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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Clarification of conditional propositions [duplicate]

I am studying first order logic and we have been introduced to conditional propositions.$(p \Rightarrow q)\;$ The truth table for $p \Rightarrow q$ is this: ...
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2answers
59 views

Formula that's only satisfiable in infinite structures

What formula in first order logic can I write that's only satisfiable over infinite structures, over a dictionary without the = sign?
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3answers
63 views

Please help me understand the proof

Doubt In third line of the proof, why is Q $\rightarrow$ R ? Thanks
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1answer
55 views

What is the definition of a definable set of statements, and what is a constructive way to think of this regarding Tarski's Undefinability Theorem?

Logic and model theory are not my area so my thinking is probably off, but I am curious about this so please go ahead and set me straight. A definable set is one for which there is a formula that is ...
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0answers
46 views

Algorithm to answer existential questions - Reduction

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 ...
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1answer
60 views

Meaning of “such that”

The use of the term "such that" confuses me I've seen this like $A=\{(x,y) :x,y\in\Bbb R\ \text{and } P(x,y) \}$ and $B=\{(x,y)\in \Bbb R^2:P(x,y)\}$ for some predicate $P$. Is there any difference ...
3
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2answers
62 views

A simple expression to map $\mathbb N^*$ bijectively to $\mathbb N$

Let $\mathbb N = \{ 1,2,3,\ldots \}$, then by the well-known "Cantor"-Scheme we have $\mathbb N \times \mathbb N \cong \mathbb N$. But even nicer is that we can write this scheme $\varphi : \mathbb N ...
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2answers
37 views

Testing whether Argument is valid or not

I am to determine if argument is valid by making truth table ATTEMPT Let W= Warning lights will come on P= Pressure is high R=Relief valve is clogged Then i have premises as W ...
2
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2answers
44 views

Order of statements in implication

The question is from Exercise 13 of part 1.4 in Rosen's "Discrete Mathematics and Its Applications" (5th edition): "let $M(x,y)$ be "$x$ has sent y an e-mail message", where the universe of discourse ...
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2answers
32 views

Boolean Algebra Problem ABCC'

Hi I just want to ask the answer of this Boolean Algebra problem.. $$ABCC' + B + A'B $$ How to simplify that one?
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1answer
41 views

Proof outline of a certain sentence (Introductory course on logic, proof writing et al.)

The exercise asks me to outline a direct proof that if $\mathbf A$ is a diagonal matrix, then $\mathbf A$ is invertible whenever all its diagonal entries are nonzero. To me this sounds like ...
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2answers
65 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 ...
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2answers
56 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) $?
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1answer
36 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 ...
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0answers
26 views

prove that ¬[P ∨ (L)→M |- M [on hold]

prove that need help to prove that examples
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3answers
62 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 ...
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1answer
28 views

prove [(¬M∧R)∧Q |- Q∨T [on hold]

prove [(¬M∧R→Q |- Q∨T really confused :(
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1answer
35 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 ...
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2answers
41 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 ...
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0answers
61 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
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1answer
90 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 $\phi$?
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4answers
117 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 ...
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3answers
72 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 ...
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1answer
25 views

Logic question and proportions [closed]

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
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1answer
110 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
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1answer
38 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
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1answer
67 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
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2answers
105 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
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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
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0answers
59 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 ...
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2answers
40 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
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1answer
32 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 ...
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1answer
27 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
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1answer
128 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
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3answers
50 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 ...
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0answers
12 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 ...
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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 ...
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0answers
18 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
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1answer
28 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 ...
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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
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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, ...
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1answer
62 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 ...
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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
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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 ...
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0answers
58 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
62 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, ...
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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 ...
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2answers
54 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 ...
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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: ...
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3answers
171 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 ...