Timeline for Arithmetization of Syntax: Can any semantic be encoded as syntax?
Current License: CC BY-SA 4.0
9 events
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| Nov 6, 2019 at 4:57 | comment | added | TCP | @AlexKruckman To handle countably infinitely many variables we could provide a recursively enumerable procedure for the mapping. So rather than providing an infinite explicit mapping with syntax, we provide the syntax for a procedure of which the result is an infinite explicit mapping with syntax. For the arbitrary case, we would provide the syntax for a countably infinite procedure who requires (depends on) an input of a certain type. And thank you, this line of questioning has exposed limitations in my thought process! I used an assertive tone for brevity, but its clear I may be mistaken. | |
| Nov 4, 2019 at 0:53 | comment | added | Alex Kruckman | It's true that in propositional logic with finitely many propositional variables, any model can be written down explicitly (by writing down the function from the finite set of propositional variables to the set of truth values). But what about a propositional language with infinitely many variables? Now there are (at least) continuum-many truth assignments. Or what about a first-order language, in which a model consists of interpretations of the function and relation symbols as arbitrary functions and relations on an arbitrary set. In what sense are these models syntactic? | |
| Oct 31, 2019 at 3:45 | comment | added | TCP | @AlexKruckman If we had one proposition p in our theory then there are two possible interpretations onto the structure {True, False}. One is f(p) = True, the other is f(p) = False. So the "assignment is expressed with syntax" in the form of a function where the domain is propositions in our theory and the codomain is elements in the set { True, False } | |
| Oct 31, 2019 at 2:56 | comment | added | Alex Kruckman | I'm still not sure I understand what you mean. Can you be more precise about what you mean when you say "that assignment is expressed with syntax"? Maybe with an example? | |
| Oct 31, 2019 at 2:39 | comment | added | TCP | @AlexKruckman Say we start solely with a formal theory T. An interpretation in model theory is an assignment of the meaning of the syntax of T and that assignment is expressed with syntax -- we have a systematic notation and we write it down and other people understand it. The structure that the interpretation assigns meaning to is also expressed with syntax. This is an example of us taking a semantic (in this case it is the semantic of semantic) and expressing it with syntax. In this way, model theory (or this aspect of it) is itself the act of representing a semantic with syntax. | |
| Oct 30, 2019 at 20:32 | comment | added | Alex Kruckman | "The semantics of any logical system are able to be represented as syntax via the (orthodox) model-theoretic interpretation." How does the model-theoretic interpretation give a "representation as syntax"? | |
| Oct 30, 2019 at 20:30 | comment | added | Alex Kruckman | "satisfiable is defined with syntactic rules" - what does this mean? Satisfiability is a semantic notion, in that the definition refers to models. | |
| Oct 29, 2019 at 23:00 | review | First posts | |||
| Oct 29, 2019 at 23:09 | |||||
| Oct 29, 2019 at 22:55 | history | asked | TCP | CC BY-SA 4.0 |