John S, Jon A, List, It is quite helpful to me to see these kinds of points about how and why one might go about setting up a system of logic one way or another to handle such things as temporal change made more explicit. While I am familiar with different ways in which temporal logics have been developed, those who are on the outside don't get to hear the reasons that guided the choices that were made by the logicians in setting things up one way or another.
John S, thanks for sharing the references. Those look to be right on topic for what I've been hoping to see explained in greater detail. --Jeff Jeffrey Downard Associate Professor Department of Philosophy Northern Arizona University (o) 928 523-8354 ________________________________________ From: John F Sowa [[email protected]] Sent: Tuesday, November 15, 2016 9:52 AM To: [email protected] Subject: Re: [PEIRCE-L] Re: Time, Topology, Differential Logic On 11/15/2016 8:40 AM, Jon Awbrey wrote: > Differential logic is simply the qualitative analogue of the > differential and integral calculus. That's an analogy that might inspire an abduction. But abductions are hypotheses or guesses that may be useful -- or not. In any case, their acceptance must be determined by the predictions they imply. > Both are called upon as we pass from the description of static > situations to dealing with differences between and transformations > among multiple situations, those that occur in different modes of > being or different points in time. There are many ways of representing time. Temporal logic (by Arthur Prior, who was inspired by CSP) uses special operators, such as 'always', 'sometimes', and of course 'prior-to'. Every version of temporal logic, dynamic logic, etc. can be mapped to first-order logic with explicit quantifiers that range over time: 1. 'always' => for every time t 2. 'sometimes' => there exists a time t 3. 'prior-to' for a given t => there exists t' and t' < t 4. Map every N-adic relation R(x1, ..., xN) to an N+1 adic relation with t at the end: R'(x1, ..., xN, t). If you want to deal with both times and situations, quantify over a variable s and add two arguments (t and s) to every N-adic relation. To reduce the number of arguments in your relations, you could take situations as primitive and introduce time as a function of the situation: t = when(s). To avoid increasing the number of arguments in your relations, you could introduce nested contexts: Each context c would contain a collection of propositions expressed in FOL. The following paper is a short (6 page) overview of the issues and their relation to Peirce. The longer worlds.pdf and laws.htm go into more detail. "Answers to five questions on epistemic logic" (2010) http://www.jfsowa.com/pubs/5qelogic.pdf "Worlds, models, and descriptions" (2006) http://www.jfsowa.com/pubs/worlds.pdf "Laws, facts, and contexts: Foundations for multimodal reasoning" (2003) http://www.jfsowa.com/pubs/laws.pdf John
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