Extended quantifiers and definates v 2.0.0

Classical logic considered propositions like “All men are mortal”. Such were later modified into predicate calculus that uses the ∀ and ∃ quantifiers,  and which is used extensively in computer science. The previous statement can be written symbolically as:

∀m∈Men⋅Mortal(m)

I wanted a shorthand that extended the quantifiers yet reduced the symbolism further: unknown, none, one, some, all, not all, some but not all, all but one, etc. In addition, I wanted to denote the scope of the set, which I call “definates”: none, one, some, unknown. The idea is to allow expressions for noams, memes and demes which could express such as:

Some but not all of a set of definitely more than one are true

This would work on a string of binary digits such as:

01101000

The notation ended up looking like set notation with accents above and below a letter. The diacritic above the set is the quantifier – how many are true; the one below is the definite – the size of the set. Not all of the symbol combinations are convenient in WordPress and take some digging out of Unicode. They are primarily intended for doodling diagrams quickly rather than neatly formatted documentation.

The symbols chosen for the quantifiers are:

• ∅ – none of the set are true
• ⋅ – one and only one of the set are true
• ∧ – all the set are true (logical and symbol)
• ∨ – one or more of the set are true (logical or symbol)
• O – one or more of the set are not true (inversion of the dot)
• * – some but not all of the set are true (a combination of some and not all)
• “not all” comes with a bar above the ∧
• an unknown quantity is left blank

The symbols chosen for the definates are

• ∅ – an empty set
• ⋅ – a singleton set
• ~ – a set of more than one element
• an unknown quantity is left blank

  

The expression above would consist of an asterisk (some but not all) above, and a tilde (more than one element) below, which is somewhat simpler than the predicate calculus equivalent notation.

*

S

~

Clearly, there is a grid of quantifiers and definates. Some of the possibilities are equal to others, some are implied by others, and some are logical impossibilities. The interrelationships are left for philosophers, but resembles the square of opposites.

The principle is extensible in what quantifiers and definates could be incorporated, using other diacratic marks, and could be applied to other collections such as lists, or other logics such as modal or doxastic. For example À could represent a list excluding the first element. Such further extensions remain to be formalised if they turn out to be useful.