Tagged Questions
2
votes
2answers
63 views
How do you define definition symbol :=?
How is "$:=$" defined formally and why? "$\iff$", "$=$", ...?
0
votes
0answers
51 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
39 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 ...
1
vote
3answers
61 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 ...
2
votes
1answer
39 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
36 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
241 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
94 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.
...
4
votes
6answers
154 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 $\le$?
12
votes
5answers
392 views
Notation Question: What does $\vdash$ mean in logic?
In a "math structures" class at the community college I'm attending (uses the book Discrete Math by Epp, and is basically a discrete math "light" edition), we've been covering some basic logic.
I've ...
1
vote
1answer
119 views
Logic about systems?
In Godel's Incompleteness Theorem, his theorem is about a system of logic. Where can I find more about this study, especially the notation?
EDIT
I mean logic about systems in general. I worded the ...
1
vote
2answers
95 views
Logic Notation question (specifically about logical equivalences)
$$\text{Equivalence}$$
$p \land T \equiv p\tag{Identity law 1}$
$$p\lor F \equiv p\tag {Identity law 2}$$
$$p\lor T \equiv T\tag{Domination law 1}$$
$$p\land F \equiv F\tag{Domination law 2}$$
...
1
vote
1answer
57 views
ZFC Union axiom
Maybe I'm just need to buff up on my logic notation, but I don't fully understand the following:
$$\exists y\forall z \left(\exists w(z\in w\wedge w\in x)\implies z\in y\right)$$
How should I ...
6
votes
3answers
344 views
Difference between $\implies$ and $\;\therefore\;\;$?
I've seen both symbols used to mean "therefore" or logical implication. It seems like $\therefore$ is more frequently used when reaching the conclusion of an argument, while $\implies$ is for ...
0
votes
2answers
76 views
$\leq$ operation and logical error
We define $x \leq y$ operation as '$x<y \ \ or \ \ x=y$'. But this is false when both $x<y$ and $x=y$ as both can not be true at the same time. But I read text books that use this expression in ...
1
vote
3answers
128 views
Simple predicate calculus
All I want to do is write the following things into notation.
My trouble is in inserting the correct order and understand where to use the same variables.
The predicates are Students, Answers, and ...
0
votes
0answers
152 views
Questions regarding set theory notation
I’ve got some questions regarding set theory. I am struggling to find the right notation in order to express a number of conditions. I have a set named A that ...
5
votes
2answers
120 views
Symbols and terminology for distinguishing derivability from sequents
First some definitions to make it clear what I'm talking about:
A deductive system is a set $J$ of judgments together with a set $R$ of inference rules each of the form
$$ j_0 \leftarrow j_1, ...
4
votes
1answer
75 views
“Sum” over logical and?
Given a continuous sequence of integers $(a, a+1, a+2, \dots, b)$ I want to write:
$P_a \wedge P_{a+1} \wedge P_{a+2} \wedge \dots \wedge P_b$
Where $P_i$ is some logical statement parametrized by ...
2
votes
1answer
179 views
Exponentiation and Set of Functions Notation.
Given arbitrary sets $A$ and $B$, the notation $A^{B}$ is mostly clear from context to mean $A^{B} = \{f : f : B \rightarrow A\}$.
However, when these sets are ordinal or cardinals, especially ...
3
votes
1answer
191 views
Is there notation for “some two of the three statements are true”?
There are three propositions A, B, C and another condition "some two of these propositions are true and the third one is false", or, in other words, "exactly 2 of 3 propositions are true". Using truth ...
4
votes
1answer
161 views
What is the $\tau$ symbol in the Bourbaki text?
I'm reading the first book by Bourbaki (set theory) and he introduces this logical symbol $\tau$ to later define the quantifiers with it. It is such that if $A$ is an assembly possibly contianing $x$ ...
1
vote
1answer
66 views
How can the set of the deducible WFF $\{\phi\mid\Sigma\vdash_K\phi\}$ be denoted?
I am relatively new to the formalism of Mathematical Logic, and don't know how denote the set of the wff logically deducible by a given set of premises $\Sigma$ in a predicate calculus $K.$ I have ...
3
votes
1answer
127 views
mathematical notation for a logical statement
The proof of the statement below is a homework question, however I did not tag this question as such since I don't need the actual proof: I have already proved the statement wring and don't need a ...
4
votes
1answer
76 views
Notation in Sacks' 'Higher Recursion Theory'
I'm having trouble with the notation in Sacks' Higher Recursion Theory. I've asked specific questions from page 4. Instead of reading my question in detail and trying to understand my confusion (which ...
0
votes
1answer
485 views
Boolean algebra operation precedence?
In my discrete mathematics class we wrote down the truth table for some Boolean functions and in that table they go in the following order:
¬, ∧, ∨, →, ~, ⊕, |, ↓
So, I assumed that this is the ...
1
vote
1answer
370 views
Right and Left arrow notation in proof.
I'm studying vector spaces and I'm reading a proof where the authour uses the symbols
$$(\Rightarrow)$$
and
$$(\Leftarrow)$$
when proving a theorem. He doesn't use them in context, but rather ...
2
votes
1answer
182 views
Predicate logic notation: where to put the parentheses, etc.
