Questions about logic and 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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2
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1answer
13 views

Difference between necessary and necessary but not sufficient?

This is from Discrete Mathematics and its applications I read up on necessary and sufficient from here What is the difference between necessary and sufficient conditions? If p->q (p implies ...
0
votes
1answer
19 views

Question about mathematical logic ∀x ∈ S, ∀z ∈ S, ∃y ∈ C

∀x ∈ S, ∀z ∈ S, ∃y ∈ C,(x != z) ⇒ ¬(T(x, y) ∧ T(z, y)) I'm trying to express this in English, but I can't use the variables x or y in my sentence. Basically it means for elements x in S, and all ...
0
votes
1answer
12 views

Expressing the converse, contra-positive, and inverse of conditional statements

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 ...
1
vote
2answers
24 views

Proving contradiction with logical identities

We know that p → q is not equivalent to q → p. But suppose we make a proof system that has all the rules of logical identities plus the rule (“commutativity of implies”) p → q ≃ q → p. (We are using ...
1
vote
3answers
201 views

Is there a quicker way to check if this proposition is self contradictory?

I have been trying to refresh my memory with regards to classical logic. As a result, I am currently going over the basics. The following proposition seems to be false in all possible worlds. ...
-2
votes
0answers
12 views

Give an example of two relational systems $A$ and $B$ and a homomorphism $h : A\rightarrow B$, which is not a strong homomorphism. [on hold]

Give an example of two relational systems $A$ and $B$ and a homomorphism $h : A\rightarrow B$, which is not a strong homomorphism.
-1
votes
2answers
39 views

Can anyone help me with a solution? [on hold]

Write down the assumptions in a form of classes and give a resolution proof that the formula $$\Big((p \rightarrow q) \land ( q \rightarrow r) \land p \Big) \rightarrow ...
6
votes
8answers
522 views

An easy example of a non-constructive proof without an obvious “fix”?

I wanted to give an easy example of a non-constructive proof, or, more precisely, of a proof which states that an object exists, but gives no obvious recipe to create/find it. Euclid's proof of the ...
2
votes
1answer
15 views

What is the difference between weak and strong completeness in many valued logic?

I know a bunch of facts about weak and strong completeness in many valued logic, that there is strong completeness for the finite mv logic, and that for the infinite ones you can either only have weak ...
0
votes
2answers
48 views

Can someone verify my assertion from this english sentence? [duplicate]

This is from Discrete Mathematics and its Applications This is the book means when mentions a list of common ways to express conditional statements After going through the list, I immediately ...
2
votes
4answers
53 views

Clarifying on how if p,q is logically equivalent to p only if q [duplicate]

Here is what my book says about the different ways implications are worded I am struggling with how "if p, then q" is logically equivalent to "p only if q" The example I came up with With "if ...
0
votes
1answer
27 views

How to tell the difference between interval and coordinate notation from context?

I am working on a practice problem with sets. (the answer key) At first I was confused by the notation Ai = (0,i), i is a natural number. I looked up the use of paranthesis and saw that they could ...
2
votes
1answer
39 views

Please help me to understand domain of interpretation

In the literature on Description Logic, when interpretations are explained, we encounter expressions like, $$\mathcal{I} = (\Delta^\mathcal{I}, \cdot^\mathcal{I})$$ (Actually, I am talking about, ...
1
vote
1answer
26 views

Negation of a Statement with Quantifiers — If Then?

I need to find the negation of a statement on my homework, specifically problem 19 of secton 3.2 in Discrete Mathematics with Applications by Susanna Epp. The problem is as follows: \begin{align} ...
2
votes
1answer
37 views

How to adapt proof by contradiction showing that a sqrt(2) is irrational for sqrt(20)?

This example is from Discrete Math and its Applications I understand the steps the author is taking. First he assumes sqrt(2) is rational meaning that there exists integers a, and b such that ...
2
votes
1answer
35 views

Show that the conditional statement is a tautology without using a truth table

I have been attempting to use identities to get to the answer but I am unable to get anywhere. Here is the equation I am trying to prove tautological without using truth tables: $[(p\rightarrow q) ...
0
votes
0answers
16 views

Is it necessary to write out the whole truth table to show system specification is consistent?

