr 04 '16 at 19:50

2 Answers2

3

The "for" here is not a symbol of the formal language, which is why it is not a connective of that language. "Undefined notions" refer to symbols in the formal language being studied. The "for" is simply English being used to describe what is going on. It is part of the metalanguage, so to speak. In those examples the metalanguage is English.

It is somewhat common to use other symbols, besides $=$ and $\Leftrightarrow$, to denote this kind of thing. Two somewhat common examples are $=_{\text{def}}$ and $\overset{\Delta}{=}$, which are used to mean "the left side is defined to be the right side".

Similarly, when I define a formula by explicitly writing it out, I often use $\equiv$ to separate the name of the formula from its definition, e.g. $$ \phi(n) \equiv (\exists m)[m+ m = n]. $$ This is because $\equiv$ is not a symbol of the formal language, while $=$ and $\Leftrightarrow$ are.

Carl Mummert
  • 81,604
3

We may have two different (but similar) situations:

(i) the language has the connectives: $\lnot, \land, \lor, \Rightarrow, \iff$.

In this case we prove "inside" the calculus that, e.g.

$(a \lor b) \iff \lnot (\lnot a \land \lnot b)$,

and so on.

(ii) the language has the connectives: $\lnot, \land$

and we define the abbreviations:

$(a \lor b)$ "stands for" $\lnot (\lnot a \land \lnot b)$,

and so on.

In this case, the definition of the abbreviation is not a formula of the calculus, but a statement in the meta-language.

In this case, "stands for" has the same "meaning" of iff, i.e. $\iff$, but we usually avoid to use the symbols for the connectives in te meta-language.

Mauro ALLEGRANZA
  • 94,169