My math professor tends to write $\exists x\in\mathbf{X} \ni P(x)$. Is this a correct use of the such that symbol $\ni$? If not, what is the use of that symbol? Isn't it better to write $\exists ...
3
votes
2answers
329 views
How to write “let” in symbolic logic
How do I write let in symbolic logic? For example, if I am in the middle of a proof and there is a variable which I can assign to an arbitrary value, what would I write? My best guess is:
$$ x := a ...
1
vote
1answer
357 views
Explicit description in set-builder notation of an arbitrary open set of the product topology
Short version: Is it possible to explicitly describe the open sets of the product topology (of arbitrary topological spaces) via set-builder notation? (Or differently formulated: What do to if set set ...
6
votes
3answers
526 views
What's the difference between “$\to$” (implication) and “$\vdash$” (therefore)?
In Wikipedia, here in the last axiom of the Natural deduction system, it says "From [accepting $p$ allows a proof of $q$], infer $(p \to q)$." Isn't that a tautology? In the big table "Basic and ...
0
votes
1answer
88 views
Clarification regarding white space in rules of inference
What is the meaning of the white space in the following notation (or what is meant by the rules themselves)?
$$\frac{ }{x\ \ \ x}$$
or
$$\frac{y\ \ \ t_i}{y\ \ \ ft_1...t_n}$$
Where the white ...
2
votes
3answers
136 views
Is there a compact notation for indicating the reasons for an implication?
Is there a compact, commonly used notation for indicating the reasons for an implication? For example, suppose I have previously established or been given $P$, and can use it to show that $B$ follows ...
0
votes
2answers
210 views
distributive law in polish notation
On page 18 "Logic as Algebra" Halmos&Givant wrote the distributive law in Polish notation as
$$ = \times a + bc + \times ab \times ac $$
I fail to see anything remarkable here, is there a ...
3
votes
2answers
188 views
Different standards for writing down expressions in a formal way
What are standard ways to write mathematical expressions in a (semi)formal way ?
In different posts of mine concerning similar question I have encountered for a generic expression of the type "for all ...
2
votes
2answers
245 views
Catalog of named tautologies
Some propositional taulogies have names, for example, Modus Ponens, Modus Tollens, Contrapositon, ...
Is there a catalog of all named propositional taulogies?
In particular, does the following ...
1
vote
1answer
218 views
What is the origin of the prefix logic notation used in WFF 'N PROOF?
The classic "modern logic" game of WFF 'N PROOF uses a set of symbols to represent logical relations that I've seen used nowhere else: $C$ for then; $A$ for or; $K$ for and; $E$ for if and only if; ...
1
vote
1answer
246 views
What are notations to express uniqueness in formulae and diagrams?
I am familiar with the notation $\exists\,!$ to express both existence and uniqueness. For example $$\;\;\exists\,!x\!:\!P\,(x)\;\;$$ means "there exists a unique $x$ such that $P\,(x)$ holds", for ...
2
votes
3answers
2k views
How to represent XOR of two decimal Numbers with Arithmetic Operators
Is there any way to represent XOR of two decimal Numbers using Arithmetic Operators (+,-,*,/,%).
-7
votes
1answer
399 views
How Can One Legitimately Use the Rules of Substitution and Repalcement in a Non-Mechanical Fashion?
Charles Stewart said I might ask the following as a question here.
If an author doesn't use parentheses for logical expressions like x->y, say in propositional calculus, I don't see how the ...
22
votes
4answers
987 views
Who invented $\vee$ and $\wedge$, $\forall$ and $\exists$?
I can rather easily imagine that some mathematician/logician had the idea to symbolize "it E xists" by $\exists$ - a reversed E - and after that some other (imitative) mathematician/logician had the ...
4
votes
2answers
355 views
How do you read this logical statement aloud, and how do you notate it in symbols?
Harry Potter and the Methods of Rationality is a wonderful work of fan fiction by AI researcher and decision theorist Eliezer Yudkowsky. In Chapter 39, this exchange takes place between Dumbledore and ...
1
vote
3answers
188 views
How to manage without specifying a particular algebraic system?
My long standing question:
How to eliminate writing $\cap^L$ instead of plain $\cap$ when we deal with more than one lattice? (and likewise with other (finite and infinite) structures) It is ...
7
votes
5answers
438 views
'is odd' / 'is even' notation
I would like to write down that $x$ is $true$ if $n$ is odd and $false$ if $n$ is even.
So far I made this up:
$x = ( n - 2⌊\frac{n}{2}⌋ = 1)$
However, I was wondering whether this can be expressed ...
9
votes
2answers
362 views
Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials
[disclaimer: I've studied a lot of logic but never been good at analysis, so that's the angle I'm coming from below]
in my attempt to find a precise version of the 'definitions' usually given when ...
3
votes
2answers
177 views
n out of m theorems (some imply the rest)
Is there symbolism (or even a name) for groups of statements in which any fixed-number of them imply all the rest?
For example, in linear algebra, a basis is sometimes defined as a set of ...
10
votes
4answers
2k views
What is the meaning of this symbol ($\models$)?
What's the meaning of this symbol in mathematical notation? :
$\models$
