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:

…

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 :

Soap warts?

INSCITIA:

Or should I call them soaplactites?

I'm talking about the way soap, left sitting in a moist place, gradually produces little bumps on its surface. I can't explain this. I failed Soapology in college. I don't know anything about the chemical structure of soap and I'm not inclined to google my way to wisdom. But here's my gambit at climbing Mt. Ignorance (scandendare mons inscitiae … or is it mountin' ignorance?).

COGITATIO:

Moisture in the air settles on the soap and softens tiny rivulets around areas that happen to be slightly harder, less soluble, than other areas. The softer areas dissolve away, flowing to some other patch of the some, even if only micrometers away, and the harder nodes are left as soaplactites. I suspect some mathematician could describe this phenomenon with chaos theory or some kind of fractal analysis. But what do I know?

RESPONSUM:

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