#first order logic

[criminal-delirium]

this is my homework now.

[nietp]

since everyone on this website is an annoying nerd who cant shut up, is there actually one of you who wants to be useful for once and help me with first order predicate calculus please

[thelastmemeera]

∃x ( Have( x , you ) ) ?

∃x ( Have( x , you ) ∧ Lose( x, you) ) ?

∃x ( Have( x , you ) ∧ Lose( x, you) ∧ ¬ Lose ( x, mind ) ) ?

[tilde-he]

FOL+Nat?

So, First Order Logic (FOL) can’t uniquely specify the natural numbers. There are non-standard models of first-order arithmetic.

Is there maybe a way to talk about like, “First order logic, except that we have access to the standard model of arithmetic”? And like, talk about what structures can be uniquely defined in FOL+(access to standard model of arithmetic) ?

I suspect the answer is yes.

(By “standard model of arithmetic” I mean standard model of the natural numbers, so, not including negative integers, not that it particularly matters which one I meant.)

So, like, I’d think that in such a “FOL+Nat” there would be a way to uniquely define the integers, the rationals, etc. These would only be uniquely defined given the standard model of the natural numbers, but I think that would be enough in many cases.

I guess one could define it as just like,

“A theory in FOL+Nat consists of a theory in FOL which has the theory of first order arithmetic as a designated part of it, and a model of such a theory in FOL+Nat is a model of the theory such that when restricted to the sort that represents the natural numbers, is the standard model of the natural numbers.”

That seems like it should work fine?

I think it should work fine.

One question then, would be, how does the Löwenheim–Skolem theorem (which says that for any countable first-order theory, if it has an infinite model, then it has a model of every infinite cardinality), change when we look at, instead of models of theories considered as theories in FOL, instead look at models of theories considered as theories in FOL+Nat ? Can we classify the “theories in FOL+Nat which have exactly one model”? (Where, “only one model” means in the sense of “a model of a theory in FOL+Nat, not in the sense of “a model of a theory in FOL”, which is to say, it only counts if the part of the model which is modeling the part of the theory which is first-order arithmetic, is the standard model of arithmetic.)

So, this is asking, essentially, “what theories in FOL which include first-order arithmetic as part of them, would uniquely specify a model, under the assumption that the natural numbers in the model are the standard natural numbers?” .

It would be cool if the answer was “basically most of the ones that you would want.” , but I suspect that may be a little too optimistic. But, it does seem likely enough that the answer is, “oh, lots, and many of the ones you would want.” . But, I haven’t really taken a class or anything on this kind of topic so my intuition there could easily be wrong.

[lightinglamps]

🖼️

[kramergotchi]

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[ijtsrd]

Dynamic Question Answer Generator An Enhanced Approach to Question Generation

by Rahul Bhatia | Vishakha Gautam | Yash Kumar | Ankush Garg "Dynamic Question Answer Generator: An Enhanced Approach to Question Generation" 

Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019, 

URL: https://www.ijtsrd.com/papers/ijtsrd23730.pdf

Paper URL: https://www.ijtsrd.com/computer-science/artificial-intelligence/23730/dynamic-question-answer-generator-an-enhanced-approach-to-question-generation/rahul-bhatia

call for paper Artificial Intelligence, international journal Artificial Intelligence, ugc approved journals Artificial Intelligence

Teachers and educational institutions seek new questions with different difficulty levels for setting up tests for their students. Also, students long for distinct and new questions to practice for their tests as redundant questions are found everywhere. However, setting up new questions every time is a tedious task for teachers. To overcome this conundrum, we have concocted an artificially intelligent system which generates questions and answers for the mathematical topic –Quadratic equations. The system uses i Randomization technique for generating unique questions each time and ii First order logic and Automated deduction to produce solution for the generated question. The goal was achieved and the system works efficiently. It is robust, reliable and helpful for teachers, students and other organizations for retrieving Quadratic equations questions, hassle free. 

[dowereallyneedthis]

for every object x in our universe of discourse, if x glitters, then x is gold. if and only if an object x is a shooting star, then the object x can make the mold.