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Wednesday, July 19, 2006

Dots as Brackets in Formulas

Ever tried reading logical texts from the 20s or before (e.g., C. I. Lewis's Symbolic Logic)? Confused by the absence of parentheses and all the dots and colons? Here's Carnap's explanation of the notation (from Abriss der Logistik):
4 c. The Dot Rules

The dot symbols (. : :. :: etc.) replace the bracketing of propositions. The dot signs fall into three distinct levels, depending on whether they occur
  1. between two propositions in a conjunction,
  2. after an operator (x), (∃ x), [(ιx)(φx)],
  3. after |-, before and after the sign ⊃, ≡ ∨, |, =Df.
Dot rules for reading: The scope of a dot symbol (for 1, to the left and to the right, for 2 to the right, for 3 to left or right, depending) extends either to the end of the proposition or to a dot symbol with more dots or to a symbol of the same or a higher level with the same number of dots.

Dot rules for writing: If the scope of a dot sybol is to extend beyond that of another, it must, if it is of a higher level than the latter, contain at least as many dots, and otherwise more dots.

Examples:

p ∨ . q . rmeansp ∨ (q . r)
|- : (p, q) : pq . ⊃ . qp"|- {(p, q) . [(pq) ⊃ (qp)]}
p : ∨ : q . ⊃ . qp"p ∨ [q ⊃ (qp)]
(x) . φx . ⊃ . pq"[(x) . φx] ⊃ (pq)
(x) : φx . ⊃ . pq"(x) . [φx ⊃ (pq)]
(x) : φxp . ∨ q"(x) . [(φxp) ∨ q]
(x) : φxp : ∨ q"[(x) . (φxp)] ∨ q

6 Comments:

At July 19, 2006 3:44 PM , Anonymous Anonymous said...

That's great that you've put these rules on the web! 

Posted by Kenny Easwaran

 
At July 19, 2006 7:28 PM , Anonymous Anonymous said...

Hi Richard!

Been meaning to comment on your posts for a while---great resource, this blog.
Just wanted to add that there are still some authors that (more or less) use dots for parentheses--which I find quite painful. Adrian Mathias is a good example. 

Posted by Andres Caicedo

 
At July 20, 2006 12:21 PM , Anonymous Anonymous said...

That looks like the Peano-Whitehead-Russell use of dots. Church in "Introduction to Mathematical Logic" uses an adapted version (pp. 75).

I find formulas hard to read when conjunction is denoted by a dot as well...but still easier than Polish notation! 

Posted by lumpy pea coat

 
At July 22, 2006 11:32 PM , Anonymous Anonymous said...

Often when reading blogs or academic papers about Logic, I've been struck by how disconnected various aspects are from one another. I've noticed this is especially true regarding "formal" systems of (mathematical/philosophic) logic in relation to symbolic representation. This causes a great deal of unnecessary confusion when cross-disciplinary discussions and/or instructions take place between those who're interested in Logic in a philosophical sense, and those who're interested in Logic as it relates to their profession (like electrical engineers or computer programmers). For a philosophy student, the study of different types of symbolic representations themselves can be rewarding. But for engineering or CS students who are entering fields where there are established common lexicons, standards, and representation methods, learning a PM or SD type method of formal logic isn't very practical.

Assuming that philosophy and mathematics students have a practical interest in learning technological and CS logical systems as well (as most use the electronics and computer programs with these systems as a design basis), do you all believe that there is grounds for adopting these as the initial systems of formal Logics taught to everyone?

If basic computer programming methods and symbols were focused on in mathematical and philosophic Logic courses, there would be a common resource for TA's and GSI's who typically instruct undergraduate Logic that would be applicable for almost every field, be it academic or scientific or professional. Because electronic gate logics and programming languages are very rigorously defined, the process of learning these initially would improve the overall analytic ability of undergraduate students in general.

Anyway... it's just a thought.
 

Posted by A. Scott Crawford

 
At July 28, 2006 9:53 AM , Anonymous Anonymous said...

Very useful. Thanks! 

Posted by Ole Thomassen Hjortland

 
At August 17, 2006 9:49 AM , Anonymous Anonymous said...

Richard

Great site. Thanks especially for the piece on dots as brackets. I was reading a new book on Wittgenstein which seems v. useful but it suddenly, without explanation included expressions with dot brackets. As an old-timer I knew what was ghappening at least dimly but i am sure young students will be baffled.

Cheers

Neil Paterson 

Posted by Neil Paterson

 

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