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 ...

1answer
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A Puzzle on Infinity: How to guess the color of hats? [duplicate]

Infinitely many (i.e. $\omega$ - many) people each have either a white hat or black hat on their heads. Each person can see everyone's hats except their own. Each person simultaneously announces a ...
1answer
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Is this non-constant function periodic for every definable number?

Given the set $\mathbb{D}$ which contains all definable real numbers. The definition must not be infinite long. E.g. it contains $12$, $-3$, $\frac{1}{12}$, $\sqrt{2}$, $\pi^2$, $i+e$, Chaitin's ...
1answer
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Contradiction in Davis–Putnam–Logemann–Loveland (DPLL) Method?!

As we see on page $10,11$ and $12$ on Google Books we know about Unit Clause (UC) and Pure Literal (PL) in DPLL Algorithms. in the following example if assign value $0$ to variables is prior to ...
3answers
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What does “consistency” mean if formal systems are inherently meaningless?

In the book Gödel's Proof by Ernest Nagel and James R. Newman, the authors insist that formal systems are to be considered as meaningless mechanical systems, which yield theorems by merely applying ...
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Software for solving first-order logic

Is there any class of software that can help me with the following problem in first order logic: given $\phi$ a formula with a "hole" in it (a subformula which is undetermined) and a particular set of ...
2answers
430 views

Prove using a proof sequence and justify each step

Prove using a proof sequence that the argument is valid [ A --> (B ∨ C) ] ∧ B' ∧ C' --> A' I'm having some trouble figuring the proof out here. Here is what I have so far. Is this on the right ...
2answers
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Was Smullyan really wrong?

EDIT: the OP has since edited the question fixing all the issues mentioned here. Yay! There was a question asked on Puzzling recently, titled ...
1answer
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vacuous truth -> empty set is both included and not included in every set?

I understand the concept of vacuous truth and its use in showing that the empty set is a subset of every set. Based on my understanding of vacuous truth (for example https://en.wikipedia.org/wiki/...
0answers
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Henkin theory follow complete

Assume that Γ is a a Henkin theory. For any two constants c,d, either $\Gamma \vdash c=d$ or $\Gamma \vdash c \neq d$. There are two constants a,b such that $\Gamma \vdash a\neq b$.Show that Γ is a ...
1answer
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Formulating a problem in terms of set theory

Here is one problem I was trying to solve just by trial-and-error method. However, I was thinking about how to write the clear solution using set theory. Problem: A notebook contains exactly $100$...
3answers
115 views

$1+1=2$…but Why? [duplicate]

A study that was carried on recently showed that even babies at the age of few months know that $1+1=2$. My question is : is this a fact that can be proved, or is it a just a postulate as those in ...
2answers
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How to simplify using algebra laws

Simplify the following by using algebra laws. (i) X’.Y’ + X.Y.Z. + X’.Y + X.Y My attempt: X’.Y’ + Y(X.Y.Z + X'Y + X.Y) X’.Y’ + (X.Z + X' + X) X’(X’.Y’ + X') + X.Z + X Y’ + X' + X.Z + X Y’ + X' +...
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How mathematics would be different if the first derivations, conjectures and theorems would be others? [on hold]

I've realised that mathematics is nothing else that an implication of some assumptions (plus the assumptions themselves, of course). We have axioms and we derive new "things", new rules, ideas, ...
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Prove $( \lnot C \implies \lnot B) \implies (B \implies C)$ without the Deduction Theorem

The issue is Exercise 1.47 (d) in Elliot Mendelson's "Mathematical Logic". The exercise is to prove $(\lnot C\implies\lnot B)\implies(B\implies C)$ by using the three axioms $(A1,A2,A3)$ without using ...
1answer
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can you help me to transform ∀x FO logical formula to it equivalent ¬∃ formula?

i have this formula ∀x ∀y (A(x,y) V A(y,x) → B(x,c1) ∧ B(y,c2) ∧ c1≠c2) to the equivalent formula that start by ¬∃x ¬∃ y ? you will find the question here ...
2answers
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Gödel's ontological proof and “modal collapses”

Recent findings on Gödel's ontological argument allowed to ultimately establish a couple of things: Gödel's original axiomata are inconsistent Scott's variation instead is consistent Scott's axioms ...
1answer
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An infinite set of axioms in ZF? What does that mean?

Before write this question, I lookeded around enough in this forum for a possible answer and although there are many similar questions, I couldn't find one answer which understand or satisfies me. I ...
1answer
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Answer key to Peter Smith, “An Introduction to Formal Logic”, exercise 13.C.11

If A, B are tautologically inconsistent, then so are $\neg A$ and $\neg B$ This statement is from question C11 at http://www.logicmatters.net/resources/pdfs/answers/Exercises13.pdf, which the answer ...
1answer
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True or falsehood of open formula under a fixed interpretation

Given the open formula: $\alpha =(\exists{{x}_{2}})({P}^{1}({x}_{1},{x}_{2}))$ And consider the interpretation $I$ where the domain is the natural numbers, and ${P}^{1}$ means equality. Is $\alpha$ ...
1answer
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Reasoning informally about $\{x \in B \mid x \notin C\} \in \mathscr P(A)$

Attempting to apply more flexible, informal reasoning to predicate logic as demonstrated helpfully to me by another user in answer to my last question. $\{x \in B \mid x \notin C\} \in \mathscr P(A)$ ...
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