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15.541 logic questions.

Is there a symbol in mathematical logic expressing a null statement? That is an equivalent expression to: $$p\land\lnot p$$

Are there any normal interpretations of $(=, <)$ with countable domain that are elementary equivalent but not isomorphic to $\mathbb{Q}$? To $\mathbb{Q}^+ = \mathbb{Q} \cap [0; +\infty)$? First, I ...

I found in a book , this phrase :There is a man who loves all women. the logic predicate is : $\exists x(m(x)\wedge\forall y(w(y)\wedge love(x,y)))$ or not

I would like someone to give me some advice on whether I did these right or not. Here are the two sentences: For every positive number K there is a positive number M for which |f (x)-b| < K, ...

Let's start by looking at the following example from ring theory: Let $A$ and $B$ be commutative rings with identity and let $f:A\to B$ be a ring homomorphism. For every ideal $I$ of $A$, the ...

What's the difference between a predicate and a relation? I read the definition that an $n$-ary predicate on a set $X$ is a function $X^n\to \{\text{true}, \text{false}\}$ where $\{\text{true}, \text{...

I am following Epstein & Carnielli's book on Computability and am puzzled by their reasoning: on p. 131, Theorem 2, they define a universal computation predicate C(n,b,r,q) by induction on n. I ...

Defining the new proof system $N$ as this: We have 2 Axioms - $$A \rightarrow (A \lor B)$$ $$A \rightarrow (B \rightarrow A)$$ A new deduction rule: $$\bullet \frac{(A \rightarrow B)}{A \rightarrow (...

I'm presented with a Hilbert system with just one inference rule (MP) and these axiom schemes: $$A \supset (B \supset A)$$ $$(A \supset (B \supset C)) \supset ((A \supset B) \supset (A \supset C))$$ $...

I'm not sure if I find the correct answer to this question. Using De Morgan's law; ¬(a∧b)=(¬a)∨(¬b) so it becomes: X is not odd OR Y is not even or also we can say, X is even OR Y is ODD. Can ...

I have searched and find this: In classical logic, why is $(p \Rightarrow q) = T$ if $p = F$ and $q = F$? Is the only way to prove that is not a tautology by having $T \to F$, which gives $F$?

Using predicates $Px$ for "x is a person" and $Rx,y$ for "x is the father of y" and secondly taking reference to people as implicit and using only the two-place predicate Rx,y. No one has no ...

It seems as though the deduction theorem can fail in natural language, if we think of a valid inference as one that preserves certainty, rather than truth. What I mean is that if we are certain of (...

In the PDF textbook "A Friendly Introduction to Mathematical Logic 2nd Edition" by Christopher C. Leary and Lars Kristiansen, page 53, the first quantifier inference rule (QR) is defined by the ...

The Wikipedia page on the propositional calculus says that another name for it is "zeroth-order logic." This sort of makes sense to me, but appears to break down in some pretty crucial ways with ...

I want to prove (φ→ρ) → ((ψ→ρ) → ((φ∨ψ)→ρ)) I made a proof but don`t know doing right may you check it whether doing right or wrong?

I was trying to simplify this propositional form: $$ \{(p \lor q) \land [¬q \lor (p \lor t)] \land (¬t \lor p) \} \lor (p \land ¬t)$$ but ended up with: $$\ (q \lor p) \lor (p \land ¬t)$$ My steps ...

In a family, each child has an at least 3 brothers and 2 sisters. Which is the least number of children in the family setup.

The problem: Prove using Zorn's lemma that there exists a set $A \subseteq \mathbb{R}$ such that: for every non-zero polynomial $P(x_1,\dots, x_n)$ with rational coefficients and every $a_1, \dots ...

Boolos, Burgess, and Jeffrey in "Computability and Logic" on page 147 define A set $\Gamma$ is denumerably categorical if any two denumerable models are isomorphic. What does this ...

From Quantifier (logic) - Wikipedia: Two fundamental kinds of quantification in predicate logic are universal quantification and existential quantification. But why do they become fundamental? ...

The problem Let $(X, \leq_X)$ be a linear ordening. We call $x \in X$ a left-dense point if for every $y \in X$ met $y <_X x$ there exists a $z \in X$ sucht that $y <_X z <_X x$. Let L be a ...

Let C be a category and let us have many diagrams, that give properties of this category, e.g. existence of limits, existence of pentagon identities, existence of monoidal or other algebra like ...

