2
votes
2answers
77 views

$\forall$ At the beginning or at the end?

I have a set of real numbers $x_1, x_2, \ldots, x_n$ and two functions $f:\mathbb{R} \rightarrow \mathbb{R}$ and $g:\mathbb{R} \rightarrow \mathbb{R}$. What are the differences between the following ...
0
votes
0answers
29 views

Italic notation in logic

I have seen some books on formal logic where variables are written in italic, while statements are upright. Hence a statement could like like $\mathrm A(x_1, \ldots, x_n)$. How much of a standard is ...
0
votes
1answer
72 views

Mathematical notation for expressing the top n elements

I would like to know what is the mathematical notation to express the top n elements. Look at the equation below. Here $x_w$ is a feature vector representing the contribution of a particular word $w$. ...
1
vote
1answer
18 views

Notation for Model-Relation of formulae with free variables

Lets assume we have a formula $\mathsf{path}(x,y)$ with free variables $x,y$, and $\mathsf{acyclic}$ with no free variables on the signature $\tau = \{E\}$ (i.e. Graphs). Informally, what the formula ...
1
vote
1answer
28 views

Representing a for loop with modulus in formal notation

I have the following section of code I am writing for research. Basically I need to formally represent the mathematical notation behind a set that follows: ...
1
vote
2answers
92 views

Can I use “: ,” instead of “, implies” for this example?

I've to write this statement in a formal manner: if $x>1$ then $x^{2}>1$. Writing the result of the exercise I face this problem, I wonder if these two statements are equivalent: ...
0
votes
1answer
48 views

Notation for the horizontal line found in sequent calculus for linear writing

Is there any known Unicode character which can be used as a substitute for the horizontal line which is normally used in texts presenting sequent calculus? I actually use the oblique bar ...
1
vote
1answer
92 views

What does the dot symbol mean?

A = B ⇔ (∀x.x ∈ A ⇔ x ∈ B) What does "∀x.x" mean? This expression means: saying 2 sets are equal, is equivalent to saying they're the same set, right? Thank you.
1
vote
2answers
40 views

Help with understanding the definition of operation

I'm having trouble understanding this excerpt from Wikipedia, which defines an operation: Mainly, I don't understand what is meant by $V \subset X_1 \times...\times X_k$. Why does an operation ...
3
votes
2answers
256 views

How do commas and brackets affect the meaning of quantifiers?

My logic class didn't introduce us to multiple quantifiers. I've seen a few variations that seem to have distinct meanings: $$ \forall x, \forall y(...) $$ $$ \forall x \forall y(...) $$ $$ \left( ...
1
vote
1answer
36 views

Split long relation over two line using boolean operator

Normally, when you have a long equation, you can split it on two lines. Suppose that $a$ and $b$ are very long expression. Then, for example: $$ x = a - b$$ can be rewritten as $$ x = a + $$ ...
0
votes
3answers
60 views

Give the truth table of a single binary connective which is adequate.

This might be a silly question, but I am confused. I know there is a theorem saying the only single binary connectives which are adequate are NOR or NAND, so I could use either of them. And then the ...
1
vote
2answers
88 views

Sigma hierarchy of logical formulae

In some papers on mathematical logic I've found references to hierarchy like $\Sigma_1^0$-sentence and so on. What does it mean? What is $\Sigma_n^m$, what is $n$ and $m$ here?
0
votes
1answer
25 views

How to describe a set of coordinates of variable length?

I need to describe a set of coordinates with up to 8 dimensions. A problem is asking me to describe an event from a experiment involving sampling. The catch is that the experiment doesn't end until a ...
0
votes
2answers
86 views

Can logic and mathematics be used together?

I have been wondering if logical symbols ($\to$, $\sim$, etc.) can be used with traditional mathematical notation ($+$, $/$, etc.) in the same equation. For example, would the following equation be ...
14
votes
3answers
284 views

Mathematical Notation and its importance

You can see how mathematical notation evolved during the last centuries here. I think everyone here knows that a bad notation can change an otherwise elementar problem into a difficult problem. Just ...
1
vote
1answer
36 views

Other than $\models$, is there standardized notation for semantic consequence?

It is common practice to use $\models$ both for the satisfaction relation between models and sentences, and for the corresponding semantic consequence relation. Question. Suppose I don't want to use ...
10
votes
5answers
508 views

What are reasons why some symbols in mathematical logic are not standardized?

