**15.309 logic questions.**

If an alphabet $= \{a, b\}$
I believe an example of a finite language over that alphabet with positive cardinality would be the set equal to $(a+b)^4$
An example of a countably infinite language ...

Any employee who does not participates for the strike or work in contract basis will
report for work. Some employees in contact basis will participate for the meeting. All
employees who do the strike ...

Here are four sentences:
If Jessy moves his truck, Irene will play her guitar
Irene will only move her car, if Jessy moves his garbage cans
It is not the case, that Jessy will move his ...

Consider the following statement: One plus the square of an odd natural number is divisible by a product of 2 consecutive natural numbers.
Write this in symbolic language. Is it true or false?

The principle of explosion is this:
$$ X \land \lnot X \rightarrow Y $$
And similarly, the rule of weakening/monotonicity of entailment is this:
$$ (X \rightarrow Y) \rightarrow (X \land A \...

Can a set that is closed under a binary operation have inverses for all of its elements.. without the existence of an identity element?
My mind is telling me that in order to even have inverses, ...

∀x (¬Strike(x) ∨ Contract(x)) ⇒Report(x) Eliminate ⇒∀x ¬(¬Strike(x) ∨ Contract(x))∨Report(x) Push negation inside <...

I'm new to everything involving languages and I'm having trouble proving properties with quantifier elimination. In particular what's below.
Suppose we know the following:
$Lemma$ (L1): Let $N \leq ...

If I am late for my interview, then I will not get the job. I will get the job if I interview well.
I will not interview well if I am late.
If I catch a taxi, I will not be late for my interview.
I ...

Let P[1 … n, 1 … m] be a 2‐dimensional array of the pixels of a black‐and‐white image: for every x and y, the value of P[x, y] = 1 if the 〈x, y〉th pixel is black, and P[x, y] = 1 if it's white. ...

I'm looking for a simple proof (or a reference) for the following claim:
The set of even natural numbers is not definable in the structure $\langle \mathbb{N},<\rangle$.
Thanks

I've tried making the antecedent and the conclusion implications on the LHS expression, and I've tried making the implication into an OR but I'm stumped as to how to derive logical equivalence from ...

Let $P$ be a quantifier-free decidable formula, i.e. one can prove $P \lor \lnot P$.
Does it follow that $\lnot \lnot P \to P$ intuitionistically?
Informally, a decidability of a formula means ...

Can someone help me understand this logic problem:
What are the truth values of these statements?
a) $\exists !\,x\;P(x) \implies \exists x\;P(x)$
b) $\forall x\;P(x) \implies\exists !x\;P(x)$
c) $...

Is there a mathematical symbol to represent "else"; the way it's used in computer science and propositional logic? Is it simply "or"?

I am asked to verify whether or not you can axiomatise the theory of countable dense total orderings in the language of partially ordered sets.
We have the Upward Lowenheim-Skolem Theorem, telling ...

Question: Translate the following statement into the notation of predicate
logic.
P: Not all natural numbers are even.
My solution:
P: ~[∀x ∈ N, x/2 ∉ Z]
P: ∃x ∈ N, x/2 ∈ Z
Is my solution ...

I'm trying to understand the use of the Compactness Theorem to proof certain properties for theories in languages. I've tried to prove the following:
If $\phi$ holds in every model of the theory $T$, ...

It is well-known that validity in Peano Arithmetic is undecidable. It is less well-known that validity is already undecidable in True Arithmetic (the theory of the standard model of Peano Arithmetic). ...

So i have started chapter 2 of Jech's Set Theory on Transitive models of ZFC. There are many parts which i don't understand. First let me give some context:
We define restricted quantifiers to be all ...

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 ...

For a first-order sentence $\varphi$ the spectrum of $\varphi$ is the set of cardinalities of its finite models, i.e.
$$\operatorname{spec}(\varphi):=\{n\in\mathbb{N} \mid \text{ There is a model } \...

Say we have some theory $T$ such that $Th(A_E) \subseteq T$ where $A_E$ are the axioms of arithmetic. How do I show that
(1) there are sentences $\varphi_1$ and $\varphi_2$ such that $Th(A_E) \vdash ...

I came across this paper by Scott Aaronson and though I understand nothing of quantum computing, the fact that there was an (even hypothetical and probably unrealizable) model of ...

I would like to be able to prove $(*)$ $\vDash_\mathbf K$ $\Diamond (A \lor B) \equiv (\Diamond A \lor \Diamond B)$
My initial attempts have been to use the following two tautologies to get ...

Pablo has square chocolate box of size n x n in which a variety of healthy chocolates are present denoted by 'H' initially but he finds out that some of the chocolates are rotten and unhealthy denoted ...