This is an example from Discrete Mathematics and its Applications Basically the way I see this problem is "is there a combination of propositions that will make all of these specifications true". ...
5
votes
1answer
86 views

How can you come to the truth of a statement without proving it?

I was reading a bit about Gödel's incompleteness theorems. I haven't took the time to really study it, but I'm very curious about statements like these: In other words, if our axioms are ...
1
vote
1answer
37 views

What is the difference from a theorem and a meta-theorem?

I'm confused about what a meta-theorem exactly is and if a meta-theorem can be used to prove a theorem. To illustrate my confusion i give an example. Given the three statements: Every vector space ...
2
votes
1answer
31 views

odd logical structures

How you find contrapositive and converse of these sentences. Only if John chops down the tree, will he be a lumberjack. You can't win if you don't fight. All people that root for the Ducks are from ...
0
votes
1answer
18 views

What is a predicate exactly in predicate logic?

I have been reading Predicate Logic couple of days and while everything has been pretty intuitive so far I understood that I do not exactly understand what the predicate is. This became clear after I ...
0
votes
0answers
31 views

Use rules of inferential logic for the following problem..

Here I have such a question related to laws of inference. The question asks to prove using the laws of inference (these rules) that the following facts give a certain conclusion. So the question is: ...
0
votes
0answers
53 views

On correctness of induction proof

I want to prove a certain property $\mathsf{P}$ on every multiaffine polynomial in $\Bbb R[x_1,x_2,\dots,x_{n-1},x_n]$. Supposing I show property $\mathsf{P}$ to be valid at $n\geq9$ variable ...
1
vote
2answers
38 views

Propositional Logic : Absorption - Why is it so?

Why is the Absorption Law of Propositional Logic so ? p $\lor (p \land q) \equiv$ p Would appreciate an intuitive explanation and not one using a Truth Table
-5
votes
1answer
25 views

Prove the statement. Logic and Set Theory. [on hold]

There are no natural numbers that are squares and differ 5.
3
votes
2answers
33 views

Is my deduction of $t$ being true logically correct?

According to the problem on my homework (yes, this is my homework), number 42 in chapter 2.3 of Discrete Mathematics with Applications by Susanna S. Epp, the following are true: \begin{align} ...
0
votes
0answers
28 views

Does the class of all periodic subsets of $\mathbb{Z}$ of peroid greater than $k$ form a field of sets?

We say that a subset $X\subseteq \mathbb{Z}$ is a periodic subset of $\mathbb{Z}$ of period $k$ if the set obtained from $X$ by adding $k$ to each element of $X$ is $X$ itself. Does the class of all ...
1
vote
1answer
33 views

Why is the set of all true first-order statements about non-negative integers in the language with only equality, $+$ and $\times$ undecidable?

Apparently Tarski and Mostowski proved this, but intuitively I'm not seeing the difference between statements in a language of non-negative integers with equality, addition, and multiplication vs ...
0
votes
2answers
36 views

What is the difference between a counter-intuitive statement and a paradox?

In mathematics and logic, what is the difference between a counter-intuitive statement and a paradox? For example, what differs something like the Banach-Tarski theorem or Gabriel's horn from ...
0
votes
1answer
49 views

L-sentence which expresses bijective function

I've stumbled upon this exercise from "Sets, Models, Proofs" and can't seem to find a solution. It goes like this: Let $L$ be a language with just one 1-place function symbol $F$. Give an ...
1
vote
1answer
29 views

Use logical equivalencies to classify as tautology, contradiction, or contingency.

Classify the following as tautologies, contradictions or contingencies using logical equivalences. Can anyone let me know what I'm missing or doing wrong? I got stuck, here is what I have so far: ...
2
votes
4answers
122 views

Meaning of symbols $\vdash$ and $ \models$

I'm confused about the use of symbols $\vdash$ and $ \models$. Reading the answers to Notation Question: What does $\vdash$ mean in logic? and What is the meaning of the double turnstile symbol ...
0
votes
0answers
25 views

Logic problems and Venn Diagrams [on hold]

In a class of 32 pupils: 5 pupils live in New Town, travel to school by bus and eat school dinners. 3 pupils live in New Town, travel to school by bus but do not eat school dinners. 9 pupils do not ...
0
votes
1answer
51 views

