Let $x$ be an individual variable. What is $x = x$?
An open sentence? Or a true proposition? Well, I think it's an open sentence, due to the presence of the free variable $x$. I do not know if I'm wrong. I ask for help.
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4If x is just free then yes it is an open sentence. But in practice you would usually universally quantify it, at which point it is a true proposition. – Ian Feb 02 '18 at 21:58
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2@imranfat Real numbers have nothing to do with this question. – Stefan Mesken Feb 02 '18 at 21:58
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@StefanMesken I beg to differ, unless more info is given (which I am sure there is more to it) – imranfat Feb 02 '18 at 21:59
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1@imranfat More information is given -- see the tags of the question. – Stefan Mesken Feb 02 '18 at 22:00
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@StefanMesken Ok then... – imranfat Feb 02 '18 at 22:00
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Wouldn't $x==x$ be a true proposition? – Yash Jain Feb 02 '18 at 22:04
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2It is a song by Laurie Anderson https://www.youtube.com/watch?v=r4_qdFyVnv8 – Will Jagy Feb 02 '18 at 22:11
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It is an open atomic formula It is satisfiable, for example if $=$ is interpreted as equality. It is not a valid formula since it is not true in every interpretation. For example, if $=$ means 'kills', it is not necessarily true that $x$ kills $x$. – orole Feb 02 '18 at 22:16
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See also the previous post are-there-really-open-axioms. – Mauro ALLEGRANZA Feb 03 '18 at 12:22
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In first-order logic, the formula $x=x$ is an open formula, with $x$ as a free variable. As such, it does not have any truth value in any model unless you substitute a specific value for $x$, or quantify over $x$. (This particular formula just happens to always turn out to be true if you do substitute a value for $x$, assuming you are working in first-order logic with equality.)
Eric Wofsey
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This assumes that models do not come with variable assignments. That is one common framework. In another common framework, a model comes with a variable assignment, and so every formula, open or not, is either true or false in the model. – Carl Mummert Feb 03 '18 at 02:31
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I'm not sure what the particular context is for your question, but one use case is in the reflexivity axiom of equality:
https://en.wikipedia.org/wiki/First-order_logic#Equality_and_its_axioms
Akababa
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