Why is so hard to find a standardisation regarding symbolism and/or terminology in Mathematical Logic ? We see again and again students asking if e.g. $\rightarrow$ and $\implies$ means the same ...
1
vote
1answer
73 views

Is identity included in the “key” in predicate logic?

So, my exam is in a few days. We've been told to practice setting up a key in predicate logic. . From what I've understood, a typical key looks something like this: $ Lxy$: $x$ likes $y$ ...
4
votes
3answers
368 views

What does this combination of symbols mean? $\exists !$

I just want to know what this combination of symbols means: $\exists !$ I know ∃ means 'there exists', but what does it mean when it is paired with a '!'? I have written down 'there exists unique" ...
0
votes
1answer
43 views

Proper way to use universal quantification with multiple arguments

Which one is correct? $\forall x \in X, y \in Y:$ some expression or $\forall x \in X \wedge y \in Y:$ some expression or $\forall x \in X \forall y \in Y:$ some expression
0
votes
0answers
28 views

Correct way to state logical constraints on variables

I need a refresher as how to express logical constraints. Here are two constraints I want to express: (1) For example, I have matrix with cells $p_{i, j}$, where $i$ is the row index, counting from ...
0
votes
2answers
91 views

Unable to understand combination of quantifiers and set notation

I know what universal and existential quantifiers are but following is confusing,may be its comibination of set notation and quantifers. What does the following statement means? ...
2
votes
1answer
54 views

Notation for signatures

A (multi-sorted) signature consists of some sort symbols, say $X$ and $Y$, together with some constant symbols, say $0,1 : X$, some function symbols, say $f : X \times Y \rightarrow X$ and $g : Y ...
0
votes
1answer
73 views

Correct interpretation of Kleene (Intro to Metamathematics) symbol $\vdash^x$ in Predicate Calculus

Rif. S.C.Kleene, IM (1952) : which is the correct interpretation (or the "modern equivalent") of the "x" used as exponent of the "turnstile" as in: $$A(x) \vdash^x \forall xA(x)$$ [see Derived Rules ...
0
votes
1answer
35 views

Notation for exists two elements in a set with properties

I'd like to say: for any x in set X, if x is colorful, there must be t1 and t2, both in set T, such that t1 < t2 and green(x,t1) and red(x,t2). I believe this is the correct notation, but I'm not ...
2
votes
2answers
156 views

Is there a proper term and/or symbol for an “agnostic” conclusion?

My question stems from the material conditional: $p \rightarrow q\\p\\\therefore\space q$ However, if $\bar p$ then the conditional is silent. I would like a way to represent this fact using, if ...
2
votes
2answers
92 views

$\underset{x}{\bigvee} \mathfrak{P}x \wedge \underset{x}{\bigwedge} \underset{y}{\bigwedge}(\mathfrak{P}x \wedge \mathfrak{P}y\rightarrow x=y)?$

I'm reading Behnke's fundamentals of mathematics, he written that the following proposition: $$\underset{x}{\bigvee} \mathfrak{P}x \wedge \underset{x}{\bigwedge} \underset{y}{\bigwedge}(\mathfrak{P}x ...
4
votes
4answers
145 views

Velleman - How to prove it - Do these two statements really mean the same thing?

Hello and thanks in advance for reading! In How to Prove it P29 Velleman writes: " In general, the statement y ∈ { x | P(x)} means the same thing as P(y), ... " In my understanding the first ...
0
votes
2answers
49 views

Are these statements in logic form correct?

Let M represent the set of all Mathematics courses and S represent the set of all students. Predicates: ...
0
votes
0answers
124 views

Definitions of different kinds of recursive reducability.

So this is just a definition check. In recursion theory we have many kinds of reduction, according to my notes we have $\leq_1,\leq_m,\leq_{tt},\leq_{wtt},\leq_{TT},\leq_{T}$. Now I can't for the ...
1
vote
2answers
54 views

How to read this logical statement in English?

Statement: ∀n ∈ Z, [(P(n) ∧¬(n=2)) ⇒ O(n)] where Z is a set of natural numbers P(n) is the predicate "n is a prime number" O(n) is the predicate "n is an odd number" I got this, but I don't think it ...
0
votes
1answer
116 views

Partial order relation

Define the relation $\leq$ on a boolean algebra $B$ by for all $x,y\in B$, $x\leq y \iff x\lor y=y$, show that $\leq$ is a partial order relation. First of all what exactly does boolean ...
1
vote
1answer
41 views

How to read structural rules

In the wikipedia page on structural rules, we have the following "weakening" rule. $$\frac{\Gamma \vdash \Sigma}{\Gamma \vdash A, \Sigma}$$ This makes no sense to me. It seems to be saying that, if, ...
2
votes
2answers
102 views

Is “iff” the same as equality if each member is a predicate?

