Questions about mathematical logic, including model theory, proof theory, computability theory (a.k.a. recursion theory), and non-standard logics. Consider using one of the following tags: (model-theory), (set-theory), (computability), (proof-theory) if they fit the question.

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2answers
58 views

What is the formal negation of the statement “There is much X in Y”. [on hold]

What is the formal negation of the statement "There is much X in Y"? The answer to me is that "It is not the case that there is much X in Y" But I want a more useful negation. Can I say that its ...
3
votes
2answers
49 views

Are the quantifiers interchangeable?

In other words, is it true that $\forall x \; \exists y\;\phi(x, y) \iff \exists y\;\forall x \; \phi(x, y) $?
2
votes
1answer
30 views

what is essentially universal or existential?

In Lambda-Prolog , I see essentially universal quantifier or essentially existential quantifier such terms, I am confused. It seems the universal quantification of a variable in program or goal is ...
-3
votes
0answers
26 views

prove that ¬[P ∨ (L)→M |- M [on hold]

prove that need help to prove that examples
1
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3answers
48 views

Distribution of universal quantifiers over implication

I want to prove that $∀x(φ(x)⟹ψ(x))$ implies $∀x(φ(x))⟹∀x(ψ(x))$. I read they are not equivalent, but I am not sure why. I tried the following: $∀x(φ(x)⟹ψ(x))$ $⟹[φ(a)⟹ψ(a)]$ is true. $⟹φ(a)$ is ...
0
votes
1answer
28 views

prove [(¬M∧R)∧Q |- Q∨T [on hold]

prove [(¬M∧R→Q |- Q∨T really confused :(
-1
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1answer
34 views

Solve the following proof : M |- M ∨ {[(Z∨S) ∧ (¬] → (C↔D)}

Solve the following proof : M |- M ∨ {[(Z∨SC↔D)} I try to proof above question with the following (F⋀Z)⋀ → (C↔D) 1 (F⋀Z)→C 2 F⋀Z 1⋀E 3 F 2⋀E really confused :( this ...
2
votes
2answers
35 views

Inference in Predicate Logic

I have stumbled upon the following reasoning, but I'm not sure if it's correct. Here it goes: Domain X $\forall x :\phi(x)⟹\gamma(x)$ Let $E\subseteq X⟹[\forall x\in E :\phi(x)⟹\gamma(x)]$ Suppose I ...
1
vote
0answers
51 views

Satisfiability/compactness theorem

I am trying to solve the following problem: Let $\mathcal{F}$ be a set of propositional formulas. Assume that for any valuation map $v$ there is some $F$ $\in$ $\mathcal{F}$ such that $v^*(F) = ...
4
votes
1answer
61 views

If $\phi$ holds for all standard models of ZF and ZF proves this, then does ZF prove $\phi$?

I apologize if this is a nonsensical question. Suppose $\phi$ holds in all standard models of ZF. Suppose further that ZF proves this. Then does ZF prove $\phi$?
3
votes
4answers
115 views

Proving that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even.

Here is the proof: Let $a,b \in \mathbb{Z}$. Prove that if $a+b$ and $ab$ are of the same parity, then $a$ and $b$ are even. When working these problems, I do try to set them up logically. My ...
1
vote
3answers
70 views

Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$

This is Velleman's exercise 3.3.4. Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$. I started reexpressing the terms in their equivalent forms ...
0
votes
1answer
25 views

Logic question and proportions [on hold]

there's something that have been bugging me. If we have quantities A, C, E And if we have quantities B, D, F And if we take the equimultiples G, H, K from A, C, E And if we take the equimultiples L, ...
4
votes
1answer
88 views

The existential theory is undecidable

Lemma 1. For any $x$ in the ring $F[t,t^{-1}]$ ($F[t,t^{-1}]$: the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$), $x$ is a power of $t$ if and only if $x$ divides $1$ and ...
1
vote
1answer
22 views

Proof of a classical Theorem of Martin-Löf on complexity dips for Kolmogorov complexity,

I have a question on the first Theorem from the article Complexity of Oscillations in Infinite Binary Sequences by P. Martin-Löf, which could be downloaded from the publisher or from here. Theorem ...
2
votes
1answer
63 views

(totally) (M,P)-generic forcing condition

We say a cardinal $\theta$ is sufficiently large for a forcing $Q$ if $\mathcal{P}(\mathcal{P}(Q)) \in H(\theta)$. And a set $M$ is a suitable model for $Q$ if $Q \in M$ and $M \prec H(\theta)$, $M$ ...
5
votes
2answers
103 views

Explicit construction of a nonmeasurable set, where only the proof of correctness uses choice?

