Re: [PEIRCE-L] Meta-languages. Re: Four branches of existential graphs: Alpha, Beta, Gamma, Delta
John, List: JFS: I am happy to say that I completely agree with Jon's note below. However, the following passage from another note is misleading about Peirce, Euclid, and mathematical practice from ancient times to the present. The quoted passage is from my same note below. JFS: In mathematical texts, it's common to say "Given A1, A2, A3..., it follows THAT T1, T2, T3... where the A's are axioms, and the T's are theorems that follow from the axioms. Note the word 'that'. It is a sign of METALANGUAGE, between two clauses of a sentence. It is not a sign of implication. The word 'follows' or more precisely 'my be proved from' indicate the steps of a proof. "Given A, it follows that T" is logically equivalent to "if A, then T," which is logically equivalent to "A implies T." A is the premiss or antecedent (e.g., postulates), and T is the conclusion or consequent (e.g., theorems). There is no need for metalanguage to express this in EGs since it is represented by a scroll or nested cuts or a ring-shaped shaded area--including a sheet with a red line drawn just inside its edges (or a shaded margin). JFS: As for the notations in R514 and L376, Peirce made another distinction: postulates are propositions on which the utterer and the interpreter agree. The choice of postulates is the result of an AGREEMENT between the utterer and the interpreter. The results inside the red line are the result of an INVESTIGATION that may be far more complex than an mathematical proof. In R 514, Peirce only states that "in the margin outside the red line, whatever is scribed is merely asserted to be possible. Thus, if the subject were geometry, I could write in that margin the postulates, and any pertinent problems stated in the form of postulates." He does not say anything about "an agreement between the utterer and the interpreter," nor about the "results inside the red line" before the text breaks off in mid-sentence. In R L376, Peirce does not say anything about "the red line" nor "an investigation," complex or otherwise. The "agreement between the utterer and the interpreter" is on the *subject *of the graphs scribed on each piece of paper, which represents a *portion *of the overall universe of discourse that "is before the common attention" of both parties at one time or another. He only discusses postulates in the paragraph right *before *the section on "The Phemic Sheet," and only for the purpose of explaining why "any valid deductive conclusion" is *not *"instantaneously evident upon an examination of the premisses." Here is what he says. CSP: My second reason is found in the peculiar character of mathematical postulates. These pronounce that certain things are possible. But these possibles are not, of course, single things, for a single thing must be more or less than possible: they embrace whole infinite series of infinite series of objects in each postulate; and it is upon the statement of the possibility of one single one of those objects or single one for each set of certain others, that some essential part of the conclusion is founded. How many demonstrations, for example, and very simple ones too, as mathematics goes, depend, each of them, upon the possibility of a single straight line; while this possibility is only asserted in the postulate that there is, or may be, a straight line through any two points of space. In that statement the possibility of every single straight line in space is asserted, including the single one whose existence is pertinent and concerning which a similar postulate directly or mediately asserts something which is an essential ingredient of the conclusion. Consistent with R 514, postulates "pronounce that certain things are possible." Moreover, the *only *kind of investigation that Peirce discusses here is a mathematical demonstration. JFS: The complexity of the investigation is the reason why Delta graphs are a completely new branch of EGs. Again, Peirce's *only *stated reason for needing "to add a *Delta *part" to EGs is "in order to deal with modals"--not for metalanguage, and not for complex investigations. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Sat, Mar 23, 2024 at 4:46 PM John F Sowa wrote: > Jerry, Jon, List, > > JLRC: If the critical concept that is under scrutiny here the issue of > “graphs of graphs” , how is this related to the arithmetical notion of > division? > > I agree with Jon's explanation below that Peirce did not use the word > "division" to mean the numerical operation of dividing two numbers. He was > talking about dividing different parts of a text. > > As for the phrase "graph of graphs", that excerpt occurred in the > introductory paragraphs of Lecture V of Peirce's Lowell lectures of 1903. > Immediately before that, he used the synonym "graphs about graphs''. Since > the word 'metalanguage' had not
Re: [PEIRCE-L] Meta-languages. Re: Four branches of existential graphs: Alpha, Beta, Gamma, Delta
