Logical Quantifiers

Question

I am wondering if there is a reference or book that clearly translates all English forms of logical quantifiers to mathematical quantifiers.

For example, when we say for any element $ x \in S$, is this equivalent to saying $\forall x \in S$? And what is implied when quantifiers are left out? For example, $s$ and $t$ are elements of $S$? Is this equivalent to $\forall s,t \in S$?

Sincerely, Frank

Answer 1

I am wondering if there is a reference or book that clearly translates all English forms of logical quantifiers to mathematical quantifiers.

Any good introductory logic text should help you out here. I confess a sneaking admiration for P*t*r Sm*th's Introduction to Formal Logic which, I'm told, students find particularly helpful on such matters of translation. But freely available on line, and (probably!) just as good, is Paul Teller's A Modern Logic Primer: you want the opening chapters of Vol. 2. See http://tellerprimer.ucdavis.edu

For example, when we say for any element x∈S, is this equivalent to saying ∀x∈S? And what is implied when quantifiers are left out? For example, s and t are elements of S? Is this equivalent to ∀s,t∈S?

When we say "for any element $x \in S$, $Px$", then yes, this is usually just the same as claiming $(\forall x \in S)Px$. But when we drop the quantifier, and just assume $s$ be an element of $S$ (or say "let $s$ be an element of $S$" or the like) we are not yet quantifying. The context typically is one where we are selecting a single arbitrary element, with a view to proving e.g. that it has some property $P$, so we can then -- because $s$ is arbitrary, i.e. we make no special assumptions about it -- we go on to generalize and infer $(\forall x \in S)Px$.

Answer 2

$x \in S$ is not the same as $\forall x \in S$.

$\in$ means "is an element of," so $x \in S$ simply denotes the membership of $x$ to $S$, while $\forall x \in S$, meaning "for all x that is an element of $S$" must modify an action like "Let $f(x)=x^2, \forall x \in S$." The two statements are not logically equivalent in many cases. For instance, you cannot replace: "Let $x \in S$" with "Let "$\forall x \in S$."

But if you state that something happens for $x \in S$, then it is implied that you mean $\forall x \in S$. e.g., "Let $f(x)=x^2, \forall x \in S$" is the same as "Let $f(x)=x^2, x \in S$".

For a comprehensive list, try these: http://www.philosophy-index.com/logic/symbolic/ http://en.wikipedia.org/wiki/List_of_logic_symbols