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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How do I prove double negation elimination in a propositional logic axiom system?

Here are my axioms: X -> (Y -> X) (X -> (Y -> Z)) -> ((X -> Y) -> (X -> Z)) (~Y -> ~X) -> ((~Y -> X) -> Y) You can use any uniform substitution of these axioms, and you can use Modus Ponens. Need ...
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0answers
21 views

Double Negation in Propositional Logic [on hold]

Given only modus ponens and the following axioms: $$ (@>(\$>@) \\ (@>(\$>\#))>((@>\$)>(@>\#)) \\ (\sim\$>\sim@)>((\sim\$>@)>\$) \\ $$ How do I show $P$ given ...
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0answers
18 views

Logic - Is it safe to state the following?

say that ∀x∃y in all possible integers (negative integers, 0 and positive integers) is x*y = x is it safe to say that ∃y∀x is also true. If not can someone explain why its not true. The way I'm ...
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1answer
59 views

How is induction justified in intuitionistic logic?

This question might be extremely naïve for which I apologise in advance. The induction principle can be stated as: If $A ⊂ ℕ$ such that $1 ∈ A$, and $ν(A) ⊂ A$ (where $ν\colon ℕ → ℕ$ is ...
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1answer
26 views

Computational tree logic satisfiability.

In the model I pasted above where $S_0$ and $S_1$ are starting states, is the $EXp$ formula satisfiable? $$M,s\vDash EXp$$ Does it have to be satisfiable for all the starting states given the $M$, ...
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3answers
68 views

If A implies B and C implies B, do A and C together imply B? [on hold]

If A implies B and C implies B, do A and C together imply B? I need a clarification regarding this question.
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1answer
26 views

How to transcribe the following statement into a predicate wff?

There was a disagreement in my college class regarding what the following statement would be in a predicate wff format: It is always a sunny day only if it is a rainy day. Where D(x) is "x is a ...
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0answers
68 views

What is a Horn Clause? [on hold]

I am not an expert in Mathematics :) thus if someone can let me know What is a Horn Clause in layman's terms? I know it is used in First order Logic but I am unable to understand what is it and how to ...
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1answer
53 views

A version of Zorn's lemma

The version of Zorn's lemma that I have found more often is Zorn's Lemma (1) If every chain belonging to the partially ordered set $S$ has an upper bound in $S$ then $S$ contains a maximal ...
2
votes
2answers
60 views

Principle of explosion: Other arguments?

I've come across a proof-theoretic argument for explosion on Wikipedia, which is as follows: $A \ \ \wedge\sim A$ $A$ $ \sim A$ $ A \lor B$ $B$ $(A \ \ \wedge \sim A) \implies B$ I've thought of ...
2
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2answers
39 views

Logic - Logically implies question

$\forall x(A(x) \rightarrow B(x))$ logically implies $\exists x(A(x) \land B(x))$ Is the above statement true or false? I have no clue on how to start figuring this out. Can someone help me please?
3
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2answers
90 views

Logic - how to write $\exists !x$ without the $\exists !$ symbol [duplicate]

What is $\exists !$ equivalent to? I need to write $\exists !x \,P(x)$ without using the $\exists !$ symbol; thus, I am wondering what the $\exists !$ symbol is equivalent to.
3
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2answers
28 views

Difference between “necessary” and “necessary but not sufficient”?

This is from Discrete Mathematics and Its Applications: Let $p, q,$ and $r$ be the propositions: $\quad p:$ Grizzly bears have been seen in the area. $\quad q:$ Hiking is safe on the ...
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1answer
33 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 ...
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1answer
23 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 ...
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2answers
43 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 ...
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3answers
313 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. ...
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0answers
14 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.
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2answers
61 views

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

Write down the assumptions in a form of clauses and give a resolution proof that the proposition $$\Big((p \rightarrow q) \land ( q \rightarrow r) \land p \Big) ...
19
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12answers
2k 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
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1answer
16 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 ...
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2answers
54 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
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4answers
58 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 ...
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1answer
28 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
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1answer
42 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, ...
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1answer
30 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
39 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
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1answer
37 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
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0answers
18 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
89 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 ...
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1answer
38 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
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1answer
33 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 ...
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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 ...
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0answers
34 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: ...
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0answers
56 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 ...
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2answers
40 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
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1answer
25 views

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

There are no natural numbers that are squares and differ 5.
3
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2answers
34 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} ...
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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 ...
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1answer
34 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 ...
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2answers
40 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 ...
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1answer
51 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 ...
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1answer
32 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
125 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 ...
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0answers
26 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 ...
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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 ...
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1answer
66 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 ...
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1answer
27 views

boolean simplification , help please [closed]

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.
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3answers
33 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.
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0answers
88 views

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

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.