Jon A., Jerry, Jeffrey, list In any notation for logic, we have to distinguish the ontology from the base logic. Peirce's basic existential graphs (Alpha + Beta) make minimal assumptions about ontology. They can be mapped to and from FOL in Peirce-Schröder-Peano algebraic notation.
Peirce's many extensions to the EG base can be viewed as different ways of (1) adding some version(s) of ontology to the base and/or (2) extending his basic EG rules of inference. For any proposed notation for logic, I recommend two exercises: (1) a translation to English or other natural language; and (2) a translation to the PSP algebraic notation. These exercises help distinguish syntactic features (which are lost in translation) from semantic features (which are independent of any notation). JA
Our inquiry now calls on the rudiments of topology, for which I turn to J.L. Kelley.
That statement introduces two kinds of ontology: sets and set-based topology. It's OK to design a notation that mixes logic and ontology, but it's important to be clear about what the ontology adds to the logic -- and what alternatives might be considered. In fact, a major complaint about the version of higher-order logic in the _Principia_ is that it mixes an ontology for Cantor's set theory with PSP notation. Common Logic (discussed below) is an alternative: it allows quantifiers to range over functions and relations without requiring uncountable sets. JFS
Every version of temporal logic, dynamic logic, etc. can be mapped to first-order logic with explicit quantifiers that range over time:
JLRC
I was a bit surprised by this statement. I am thinking of chemical and biological phenomena such as enzyme catalysis, biological reproduction and other “emergent" phenomena that require ampliative logics.
Note that I was only talking about mapping temporal and dynamic logics to FOL plus an ontology for time and processes. The following book goes into a great deal of detail on these issues: Gergely, Tamás, & László Úry (1991) First-Order Programming Theories, Berlin: Springer. As the title indicates, the book addresses issues about programming digital computers to represent time and processes. Since a Turing machine is sufficiently general to simulate any digital computer, it can simulate any theory of time and processes that can be programmed in any digital computer. Short summary: If you can simulate those biological processes on a digital computer, it is possible to simulate them on a Turing machine and map them to FOL plus an ontology for time. I say more about theories of processes and causality in the following article, which contains material from several sources, including my book _Knowledge Representation_ (2000): http://www.jfsowa.com/ontology/causal.htm JBD
The claim might hold for some formal systems (i.e., mathematical) of deductive logic, but does it hold for all? Peirce seems to suggest that second order intentional logics can't be mapped onto what we call first-order predicate logics with quantifiers.
I agree with Peirce that you need more than PSP algebraic notation (or Alpha + Beta EGs). I had been working with Pat Hayes and Chris Menzel on the ISO standard for Common Logic (CL) and its extension to the IKL logic (Interoperable Knowledge Language). For the various publications that led to CL and IKL, search for the names Hayes, Menzel, and Sowa in the following web page: http://www.jfsowa.com/ikl Peirce's EGs with extensions that he had proposed in various writings are sufficient to represent the full semantics of CL and IKL. For an overview and examples, see the following article: http://www.jfsowa.com/pubs/eg2cg.pdf I won't claim that CL and IKL are the versions of logic that Peirce would have preferred if he had the time to explore these issues. What I do claim is that Peirce's base (Alpha + Beta) together with a subset of Gamma and miscellaneous examples in his writings form a highly expressive logic with the following properties: 1. It's consistent with CL and IKL, as defined in the references in the eg2cg.pdf article. 2. It's a superset of the discourse representation structures (DRSs) that Hans Kamp developed to represent natural language semantics. And Peirce's basic EG rules for Alpha + Beta are a sound and complete deductive system for DRS. 3. Peirce's EG extensions for quantifying over relations can be translated to and from CL. His extensions for metalanguage can be translated to and from IKL. 4. Many of his Gamma graphs and his rules for reasoning with them can be translated to and from IKL. Point #4 is still an open research area. The exercise of analyzing CSP's often cryptic writings and examples may unearth some important new insights for further development. Hilary Putnam's observation about Aristotle is true about Peirce: "Whenever I become clearer about a subject, I find that Aristotle [or Peirce] has also become clearer about it." John
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