Formal ands?

I finished Mortimer Taube's Computers and Common Sense and a passing reference of his to the ability of computers to handle formal operations got me thinking…

INSCITIA:

Is 'and' definable in formal logic? More generally, are any of the basic logical operations definable in formal logic? 'And' is symbolized with a floating dot, namely, •. "I am tall and I am short" is symbolized as "p • q". "I am short" can be symbolized as M ∃ P(x), where M ∃ means "there is a man" and P(x) means "it is true that x is tall". "I am not short" would be ¬M ∃ P(x), etc ("there is no man such that…).

But can we state 'and' in formal notation? And has a meaning, so can't it be stated as a sort of proposition? Would it be • ≡ • ∀ P(x) ∪ x ∍ (p ∨ q ∌ ¬p ∨ ¬q)?

I am way too weak in formal logic to crack this right now. But in normal English I am inclined to say "'and' is materially equivalent to all cases where x is a member of the set that contains [the attributes or propositions] p, q, no negation of p, and no negation of q (as well as any further elements that can be listed with p)." But then we've used and in the definition. Well?

RESPONSUM:

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ADDENDUM:

Description : Symbol

Disjunction :

Material implication :

Material equivalence :

Negation of material equivalence :

Negation of equality :

Therefore :

Semantic consequence :

Syntactic consequence :

Existential quantifier :

Universal quantifier :

Set membership :

Denial of set membership :

Set intersection :

Set union :

Subset :

Proper subset :

One-to-one correspondence :

Aleph :

Gamma :

Delta :

Necessity :

Possibility :