By Solovay's theorem, assuming the existence of an inaccessible cardinal, the axiom of choice is necessary to prove the existence of nonmeasurable sets. In the past, I've thought that one consequence ...
2
votes
2answers
77 views

Why are some conditionals regarded false even if the antecedent is false?

In the Mendelson's logic book, there are 2 conditionals which Mendelson says they are regarded false even if their antecedent is false. One of them is the following: If this piece of iron is ...
2
votes
0answers
57 views

Is this a typo in Jech's Set Theory?

In Jech's Set Theory, p. 603 in the chapter about Proper Forcing, the proof of Theorem 31.7. In the second but last paragraph, the proof says By Theorem 8.27 (Menas), $\lbrace M \cap \lambda ...
0
votes
2answers
38 views

Max/Min to logical operator transformation and viceversa

I have some doubts in transforming conditions that involve max/min in logical operator condition and viceversa. In particular, should be (I put some examples, I would know if I'm right and the ...
2
votes
1answer
31 views

Real Closed Fields with Predicate for a Dense Subfield

Consider $M = (\mathbb{R};+,<, \times, 0, 1, K)$ where $K$ is a unary predicate which holds on $\mathbb{Q}$ (or any dense subfield of $\mathbb{R}$). Question: Is it true that the parametrically ...
1
vote
1answer
26 views

Why $C(n\mid l(n)) \ge C(n) - C(l(n))$ for Kolmogorov complexity

Denote by $C(n)$ the plain Kolmogorov complexity of $n$ and the length of a binary encoding of $n$ by $l(n)$, why do we have $$ C(n\mid l(n)) \ge C(n) - C(l(n))? $$ If I have a shortest program $p$ ...
2
votes
1answer
98 views

Is there a rule for uniform substitution of predicate symbols in FOL?

In a Hilbert-style axiomatization of first-order logic (FOL), there is a rule for variable substitution but I don't see any rule for substituting predicate symbols. Consider a theorem like: $\forall ...
2
votes
3answers
48 views

Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$

The problem is the following (Velleman's exercise 3.2.10): Suppose that $x$ and $y$ are real numbers. Prove that if $x \neq 0$, then if $ y = \frac{3x^2+2y}{x^2+2}$ then $y=3$. My approach so ...
1
vote
0answers
11 views

Each recursive approximating sequence for Kolmogorov complexity is not uniform

Denote the plain Kolmogorov complexity by $C(x)$. Let $\phi(t,x)$ be a recursive function and $\lim_{t\to\infty} \phi(t,x) = C(x)$ for all $x$. For each $t$ define $\psi_t(x) := \phi(t,x)$ for all ...
1
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2answers
36 views

Analyzing logical form of the statements

I have four statements given as exercises in the book: How to prove it. Sa : Alice and Bob are not both in the room. Sb : Alice and Bob are both not in the room. Sc : Either Alice or Bob is not ...
1
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0answers
16 views

Logical form of the statements

I have two statements taken from the book: How to prove it. S1 : We’ll have either a reading assignment or homework problems,but we won’t have both homework problems and a test. S2: You won’t go ...
3
votes
1answer
27 views

Proving the Downward Löwenheim-Skolem using monotonic operators

This is another exercise from Kees Doets Basic Model Theory. Here's the idea. It's well known that the downward Löwenheim-Skolem theorem follows as an easy corollary of the following lemma using ...
6
votes
2answers
88 views

Law of Clavius explained

Law of Clavius states $ \sim P \Rightarrow P \vdash P$ And the only explanation I sort of understand is ...
2
votes
1answer
66 views

Different definitions of a valid argument?

I have some serious problems understanding what counts as a valid argument and what does not. I have read some different definitions of what a valid argument is: (Sorry if this post is missplaced, ...
1
vote
1answer
58 views

Help me solve this (∀x)[Px⇔(∀y)[Qxy⇔¬Qyy]]⇒(∀x)[¬Px] first order logic (step by step)

This is a MCQ of a competitive exam(GATE) , defined below . I found many different -2 explanation in market books and many other sources , but there is conflict between each explanation , I found all ...
2
votes
0answers
15 views

Kolmogorov complexity of substring if string is divided according to rule

Denote the plain Kolmogorov complexity of a string $u$ by $C(u)$. Now let $u$ be a string of length $n$ with $C(u) \ge n - O(1)$ and suppose $u = u_1 \cdots u_{\log n}$, a subdivision of the ...
2
votes
2answers
31 views

find formula for $P\land Q$ using $\uparrow$

I am supposed to find a formula for $P \land Q$ using the logical connective $\uparrow$ $P \uparrow Q$ means that not both $P$ and $Q$ is true. I have already found that $P \lor Q \equiv (P\uparrow ...
2
votes
0answers
48 views
+50