Jerry, Jon, List, JLRC: If the critical concept that is under scrutiny here the issue of “graphs of graphs” , how is this related to the arithmetical notion of division? I agree with Jon's explanation below that Peirce did not use the word "division" to mean the numerical operation of dividing two numbers. He was talking about dividing different parts of a text. As for the phrase "graph of graphs", that excerpt occurred in the introductory paragraphs of Lecture V of Peirce's Lowell lectures of 1903. Immediately before that, he used the synonym "graphs about graphs''. Since the word 'metalanguage' had not yet been introduced in English, the phrase "graphs about graphs" is his best and clearest term. It he had used his Greek, he might have coined the word 'metagraph'. I am happy to say that I completely agree with Jon's note below. However, the following passage from another note is misleading about Peirce, Euclid, and mathematical practice from ancient times to the present. JAS: The "red pencil" notation (1909) is entirely different from this--a red line is drawn just inside the physical edge of the sheet, and postulates are written in the resulting margin. These are not propositions about the propositions written inside the red line (metalanguage), they are premisses (antecedent) from which the propositions written inside the red line follow necessarily as deductive conclusions (consequent). For example, if the EGs for Euclid's five postulates are scribed in the margin, then they can be iterated to the interior, where the EGs for all the theorems of Euclidean geometry can be derived from them in accordance with the usual permissions. It's true that postulates are iterated (or copied) during the process of proving a theorem. But it's also possible to iterate a statement from a that-clause of metalanguage to a collection of statements that are being discussed in other ways. In mathematical texts, it's common to say "Given A1, A2, A3..., it follows THAT T1, T2, T3... where the A's are axioms, and the T's are theorems that follow from the axioms. Note the word 'that'. It is a sign of METALANGUAGE, between two clauses of a sentence. It is not a sign of implication. The word 'follows' or more precisely 'my be proved from' indicate the steps of a proof. As for the notations in R514 and L376, Peirce made another distinction: postulates are propositions on which the utterer and the interpreter agree. The choice of postulates is the result of an AGREEMENT between the utterer and the interpreter. The results inside the red line are the result of an INVESTIGATION that may be far more complex than an mathematical proof. The complexity of the investigation is the reason why Delta graphs are a completely new branch of EGs. In summary, metalanguage is the "secrete sauce" that makes Gamma graphs a third branch. But investigation makes Delta graphs the fourth branch. That difference is very important. John From: "Jon Alan Schmidt" Jerry, List: No one is claiming that Peirce ever used the term "metalanguage," only the concept. Specifically, he provided a Gamma EG notation for asserting a proposition about a proposition--the lightly drawn (1898) or dotted (1903) oval for treating a complete proposition as a subject that fills the blank in a rheme attached to the oval by a lightly drawn (1898) or dotted (1903) line to compose another complete proposition (CP 4.560, 1906). For example, here is the Gamma EG for "A thinks that it is possible that B," where A is a person, B is a proposition, "A thinks ___" and "___ is possible" are rhemes, and each instance of "that" in English corresponds to a dotted oval/line in the graph. [image.png] The "red pencil" notation (1909) is entirely different from this--a red line is drawn just inside the physical edge of the sheet, and postulates are written in the resulting margin. These are not propositions about the propositions written inside the red line (metalanguage), they are premisses (antecedent) from which the propositions written inside the red line follow necessarily as deductive conclusions (consequent). For example, if the EGs for Euclid's five postulates are scribed in the margin, then they can be iterated to the interior, where the EGs for all the theorems of Euclidean geometry can be derived from them in accordance with the usual permissions. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Fri, Mar 22, 2024 at 11:43 PM Jerry LR Chandler wrote: Jon, List On Mar 20, 2024, at 12:46 PM, Jon Alan Schmidt wrote: - Peirce's 1898 and 1903 notations for metalanguage are identical, except that the oval and line are lightly drawn in the former and dotted in the latter. - Peirce's "red pencil" notation in R 514 has nothing to do
Re: [PEIRCE-L] Four branches of existential graphs: Alpha, Beta, Gamma, Delta