There are two weather stations, station A and station B which are independent of each other. On average, the weather forecast accuracy of station A is $80\%$ and that of station B is $90\%$. Station A ...

I am looking for study and beginner material to study mathematical logic. I understand that it is a very broad topic but I would like to know what the best path there is to learning mathematical logic....

I know only logics whose predicates and functions accept terms as arguments and all the modal operators are defined at the logical level and only modla operators and connectives can accept sentences (...

Given that a topology on a set X is defined as a subset of the powerset of X that satisfy certain conditions and the powerset of X itself is a topology over X, I conclude, that the definition is ...

I want to prove this $$(\varphi\to\psi)\to((\varphi\to\lnot\psi)\to\lnot\varphi)$$ but don`t know how to deal with because $(\varphi\to\psi)$ and $(\varphi\to\lnot\psi)$ makes question fuzzy

Okay I have the following fitch proof almost figured out, from here I know I have to move forth in conlcuding but I don't know what I am missing (2 premises) S → (R ∨ P) P → (¬R → Q) my work ...

How to below this First order logic procedure convert Convert them into Conjunctive Normal Form ? Ɐx [[employee(x) ꓥ ¬[PST(x) ꓦ PWO(x)]] → work(x)] I strive to do this below step , i. S1: ...

At Ieke Moerdijk's homepage, one can read that his research interests include "applications of topology to mathematical logic". I know very few such applications (essentially I only know topological ...

Given $Σ = \{0, 1\}$ and $w ∈ Σ^*$. The binary string $w$ is called heavy if (the number of $1$ of $w$) - (the number of $0$ of $w$) = $1$. For example, the strings $011$, $100011110$ are heavy, while ...

The problem Let $X$ be a set. We define an equivalence relation ~ on $X^{\mathbb{N}}$: $(a_n)_n$ ~ $(b_n)_n \iff \exists n_0 \in \mathbb{N}, \forall n \geq n_0 : a_n = b_n$ for $(a_n)_n, (b_n)_n \...

A set of propositions $S$ is independent if $\forall t \in S$ we have $S-\{t\}\nvdash t$. The syntactic axioms defined in our logic course are: 1) $\forall p,q \in L : p\Rightarrow (q \Rightarrow ...

We denote the Peano axioms with $\mathsf {PA}$ and $S=\{0,1,+,\cdot,<\}$ denotes the language of number theory. Let $\varphi$ be the formula $$(1+1)\cdot v_2\equiv(v_0+v_1)\cdot(v_0+v_1+1)+(1+1)\...

The problem Let $L$ be a language. A class $M$ of $L$-structures is called elementary if there is an $L$-theory $T$ such that $M$ is precisely the class of all models of $T$. Suppose that for such a ...

This problem is from Discrete Mathematics and its Applications Here is my book's definition on converse, contrapositive, and inverse And the common ways to express an implication For this problem, ...

Are non-standard models of ZF set theory by definition always not well-founded? And it seems it is, because it must be. But then, Wikipedia says that when there is a set that is a standard model of ...

I am talking about a Hilbert style system for Propositional Calculus. The only axioms and rule of inference that I can use are, $\color{crimson}{\text{Axiom 1.}}\ P\to (Q\to P)$ $\color{crimson}{\...

Probably it is very simple lemma but I cannot see it. Suppose that we have intuitionistic propositional logic (in fact, it can be classical) and $W$ is the set of all prime (or maximal - in classical ...

I have some questions about the commonly presented proof of Godel's first incompleteness theorem. First of all, what is the need for the Godel numbers in the proof? Would not the proof be valid ...

Let $Y \in \mathbb{R}^n$ be a nonempty convex set such that $0 \notin Y$ and fix $y_1,\dots,y_n$ in $Y$, where $n \ge 2$. I know that there exist $i,j$ such that $\Vert y_i \Vert > \Vert y_j\Vert$....

The problem: Let $N$ be an $L$-structure for a language $L$. The $diagram$ of $N$, $D(N)$ is the set of all quantifier-free $L_N$-sentences true in $N$. Suppose $M$ is a model of $D(N)$. Show that $M$...

Let the real numbers and the relations =, >, $\ge$ be defined as in this lecture PDF. I want to show the following statement: $$\forall x \in \mathbb{R}( x \ge 0 \wedge x \neq 0 \Longrightarrow x &...

I have the boolean expression: $$ ((\neg A \lor B) \land (\neg B \lor \neg C))\implies (A\implies \neg C) $$ I have reshaped this so far that no equivalence and no implication is included. As far ...

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