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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1answer
18 views

Not getting how to prove reverse hypothesis.

This is a theorem from Dummit & Foote text- Let $G$ be a group acting on the non-empty set $A$.The relation on $A$ defined by $a \sim b$ iff $a=g.b$ for some $g \in G$ is an ...
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0answers
20 views

Predicate Logic - Archimedes' Library

Problem: Our class takes a field-trip to Archimedes’ Library. Before entering the library, your tour guide makes you notice the sign on the main doors which reads: “Observe the Rule of Archimedes’ ...
4
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1answer
46 views

Proof of $(\neg A \supset A) \supset A$

As a (total) beginner in logic, I read this introduction : http://www.loria.fr/~roegel/cours/logique-pdf.pdf (in french). They give an exercise I couldn't achieve. Could someone help me (give an ...
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0answers
9 views

Number of Minimally Functionally Complete (adequate) ternary Operators Sets and what they are

Is there a simpler way than through trial and error to determine the number of Minimally Functionally Complete Operator Sets (MFCOS) (or adequate operator sets) for a given arity and what those ...
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2answers
38 views

Basic: Sequent definition, and-introduction, and iff

I am reading through "Mathematical Logic by Ian Chiswell & Wilfred Hodges"(amazon, and publisher) So far have it has covered $\land$-Introduction and $\land$-Elimination Sadly this text only has ...
2
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1answer
47 views

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 ...
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1answer
33 views

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 ...
2
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0answers
24 views

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 ...
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3answers
65 views

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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0answers
13 views

Showing a function, defined by bounded maximization of a parameter where another function is zero, is primitive recursive [on hold]

Let $g:\mathbb N^2 \to \mathbb N$ be a primitive recursive function and define $f: \mathbb N \to \mathbb N$ by $f(n)$ = largest $m$ such that $m \leq n$ and $g(n,m) = 0$. If there is no such $m$, set $...
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0answers
18 views

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 ...
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3answers
46 views

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/...
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3answers
120 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 ...
0
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0answers
58 views

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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2answers
57 views

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' +...
0
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1answer
30 views

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 ...
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1answer
40 views

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 ...
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1answer
27 views

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 ...
1
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1answer
51 views

How should I interpret this exercise from Chiswell & Hodges Mathematical Logic?

Exercise 5.4.7 on page 127 of Chiswell and Hodges "Mathematical Logic" is: Let $\sigma$ be a signature, $r$ a term of qf LR($\sigma$), $y$ a variable and $\phi$ a formula of qf LR($\sigma$). Let $...
1
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1answer
47 views

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 $100$...
1
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1answer
29 views

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. $\{x \in B \mid x \notin C\} \in \mathscr P(A)$ ...
3
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2answers
23 views

Rewriting $\mathscr P(\bigcup_{i \in I} A_i)\not\subset\bigcup_{i \in I} \mathscr P(A_i)$ in more fundamental terms.

Working through Velleman's "How to Prove It" and currently having a bit of difficulty with a problem where I'm asked to rewrite this: $$\mathscr P\left(\bigcup_{i\in I} A_i\right)\not\subset\bigcup_{...
1
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1answer
69 views

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 ...
0
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1answer
38 views

Why is $p{\implies}q$ defined to have a truth value if $p$ is false? [duplicate]

At first it would seem that if $p{\implies}q$ means "$p$ implies $q$", then if $p$ is false then the entire statement doesn't make sense. It looks like if we have no way of knowing whether $p$ implies ...
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0answers
12 views

Unary relation in a logical sentence

I'd appreciate help with this sentence: Let there be a language L and a structure M, and I need to prove the following sentence is logically false: $$\varphi :\exists xR(x)\rightarrow \forall yR(y)$$ ...
0
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1answer
25 views

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 $\alpha$ ...
2
votes
0answers
30 views

Show that the law of the excluded middle does not hold in a BCCC

I want to show that the law of the excluded middle do not hold in a bicartesian closed category (BCCC), interpreted as follows: In general, there need not be a morphism $1 \to A + 0^A$ for $A \in \...
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0answers
42 views
0
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1answer
26 views

Using separators as functional symbols in first order logic

Suppose we have the following definition of a term: A $term$ is: $x$, where "$x$" is a variable $c$, where "$c$" is a constant symbol $f(\tau_1,...,\tau_n)$, where "$f$" is a ...
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2answers
29 views

Predicate logic: $(\forall x\varphi \rightarrow \forall x\psi ) \nRightarrow (\forall x(\varphi \rightarrow \psi))$

Given $L$ language and $\varphi$ and $\psi$ are formulas. Needs to show that is happening in general: $$(\forall x\varphi \rightarrow \forall x\psi ) \nRightarrow (\forall x(\varphi \rightarrow \psi)...
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0answers
12 views

Logic-Calculating Cd Failures

I am working on homework and have the problem At a company every 4th CD is tested, the testing consists of 4 testing programs and the probability that they fail are as follow Program 1 : .01 Program ...
0
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3answers
31 views

Find the number of all possible valuations that will satisfy given expression.