Jerry, List: In this context, "division" simply refers to Peirce's 1903 *organization *of Existential Graphs (EGs) into distinct Alpha, Beta, and Gamma parts. Alpha implements propositional logic, Beta implements a version of first-order predicate logic by adding the line of identity, and Gamma implements various advanced logics by adding the broken cuts (modal logic), the heavy line with dotted lines along both sides (second-order logic), the dotted oval/line (metalanguage), etc. Again, the latter is equivalent to the 1898 (RLT) notation, which has a lightly drawn oval/line instead of a dotted oval/line. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Fri, Mar 22, 2024 at 11:52 PM Jerry LR Chandler < jerry_lr_chand...@icloud.com> wrote: > List, John: > > On Mar 20, 2024, at 3:16 PM, John F Sowa wrote: > > That quotation shows that Gamma graphs add one and only one *NECESSARY* > feature > to Alpha + Beta graphs: the same or equivalent metalanguage feature used > in 1898 (RLT). When Peirce referred to the *DIVISION *of Gamma graphs, > that is the only feature required.He later did much more talking about > modality and with new notations. He never again used any of the notations > that are unique to the 1903 Gamma graphs. > > I am puzzled by this paragraph. > > If the critical concept that is under scrutiny here the issue of “graphs > of graphs” , how is this related to the arithmetical notion of division? > > And what sort of mental operations would be required to assert the nature > of a division of a “graph of graphs“? > > Cheers > Jerry > _ _ _ _ _ _ _ _ _ _ ARISBE: THE PEIRCE GATEWAY is now at https://cspeirce.com and, just as well, at https://www.cspeirce.com . It'll take a while to repair / update all the links! ► 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 UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
Re: [PEIRCE-L] Meta-languages. Re: Four branches of existential graphs: Alpha, Beta, Gamma, Delta
Jerry, List: No one is claiming that Peirce ever used the *term *"metalanguage," only the *concept*. Specifically, he provided a Gamma EG notation for asserting a proposition about a proposition--the lightly drawn (1898) or dotted (1903) oval for treating a complete proposition as a *subject *that fills the blank in a rheme attached to the oval by a lightly drawn (1898) or dotted (1903) line to compose another complete proposition (CP 4.560, 1906). For example, here is the Gamma EG for "A thinks that it is possible that B," where A is a person, B is a proposition, "A thinks ___" and "___ is possible" are rhemes, and each instance of "that" in English corresponds to a dotted oval/line in the graph. [image: image.png] The "red pencil" notation (1909) is entirely different from this--a red line is drawn just inside the physical edge of the sheet, and postulates are written in the resulting margin. These are not propositions *about *the propositions written inside the red line (metalanguage), they are premisses (antecedent) from which the propositions written inside the red line follow necessarily as deductive conclusions (consequent). For example, if the EGs for Euclid's five postulates are scribed in the margin, then they can be iterated to the interior, where the EGs for all the theorems of Euclidean geometry can be derived from them in accordance with the usual permissions. Regards, Jon Alan Schmidt - Olathe, Kansas, USA Structural Engineer, Synechist Philosopher, Lutheran Christian www.LinkedIn.com/in/JonAlanSchmidt / twitter.com/JonAlanSchmidt On Fri, Mar 22, 2024 at 11:43 PM Jerry LR Chandler < jerry_lr_chand...@icloud.com> wrote: > Jon, List > > On Mar 20, 2024, at 12:46 PM, Jon Alan Schmidt > wrote: > >- Peirce's 1898 and 1903 notations for metalanguage are *identical*, >except that the oval and line are lightly drawn in the former and dotted in >the latter. >- Peirce's "red pencil" notation in R 514 has nothing to do with >metalanguage--it turns an entire sheet into nested cuts for implication, >with the antecedent (postulates) in the margin and the consequent >(theorems) inside the red line. > > In these two paragraphs, the term “metalanguage” is used. > Was I misled by the assertion (by another logician) that Tarski was the > first to use this term? > > Although this may seem as a trivial point, it becomes rather critical from > the perspectives of emergence and evolution with the putative levels of > development and the corresponding grammatical distinctions between the > social and natural sciences and current notions of “metalogics”. > > At issue is the languages in which propositions are posited. > > Cheers > Jerry > _ _ _ _ _ _ _ _ _ _ ARISBE: THE PEIRCE GATEWAY is now at https://cspeirce.com and, just as well, at https://www.cspeirce.com . It'll take a while to repair / update all the links! ► 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 UNSUBSCRIBE PEIRCE-L in the SUBJECT LINE of the message and nothing in the body. More at https://list.iupui.edu/sympa/help/user-signoff.html . ► PEIRCE-L is owned by THE PEIRCE GROUP; moderated by Gary Richmond; and co-managed by him and Ben Udell.
