Articles of proof theory

How does one prove that proof by contradiction has completely and utterly failed?

This question already has an answer here: Is Godel's modified liar an illogical statement? 1 answer

Want to show the a proof of the sequent $\forall x \forall y R(x,y) \Rightarrow R(y,y)$ must have a cut

Want to show the a proof of the sequent $\forall x \forall y R(x,y) \Rightarrow R(y,y)$ must have a cut. For this question we are in the Gentzen calculus. I am even having trouble just finding a proof of this sequent first off. There are two results I know of that may be relate, namely: […]

Non-self-referential undecidable sentences in arithmetic

Are there any known undecidable sentences for PA are neither “self-referential” (like a sentence equivalent to its own nonprovability) nor imply consistency of PA (like in the Paris Harrington theorem)?

How to prove consistency of Natural Deduction systems

In Dag Prawitz, Natural Deduction A Proof-Theoretical Study (1965), we have the system I of intuitionistic (first-order) logic based on eleven introduction- and elimination-rules : the 3 couples for the connectives $\lor$, $\land$ and $\rightarrow$, the two couples for the quantifiers and the $\bot_I$ rule [page 20]. The system C of classical logic is obtained […]

What is a proposition? Conflicting definitions.

My textbook, How to Read and Do Proofs by Daniel Solow, defines a proposition as, “… a true statement of interest that you are trying to prove”. Other people seem to define a proposition as being possibly true or false, but not both: Difference between a proposition and an assertion, https://www.math.ku.edu/~jhart/Math_Logic_Student.pdf. It seems logical to […]

Simple proof theory – Propositional Logic

When addressing the questions, which are featured below, I use the following definition and two lemmas. Definition: $\phi$ is a tautology if $[[\phi]]_{v}=1$ for all valuations $v$. Moreover, $\models \phi $ stands for $\phi$ is a tautology. Let $\Gamma$ be a set of propositions, then $\Gamma \models \phi$ if and only if for all $v$: […]

Proof negation in Gentzen system

I am provided with the L¬ and R¬ Gentzen rules for negation (besides “Cut” rule and some rules for ⋀ and →): $${\Gamma\vdash\Delta,\varphi\over \Gamma,\lnot\varphi\vdash \Delta}\ L\lnot \\[4ex] {\Gamma,\varphi\vdash\Delta\over \Gamma\vdash\Delta\lnot\varphi}\ R\lnot $$ I’m trying to proof: $${\Gamma,\lnot\lnot q\vdash\Delta\over \Gamma,q\vdash \Delta}$$ However, I did not manage to this with the provided rules. Does anyone have an idea […]

Complexity of verifying proofs

My question can be read on many levels and so I welcome answers to any reading. The general question is: What is the computational complexity of verifying a proof? One way of looking at a computational complexity class (for decision problems) is that is is a set of theories (a theory being a set of […]

Quasi-interactive proof on real numbers

[This is a cleaner and simpler restatement of a question I asked earlier on Theoretical CS forum. Please re-tag as appropriate.] Suppose you have two oracles (black boxes) that represent real numbers. Each oracle works like this: you give it integer $n$ and it returns integer $k$ such that $k/n \leq r < (k+1)/n$, where […]

Proof that SAT is NPC

I am not really sure I understand the idea behind Cook theorem (it says that SAT is a NP-complete problem). I read the proof with all its parts corresponding to the Turing machine TM solving it (TM is in a single state at any given time, only single cell is read by the head of […]