Use inference rules to prove distributive law

I'm taking an intro logic course this semester and my prof is hard to follow and not really great at clarifying things. I'm stuck on this question in my assignment, I'm just not sure how to start. I ...
0
votes
1answer
65 views

Discrete Math Predicate Logic

Consider truth assignments involving only the propositional variables $x_0, x_1, x_2, x_3$ and $y_0, y_1, y_2, y_3$. Every such truth assignment gives a value of $1$ (representing true) or ...
-3
votes
1answer
25 views

boolean simplification , help please [on hold]

If we begin with $\;\bar A\,\bar C+ \bar B\,\bar C + A\, B\;$ how can we transform to $\;\bar B \, \bar C + B \,\bar C + A\, B\;$. I'm so lost please help.
0
votes
3answers
32 views

Discrete Math Logical Equivalence

x∧ ∼ y → ∼ z is logically equivalent to x ∧ z → y. I can't figure it out, especially the negations are throwing me off.
-5
votes
0answers
86 views

There is a student who has been in at least one room of every department - formalize this [on hold]

My teacher gave me this exercise to do, but noone in my class has any idea how to solve it. So I would require some help, please, and maybe also an explanation.
1
vote
2answers
35 views

Logical form of statement

I'm reading the book How to Prove It and a question is given to write out the logical form of the below definition in set-theoretic notation. Definition: $y \in \{\sqrt[3]{x} \mid x\in\mathbb{Q}\}$ ...
1
vote
1answer
35 views

Order of quantifiers

I was reading about quantifiers from this book. I decided to jot down all implications due to different orders of quantifiers. While talking about the orders of the quantifiers the author states ...
1
vote
1answer
32 views

Convert universal quantification to existential quantification

I came across following problem "Every intelligent student is not honest." And I have to convert this in quantifiers. Straight conversion will be: ∀x [(S(x)∧I(x)) → ¬H(x)] ...(i) However the ...
0
votes
0answers
90 views

Question about the foundation of mathematics [duplicate]

I have studied mathematical logic and set theory as an undergraduate. I studied mathematical logic (propositional and predicate logics) before set theory. When I studied mathematical logic, I was a ...
0
votes
1answer
22 views

I need to relate strings of implications.

Let's say we have a string of implications $p_0\Rightarrow p_1\Rightarrow\cdots\Rightarrow p_n$. What can be said about $p_n\Rightarrow p_{n-1}\Rightarrow\cdots\Rightarrow p_0$ from the original ...
1
vote
2answers
69 views

First Order Logic vs First Order Theory

What is the difference between a First Order Logic and a First Order Theory. Can anybody please describe what each one precisely (formally) is? For a bit more elaboration on the question, I think ...
1
vote
1answer
24 views

Using quantifier get truth value

In each case below say whether the given statement is true for whcih universe $(0,1)={{(x\in R: 0<x<1})}$ $[0,1]={{(x\in R: 0\le x \le1})}$ $\exists y(\forall x( x>y)$ This means there ...
0
votes
0answers
32 views

How do we express higher arity predicates and functions in terms of membership?

It's been noted by others that higher order logic is similar to set theory. We can express the second order statement $\forall$R$\forall$x(R(x)) as a first order statement $\forall$R$\forall$x (x ...
1
vote
5answers
305 views

Which can be logically inferred from the given statements?

All women are entrepreneurs. Some women are doctors. Which of the following conclusions can be logically inferred from the above statements? (A) All women are doctors. (B) All doctors are ...
0
votes
2answers
39 views

Show this language structure models this sentence.

In an effort to educate myself, I am attempting the second problem in first chapter of the book "Model Theory" by Marker. The problem is reproduced below: Let $\mathcal{L} = \{\cdot, e\}$ be the ...
0
votes
1answer
19 views

Show every boolean combination of $\mathcal{L}$-formula is equivalent one with quantifiers.

This is part 2 of a question I asked here: Prove this claim about language and structures. The setting is that suppose $\phi_1,\ldots,\phi_n$ are $\mathcal{L}$-formulas and $\psi$ is a Boolean ...
3
votes
2answers
75 views

Proof by Iteration

It seems that I suffer the "too-much-logic-too-pedantic-too-confused"-disease. (You know? This very disease which lets you doubt everything and lets you yell for formalized proof. It's annoying, ...