Reference request: fixed point and first-order logics

I'm looking for materials on the relationship between first-order and fixed-point logics, specifically on the condition for a formula in a fixed-point logic to have an equivalent first-order formula. ...
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0answers
61 views

What is the Viewpoint of Modern Logic? [closed]

Colin McLarty has suggested that I ask about the viewpoint of modern logic. What is this viewpoint? Is there such a thing? I have a feeling I'll get referred to some history about Frege. Well, ...
1
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0answers
28 views

On Kolmogorov complexity of first and last half of a string

Denote by $C(x)$ the plain Kolmogorov complexity of $x$ and let $x$ satisfy $C(x) \ge n - O(1)$ with $n = |x|$. If $x = yz$ with $|y| = |z|$ show that $C(y), C(z) \ge n/2 - O(1)$. Any ideas how to ...
1
vote
2answers
53 views

Inconsistent theory with uniformly long refutation?

I understand that there are theorems in PA that necessarily require "very long" proofs; cmp. [1]. On the other hand it seems interesting to think about Life in an inconsistent world. So is it ...
0
votes
2answers
32 views

Symbolic predicate logic “a variable belongs to naturals but bigger than two”

I have written a function and I would like to write at the end "where the variable $x$ belong to the naturals but bigger than $2$" but be translated into symbolic predicate logic. I am writing it as: ...
1
vote
3answers
160 views

Is there any commonality between Math induction and Logic induction?

Logic induction is reasoning by probability. Math induction seems to be related to just Natural numbers and used to prove a statement for every natural number. From these definitions there is no ...
0
votes
1answer
53 views

Existential quantifier axioms in Halmos' system, equivalence proof needed

I have to refer to page 21 of the book cited in the link below. There is a list of axioms Q1-Q5, and an assertion that they are equivalent to a shorter set, namely Q1, Q2 and Q6. I am trying to derive ...
2
votes
1answer
50 views

Implication in linear logic

Linear logic abandons the structural rules of weakening and contraction. I wanted to know whether we have $p ⊸ p$ in linear logic. Can anyone help?
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1answer
30 views

Negate the following logic statements [closed]

Can anybody explain me how to negate the following statements correctly? a) All roses are either odorless or have spikes. b) $\forall x \in \mathbb{Z}: \exists y \in \mathbb{Z}: x+y=0$ c) $\forall ...
1
vote
1answer
43 views

Logical equivalence - Russell's Paradox

In 'How to Prove it' Velleman creates the following set: $R = \{A\in U| A \notin A \}$. This is, according to Velleman, equivalent to $\forall A \in U (A \notin A \iff A\in R) $. That is clear. ...
9
votes
1answer
113 views
+150

Every non-increasing sequence of polynomial towers stabilizes — Finitary proof

In this question we are concerned only with positive integers $\mathbb N$ and other finitary objects that can be encoded using integers. A term function means a total computable function $\mathbb ...
0
votes
2answers
53 views

expressing inclusive OR using exclusive OR. [closed]

On page 5 of Hamilton`s mathematical logic book, it's been stated that we can express A or B or both using XOR, as also possible to express negation and conjunction using XOR. I couldn't find any ...
1
vote
0answers
60 views

Looking for this theorem by Devlin and Shelah

This is a theorem of Devlin and Shelah which I am looking for more details and also proof: $2^{\aleph_0}=2^{\aleph_1}$ is equivalent to the following statement: There is an $F:H(\aleph_1) ...
2
votes
3answers
62 views

Examples of logical possibility

According to Wikipedia, something is logically possible if it doesn't imply a contradiction. In that case, how could a mathematical statement be false but possible? Wouldn't a false statement be false ...
3
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3answers
79 views

Why can't you prove the law of the excluded middle in intuitionistic logic (for layman)?

I am learning about the difference between booleans and classical logics in Coq, and why logical propositions are sort of a superset of booleans: Why are logical connectives and booleans separate in ...
1
vote
1answer
44 views

About the cardinality of the set of all terms/formulas when $|L| > \aleph_0$

This may be a silly question; it was occasioned by exercise 17 of Doets Basic Model Theory book, in which he asks us to prove that, for an arbitrary language $L$, there are at most $|L| + \aleph_0$ ...
2
votes
1answer
47 views

Show that $(A ∩ B) ▵ C = (A ▵ C) ▵ (A \setminus B)$

I want to show the following equality (using logical connectives, not venn diagrams) Show that: $$(A ∩ B) ▵ C = (A ▵ C) ▵ (A \setminus B)$$ $A ▵ B$ is defined as: $(A ∪ B) \setminus (A ∩ B)$ My ...