Fletcher Hanks’
Stardust
© Respective
copyright holders.
The Use of Dots –
1. To bracket off propositions.
2. To indicate the logical product of two propositions.
The general principle is that a large number of dots indicate an outside bracket and a smaller number of dots indicate an inside bracket.
Group I consists of dots adjoining a sign of implication (É), of equivalence (≡), of disjunction (v), or of equality by definition (= Df).
Group II consists of dots following brackets indicative of an apparent variable such as (x), or (x,y), or (Эx), or (Эx,y), or [(ίx) (φx)] or analogous expressions.
Group III consists of dots which stand between propositions in order to indicate a logical product. The scope of any collection of dots extends backwards or forwards beyond any smaller number of dots or equal number of dots from a group of less force until we reach the end of the proposition or a greater number of dots or an equal number of dots belonging to a group of equal or superior force.
Dots indicating a logical product have a scope which works backwards and forwards. Other dots work only away from adjacent signs of disjunction, implication or equivalence, or forward from others in Group II.
“├ : p v q . É . q v q”
shows what is asserted is the whole of what follows the assertion sign.
“├ :. p É q . É : q É r . É . p É r”
means “if p implies q, then if q implies r, p implies r.”
“p É q . É . q É r : É . p É r”
will mean “if ‘p implies q’ implies ‘q implies r’ then
p implies r.” (This is in general untrue.)
“p É q . q É r . É . p É r”
will mean “p implies q; and if q implies r then
2. To indicate the logical product of two propositions.
The general principle is that a large number of dots indicate an outside bracket and a smaller number of dots indicate an inside bracket.
Group I consists of dots adjoining a sign of implication (É), of equivalence (≡), of disjunction (v), or of equality by definition (= Df).
Group II consists of dots following brackets indicative of an apparent variable such as (x), or (x,y), or (Эx), or (Эx,y), or [(ίx) (φx)] or analogous expressions.
Group III consists of dots which stand between propositions in order to indicate a logical product. The scope of any collection of dots extends backwards or forwards beyond any smaller number of dots or equal number of dots from a group of less force until we reach the end of the proposition or a greater number of dots or an equal number of dots belonging to a group of equal or superior force.
Dots indicating a logical product have a scope which works backwards and forwards. Other dots work only away from adjacent signs of disjunction, implication or equivalence, or forward from others in Group II.
“├ : p v q . É . q v q”
shows what is asserted is the whole of what follows the assertion sign.
“├ :. p É q . É : q É r . É . p É r”
means “if p implies q, then if q implies r, p implies r.”
“p É q . É . q É r : É . p É r”
will mean “if ‘p implies q’ implies ‘q implies r’ then
p implies r.” (This is in general untrue.)
“p É q . q É r . É . p É r”
will mean “p implies q; and if q implies r then
p implies r.”
In this formula the first dot indicates a logical product, hence the second dot extends back to the beginning of the proposition.
“p É q : q É r . É . p É r”
will mean “p implies q; and if q implies r, then p implies r.” (this in general is not true). Here the two dots indicate a logical product, since the two dots do not occur anywhere else, the scope of these two dots extend backwards to the beginning of the proposition, and forwards to the end.
“p v q . É :. p . v . q É r : É p v r”
will mean “if either p or q is true, then if either p or ‘q implies r’ is true, it follows that either p or r is true.” If this is to be asserted, we must put four dots before the assertion sign thus:
“┌ :: p . v . q . É :. p . v . q É r : É . p v r.”
In this formula the first dot indicates a logical product, hence the second dot extends back to the beginning of the proposition.
“p É q : q É r . É . p É r”
will mean “p implies q; and if q implies r, then p implies r.” (this in general is not true). Here the two dots indicate a logical product, since the two dots do not occur anywhere else, the scope of these two dots extend backwards to the beginning of the proposition, and forwards to the end.
“p v q . É :. p . v . q É r : É p v r”
will mean “if either p or q is true, then if either p or ‘q implies r’ is true, it follows that either p or r is true.” If this is to be asserted, we must put four dots before the assertion sign thus:
“┌ :: p . v . q . É :. p . v . q É r : É . p v r.”
To assert an equivalent proposition to the phrase: “if either p or q is true, and either p or ‘p implies r’ is true, and either p or ‘q implies r’ is true,” we write:
“├ :. p v q : p . v . q É r : É . p v r.”
here the first pair of dots indicates a logical product, while the second pair does not. Thus the second pair of dots passes over the first pair, and back until we reach the three dots after the assertion sign.
In reading a proposition, the dots should be noticed first, as they show its structure.
In a proposition containing, several signs if implication, or equivalence, the one with the greatest number of dots before or after is the principle one: everything that goes before this one is stated by the proposition to imply or be equivalent to everything that comes after it.
The dots system and other symbols are
carefully thought out to compress the language of propositional equations to as
short a form as possible. In fact it seems the entire work is a response to the
basic problem of writing about or speaking about very complex subjects, that of
the severe limitation of the amount of time and attention the reader or
listener is willing or able to devote to taking it in. As the nomenclature is
absorbed and practiced very complex expressions may be absorbed and evaluated
instantly, or practically so, without the need for the time consuming demands
of conventional written language or speech.
Also by this system error manifests itself clearly, so one is not taxed by having to decide whether or not to believe the material presented.
Also by this system error manifests itself clearly, so one is not taxed by having to decide whether or not to believe the material presented.
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