# Tagged Questions

Questions about mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Questions which merely seek to apply logical or formal reasoning to other areas of mathematics should not use this tag. Consider using one of the ...

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### Proving a formula is valid

Let a formula $A$, and a term $t$ such that $x\in FV(t)$. Show that $\varphi = A\{t/x\}\to \exists x (x=t\to A)$ is valid. So let's assume by contradiction that the formula isn't valid. Therefore ...
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### Formalizing a self referential sentence

In The logic of provability, by G. Boolos, we are asked to ponder about this statement: If this statement is consistent, then you will have an exam tomorrow, but you cannot deduce from this ...
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### Are the conditionals equivalent: $p → q ≡ q → p$?

I know that a conditional is if $p$ then $q$, but is that equivalent to saying if $q$ then $p$? Is $p → q$ saying the same as $q → p$?
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### Negating the statement $\exists x \in \Bbb R$ so that $x$ is not an integer, $x > 2016$, and $\lfloor x^2 \rfloor = \lfloor x \rfloor^2$

There exists a real number $x$ so that $x$ is not an integer, $x > 2016$, and $\lfloor x^2 \rfloor = \lfloor x \rfloor^2$. I would like clarification on how to negate this. My idea of negation is ...
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### Meta proof-searching

Suppose you have a particular theory (ex: $ZFC$) in which you want to prove a statement $\phi$. One can attempt to find a proof of $\phi$ that can be verified, but another tactic can be to find a ...
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### Monadic signature with constant

Consider a signature $\Sigma = \{ P^1, R^1, c\}$. Where $P^1, R^1$ are unary predicates, and $c$ is a constant. Let A be a formula in FOL over $\Sigma$. Prove/Disprove: If A is satisfiable ...
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### How could we formalize the introduction of new notation?

What I am thinking about is how in a textbook/proof/theorem/discussion/definition one states that from now on a new notation will be used in the appropriate scope. Example: Let $V^*$ denote the ...
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### Sequents: if-introduction and discharging assumptions

I am reading through "Mathematical Logic by Ian Chiswell & Wilfred Hodges"(amazon, and publisher) for context I am reading through this for self-study, so I don't have the normal support of a ...
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### What is the correct definition of a group?

What is the correct definition of a group? More precisely the predicate "being a group"? According to Wikipedia A group is a set, G, together with an operation • (called the group law of G) that.....
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### Proof Sequences using inference

$(p \land (p \implies q) \land (q \implies r)) \implies r$ It is written slightly different in the text book, but this should be the equivalent form. The book is a bit unclear but I think the author ...
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### (How) can one proposition be logically contradictory?

On this page in the book Probability Theory: the Logic of Science written by E. T. Jaynes, the author says that: If A implies B then a false proposition implies all propositions, and if we tried ...
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### How can logic talk about itself? [closed]

How can there exist theorems like Goedel's Completeness theorem or Incompleteness theorem? They all make some statements about logical theories, but don't we need a certain logical scheme first to be ...