This part concerns the 256 possible truth valuations of the following eight propositional letters A, B, C, D, E, F, G, H. For each of the following expressions, say how many of the 256 valuations ...
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0answers
33 views

i dont understand how theorem calculation proofs work please help [closed]

I do not understand hilbert style proofs and how they work. Can someone please explain them to me? some things i need to know are: • Write theorem-calculations from Γ (equivalently, Γ−...
3
votes
4answers
93 views

$a^n$ even implies $a$ even

I've tried to prove that $(\forall a,n>0 \in \mathbb{N}),(a^n \text{ even} \implies a \text{ even})$, can someone tell me whether my proof is sound? Lemma 1: $a \text{ even} \implies a^2 \text{ ...
0
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1answer
47 views

The positive introspection axiom

I am studying modal logic with the textbook 'Reasoning about Knowledge' Fagin et al. 1995 The positive introspection axiom is taken as something that can be proved with the possible worlds model of ...
0
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2answers
47 views

Natural Deducion: assumptions can be used more than once?

Im trying to prove: $ \forall{x}\forall{y}(P(x,y)\rightarrow{}\sim P(y,x)) \vdash \forall{x} \sim P(x,x)$ What i have: $\forall{x}\forall{y}(P(x,y)\rightarrow{}\sim P(y,x))\;$ Premise $ \forall{y}...
2
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1answer
66 views

Why is the proof of Gödel's first incompleteness theorem no contradiction?

I consider the following version of Gödel's first incompleteness theorem: Assume $F$ is a formalized system which contains Robinson arithmetic $ Q$. Then a sentence $G_F$ of the language of $F$ can ...
2
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0answers
31 views

What's a good introduction to Second Order Logic

I'm looking for a good introduction to second order logic. Any recommendations?
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2answers
45 views

If empty set is a subset of empty set is always true , then is empty set not a subset of empty set always false? [closed]

If $\varnothing \subseteq \varnothing$ is always true , is $\varnothing \subsetneq \varnothing$ always false ? Any proofs ?
1
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1answer
46 views

What are the different ways to get a first-order formula that express the statement“$P$ is the $n$-th prime”

I know that such a $2$-predicate formula exists since Enderton's have already constructed such a formula in his text on mathematical logic but it was not easy to remember so I wonder if there is other ...
1
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0answers
23 views

Predicate calculus -help [duplicate]

I need to prove that the ∃x(R(x)→∀yR(y)) is logically valid. I'm trying to understand why this statement is true but I can't figure it out.
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0answers
40 views

What is the intutition behind the negative exponential ? in linear logic?

The positive exponential ! has a very satisfying interpretation in terms of the standard resource interpretation of linear logic. Given a resource $a$, we know that $!a$ means an infinite supply of $a$...
0
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1answer
34 views

How can I show that every $\Pi_1$ sentence consistent with Robinson Arithmetic is true in the standard model?

Let $\mathcal{N}=(\mathbb{N}, ...)$ be the standard model of Q (Robinson Arithmetic), and let $\mathcal{N}^{\ast}=(N, ...)$ be an arbitrary model. Let $\varphi$ be a $\Sigma_1$-theorem, and let $\...
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votes
1answer
49 views

How to show that Peano axioms prove that if $\varphi$ defines a non-empty set, then it has a least element? [on hold]

Show the following statement in PA $\forall v_1\dotso\forall v_k\,(\exists v_0\,\varphi\to\exists v_0(\varphi\wedge\forall v_{k+1}<v_0\,\neg\varphi^{v_0}_{v_{k+1}}))$ With $v_0, v_1, \...
0
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3answers
67 views

Is this symbolic statement impossible?

Is this statement logically impossible if x is a single real number (i.e. not a set)? $$(x<5) \land(x>7)$$ it seems to me that x cannot both be greater than 7 and less than 5 if ...
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1answer
58 views

What would be the solution to this logic puzzle? [closed]

This is the puzzle I am having trouble in understanding Also, do explain me the question along with the answer. Thank You
1
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2answers
26 views

Should multiple premises of a natural deduction inference rule always have the same context?

Consider the conjunction introduction and implication elimination rules of natural deduction: $$\frac{\Gamma\vdash\alpha \quad \Gamma\vdash\beta}{ \Gamma\vdash \alpha \land \beta} (\land I) \qquad ...