I can't tell who wrote the following quote, so I am not sure who to address here.
Many years ago linguists chewed over the issue of whether the semantic analysis of three place predicates can be broken down into a series of two place predicates and discovered that the two are not semantically or grammatically equivalent. ‘Bob gave a book to Sue' is not equivalent to e.g. ‘Bob caused Sue to have a book’ I am not sure how this would impact the argument in formal logic, since ordinary language and formal logic often part ways (e.g. ‘Bob is not unhappy’ does not equal ‘Bob is happy’), but it seems relevant in evaluating Peirce’s claim. ---begin quote--- Many logicians have correctly observed that you can replace any triadic relation by three dyadic relations plus an additional quantified variable. In a graph, the node thatte represents the variable will be linked to the three dyadic relations. For example, consider the following sentence and its translation to two different formulas in predicate calculus: x gives y to z. ∃x ∃y ∃z gives(x,y,z). ∃x ∃y ∃z ∃w (give(w) & agent(w,x) & theme(w,y) & recipient(w,z)) ---end quote--- > > On April 30, 2017 at 1:51 PM John F Sowa <s...@bestweb.net> wrote: > > Jon and Jerry, > > JA > > > > > > triadic relations extend across a threshold of complexity, such that > > relations of all higher adicities can be analyzed in terms of > > 1-adic, > > 2-adic, and 3-adic relations. > > > > > > No. Peirce never said that. Many logicians have correctly observed > that you can replace any triadic relation by three dyadic relations > plus an additional quantified variable. In a graph, the node that > represents the variable will be linked to the three dyadic relations. > > For example, consider the following sentence and its translation > to two different formulas in predicate calculus: > > x gives y to z. > ∃x ∃y ∃z gives(x,y,z). > ∃x ∃y ∃z ∃w (give(w) & agent(w,x) & theme(w,y) & recipient(w,z)). > > The second formula has a new entity named w, which is linked to three > dyadic relations. There is still an implicit triad in the formula. > > In an earlier note, I showed the sentence "Sue gives a child a book" > as two different conceptual graphs. In the attached giveEGCG.jpg, > I show that sentence translated to the same two conceptual graphs > and to their translations as existential graphs. > > To show the mappings to the algebraic formulas, I also annotated > the lines of identity: x, y, and z represent the same lines in > both EGs. But w represents a ligature of *four* lines of identity > that are connected at a "tetra-identity". > > What Peirce showed is that any connection of four or more lines may > be replaced by connections of just three lines (called teridentities). > In the diagram giveEGCG.jpg, you can replace the ligature labeled w > with a ligature of 5 lines of identity linked by two teridentities. > > JA > > > > > > In mathematics, category theory is largely based on the prevalence > > of functions in mathematical practice, and functions are dyadic > > relations. > > > > > > Not just "largely based", but "completely based". And note that the > "functions" of plus, minus, times, and divide map two arguments to > a single value. For generality, mathematicians say that functions > map elements from one domain to another, but those elements may > be pairs, N-tuples, or structures of any kind. > > I agree with Jerry: > > JLRC > > > > > > The mappings may represent a vast range of mathematical structures > > and be constrained to oriented graphs. > > > > > > Yes. Graphs are convenient because they can show some logical > connections more clearly than a linear notation. But the basic > principles are independent of notation. > > John > > ----------------------------- > PEIRCE-L subscribers: Click on "Reply List" or "Reply All" to REPLY ON > PEIRCE-L to this message. PEIRCE-L posts should go to peirce-L@list.iupui.edu > . To UNSUBSCRIBE, send a message not to PEIRCE-L but to l...@list.iupui.edu > with the line "UNSubscribe PEIRCE-L" in the BODY of the message. More at > http://www.cspeirce.com/peirce-l/peirce-l.htm . >
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