"Iff" - if and only if ($\Leftrightarrow$ or $\leftrightarrow$, although the first usually carries a "meta" meaning, something that is not evaluated) - is used in $2x=3\Leftrightarrow x=\frac32$, ...
3
votes
2answers
111 views

Why use the non-exclusive or sign in equations?

$x^2=4 \Leftrightarrow x = 2 \lor x=-2$, but $x$ can't be equal to both $2$ and $-2$, unlike what the inclusive-or symbol ($\lor$) implies. Although it may seem obvious, it's just wrong to say that ...
2
votes
1answer
64 views

Is there a symbol for a repeated logical and operation?

For example, given a predicate $P$ and a set of variables: $\{x_1,x_2,x_3,x_4,...\}$, is there a single symbol similar to $\sum_{k=1}^n$ or $\bigcap_{k=1}^n$ to denote the following? ...
4
votes
2answers
173 views

Symbols: If but not only if

I was searching for a Latex symbol that indicates $A \Rightarrow B$ and $A \not\Leftarrow B$ ($B$ if not only if $A$, $B$ ifnf $A$). I thought of using $A \Leftrightarrow B$ with the left arrow tick ...
2
votes
1answer
90 views

A few basic questions about the arithmetical hierarchy, mostly about terminology.

I was reading about the arithmetical hierarchy, and I have a few questions, mostly notational. For completeness, here's the definition given over at Wikipedia. The classifications $\Sigma_n$ and ...
2
votes
2answers
57 views

Notation in propositional logic

If in propositional logic one is trying to simplify a formula by evaluating its subformula, would it be considered an abuse of notation to actually substitute the bits $\{0,1\}$ in for the formula, to ...
3
votes
7answers
533 views

Symbol for “if and only if”: $\implies$ or $\iff$?

I was wondering about the iff sign in maths. I've never learned about it in school & see it a lot online. Usually the sign looks like this: $\implies$, but in math.stackexchange I always see this: ...
2
votes
2answers
102 views

How do you define definition symbol :=?

How is "$:=$" defined formally and why? "$\iff$", "$=$", ...?
3
votes
0answers
283 views

What is the conventional notation for these logic statements?

When I studied chemical engineering I often found the need to rewrite lecture notes, handouts and books in order to gain a thorough understanding of the subject I was reading. As much as time ...
0
votes
1answer
82 views

Confusing symbol in papers on hybrid logic

In literature about hybrid logic I'm reading for my thesis I've come across the following symbol: ::= Now, I've never seen this notation before. I can also not ...
3
votes
3answers
127 views

Using $p\supset q$ instead of $p\implies q$

I saw that a use for the notation $p\supset q$ instead of $p\implies q$ that got me a bit confused. One occurrences is in this Wikipedia link. It seems to me opposite than what it should be, let me ...
3
votes
1answer
59 views

Is it a standard to say that $a \oplus a_{\small 1}=0$ or $a \veebar a_{\small 1}=0$?

I am trying to express the following: $a$ or $a_{\small 1}=0$ but only one of them equals zero. so if $a=0$ then $a_{\small 1}\neq 0$ and if $a\neq 0$ then $a_{\small 1}=0$. And I'm ...
3
votes
0answers
107 views

Definition(s) for variable binding in first-order logic

The following statement made me realize that variable binding can be defined in first-order logic: The same holds for λ terms to define functions. There is no reason that they could not be ...
6
votes
1answer
322 views

What are $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$?

Sometimes reading on wikipedia or in this site (and in very different context like topology, arithmetic and logic) I have found these symbols $\Sigma _n^i$, $\Pi _n^i$ and $\Delta _n^i$. They are ...
2
votes
1answer
278 views

In Logic is ⇒, →, and ⊃ basically the same symbol?

I need to create a few truth tables and I got confused by the logic symbols as some of the questions use either one or the other which is really confusing especially if they all mean the same thing. ...
6
votes
5answers
3k views

Negating A Mathematical Statement

Regard this statement $ x \ge 0$. According to my teacher, by negating this statement, it will become $ x < 0$. Why is this so; why does the $\ge$ morph into $<$, and not into $\le$?