I'm struggling to adapt the following logical proposition to a CNF formula
$$\bigwedge\limits_{\substack{g_1,g_2\in Stations \\ t \in Trains \\ m \in Minutes\\ g_1 \neq g_2, t_1\neq t_2}} \bigvee\...

I have a problem with an argument in Fine structure and iteration trees by Mitchell and Steel. Let $E$ be a $(\kappa, \lambda)$-extender. Let $\dot E^{\mathcal{M}}$ the a unary predicate with is ...

I'm a high school student new to propositional logic. This is only my second attempt at writing a proof. I originally started with trying to prove the associativity of addition, but that was a bit ...

I know we can derive something like this from =Elim rule:
$$\varphi(a,b)$$
$$b=c$$
$$\therefore\varphi(a,c//b)$$
But I want to know if we can also go ahead and derive something like this if there was ...

I only have proved that if graf($f$) is recursive set then is recursively enumerable set by $(n_0, m_0) \in Graf(f) $
$$h(n):= \begin{cases} (n, m) \textit{ If } (n,m) \in Graf(f) \\(n_0, m_0) \...

I am a high school student interested in propositional logic. I have been studying many examples and decided to attempt to write my own. I feel like it's very badly done and would appreciate some ...

Disclaimer: I'm not yet really comfortable with internal logics, though I know the basics, so the best answer would not be the most technical one.
Suppose you want to prove a diagram lemma (say the ...

I'm studying backwards chaining for an exam in AI and I understand the overall idea. However, I'm confused about the following slides from one of the lectures:
where $KB = \text{knowledge base}$.
...

"A dragon is happy only if it has a green child".
Have I translated this statement correctly into logic (below)?
$\forall X \cdot dragon(X) \wedge happy(X) \Rightarrow \exists Y \cdot childOf(Y,X) \...

If we were given the task of proving Euler's identity using a formal logic system, which logic system out there would be the most convenient for such a task? And more or less what would the proof look ...

I'm new to set theory, and the axiom of regularity has been giving me some trouble. It states that every non-empty set A has an element B such that A and B are disjoint sets. Apparently, this axiom ...

In the book Gödel's Proof, by Nagel and Newman, the authors write (p. 119):
There is a theorem in logic which reads:
$$(p \cdot q) \supset (p \supset q)$$
or when translated, 'If both $p$ ...

If I solve an equation, for instance, if I assume the premise 5 * x = 10, I can logically conclude that x = 10 / 5 = 2. This is a trivial example, but even for more complex equations, a correct ...

I'm trying to understand the distributive law of Boolean algebra by relating it to what I know from ordinary algebra, and it seems to only work for one form of the law.
Ordinary algebra:
$x(y+z) = ...

Call a function $f : \mathcal{N} \rightarrow \mathcal{N}$ repetitive if for every finite sequence of natural numbers $(a_1, a_2, \cdots,a_n)$ there exists a number $k \in \mathcal{N}$ satisfying
$f(k)...

While studying the internal language of categories I came across the following result on Johnstone's Elephant (pag. 838, D1.3.12):
(he deliberately denotes by $f$ the corresponding function symbol $\...

I am confused about Church's simply typed lambda calculus and the Curry-Howard isomorphism.
Church's simply typed lambda calculus in the paper cited above is given a classical proof theory, in that ...

First, I will present the notations of the exercise $4.15$, which can be founded at the bottom of the page $37$ of Ebbinghaus's Mathematical Logic:
Now, the question:
I know that it's necessary to ...

How much of non-trivial formal logic is necessary in the mathematics used in, or implicitly underpinning, theoretical physics?
I am looking for the explicit logical proofs used. I need the actual ...

I am given a truth table of a function as follows:
\begin{array}{rrrr|r}
& x & y & z & f \\ \hline
& 0 & 0 & 0 & 1 \\
& 0 &...

I'm reading Enderton's logic book and have arrived to his deductive calculus for first order logic. After defining it, he presents the following theorem:
$\Gamma\vdash \varphi$ iff $\Gamma\cup \...

I'm having trouble symbolizing this sentence:
"Every giraffe loves only Alfred."
My thought process was "Every x such that x is a giraffe then it loves something y and y is Alfred."
∀x (Gx → ∃y (L(...

So let's say you have a set of consistent axioms A. GIT states that there are some true statements that cannot be proven from A.
But what is truth? Truth is by definition anything that is a logical ...

Any employee who does not participates for the strike or work in contract basis will report for work. Some employees in contact basis will participate for the meeting. All employees who do the strike ...

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