# Questions tagged [formal-proofs]

For questions about proofs within a formal system, such as natural deduction or Hilbert system.

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### Proof by contradiction - Getting my head around it

Hey there Math community! I have a general question on contradiction and it's getting difficult to get my head around it. Notes: I have some background in math and I have read several proofs by ...
55 views

### Prove distribution of or over implies knowing the implication is always true

I was given a task to construct a Hilbert-style proof for the following: $A → B ⊢ C ∨ A → C ∨ B$ I figured I could use the axiom $A→B≡A∨B≡B$, but this leads me nowhere since I don't think I can use ...
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### prove commuting quadratic functions of real numbers are equal

Suppose that $$f(x) = ax^2 +b$$ is a quadratic function, where $(a, b) \in \mathbb R^2$ and $a \neq 0.$ If $$g(x) = cx^2 +d,$$ where $(c, d) \in \mathbb R^2$ and $c \neq 0,$ is another quadratic ...
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### Reasons for formalizing mathematics

What is the motivation behind formalizing a piece of mathematics in a system like Mizar? I ask as someone interested in the process. I mean it's not like anyone is going to read those formal proofs. ...
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### Proving $\exists ! x (t = x)$ constructively without double negation axiom

I am wondering how one would go about this. I am using Hilbert style proof system as described in Kleene's "Introduction to Metamathematics" or "Mathematical logic". I am pretty sure that if you can ...
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### Formal Proof - premises and conclusions

So I'm learning about formal proof and understand the beginning steps. However, after I'm given an argument and conclusion, I then don't understand how to do the actual formal proving. For example: ...
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### Formal proof method for predicate logics

I am looking for the official name for a proof method, The method consists of proving the INconsistency of a theory. This was done using trees. We call it classic-elimination method but I don't know ...
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### How to prove the following formula using an indirect proof

I need to prove that the premise $A \to (B \vee C)$ leads to the conclusion $(A \to B) \vee (A \to C)$. Here's what I have so far. From here I'm stuck (and I'm not even sure if this is correct). My ...
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### How can I prove the following with natural deduction rules? ¬∀x∃yP(x,y) ⊢ ∃x∀y¬P(x,y)

I have been stuck with this problem for a long time, I tried reductio ad absurdum and I got the hypothesys [¬∃x∀y¬P(x,y)], then I try to eliminate the negation of the premise, but I have to prove ∀x∃...
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### How to use natural deduction to show $\neg (P \land Q) \vdash \neg P \lor \neg Q$?

How to use natural deduction to show $\lnot (P \land Q) \vdash \lnot P \lor \lnot Q$? I think I need to first assume $\neg(\neg P \lor \neg Q)$ and then find a contradiction but I cannot see how to do ...
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### Show that the proof rule is not sound and proof question

I'm asked to show that the proof rule $$\dfrac{\varphi \to \psi}{\lnot \varphi \to \lnot \psi}$$ is not sound. To show this would I just make the truth tables for the ...
87 views

### Proof using natural deduction (Tautology)

I've been asked to prove the following tautology via natural deduction: $\forall x \, (\lnot Px \lor Qx) \rightarrow (\forall y \, Py \rightarrow \forall z \,Qz)$ I normally use tree proofs, but I ...
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### In proof writing, is it mathematically sound to prove uniqueness before proving existence?

As stated in the title, I'd like to find out is whether or not it is always mathematically sound to prove the uniqueness of something before proving the existence of said something. I am still ...
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### Natural deduction proof of $(A \to \lnot B \lor C), ((\lnot D \land A) \to B), (\lnot E \to A) \vdash D \lor (C \lor E)$

I'm struggling to proof this both if I use or introduction rule $\lor_{I_1}$ (to work on $D$) or or introduction rule $\lor_{I_2}$ (to work on $C \lor E$). Could you help me?
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### When should I use RAA in natural deduction proofs?

I can't understand exactly when should I use RAA (reductio ad absurdum) rule in natural deduction proofs? What situation should "trigger" me to think "Now it's time to use RAA"?
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### Predicate Logic Natural Deduction: $∃x P(x) ⊢P(x)$

I am really puzzled right now. To solve the issue, I need to prove this formular: $$\exists x P(x) \vdash P(x)$$ with the natural deduction rules for propositional and predicate logic. I am ...
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### Trouble understanding proof to $\vdash ((\neg(\phi\rightarrow \psi))\rightarrow\phi)$?

I am having trouble understanding the natural deduction proof of $\vdash ((\neg(\phi\rightarrow \psi))\rightarrow\phi)$ (question 2.6.2 (b)) in Hodges and Chiswell's Mathemaical Logic. The natural ...
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### Is $A \& B \multimap A$ derivable?

Intuitively, the sentence $A \& B \multimap A$ seems to mean "Using a choice between $A$ and $B$, get an $A$." This feels like it should be derivable for any $A$ and $B$, but I haven't found any ...
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### Proving a logical implication using modus ponens and metatheorems [duplicate]

Using the law of inference the axiomatic system and metatheorems prove that $${(\neg A > B),(A > C),(B > D)} \vdash (\neg C > D)$$ Where > is 'implies' and ~ negation. I know how to use ...
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### Induction with two variables in PA

This probably has been asked before, but apologies, I don't know how to locate it. I want to prove $\forall x,y: P(x, y)$. My premises are: P(0, 0) \wedge \\ [\forall x: P(x, 0)] \wedge \\ [\...
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### Natural deduction: predicate logic proof (Prenex form)

I'm pretty familiar with proofs in propositional logic, but not so much with predicate logic. I'm trying to prove the following (which can be used during construction of prenex normal form). If $x$ ...
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### show that for every consistent theory there is a complete consistent theory

Let $\mathcal{L}$ be any language of predicate logic, $\Sigma_0$ a consistent theory in $\mathcal{L}$. Let P be the set of all consistent theories $\Sigma \supseteq \Sigma_0$ in $\mathcal{L}$. With ...
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### Let f : R → R be a function, such that $|f(x)−f(y)|≥5|x−y| \:\forall \:x, y\in \mathbb{R}$. Show that $f$ is injective. [closed]

Intro to Math Proofs course Know basic concepts of Injection functions (one-to-one)
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### Predicate Logic Hilbert Proof

In the Hilbert proof system for predicate logic, prove that the formula: $\exists x~\big(B(x)\to C(x)\big)\to\big(\forall x~B(x)\to\exists x~C(x)\big)$ I'm awful with Hilbert Proofs and have no idea ...
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### Starting with a false statement, how can one prove anything is true? [duplicate]

So I've been learning a bit of logic for class and heard that if you begin with a false statement, you can then prove anything to be true, however I don't entirely understand what this means or how to ...