0
votes
2answers
44 views

What subject in mathematics investigates the type of problems that constitute the LSAT “logic games” (example given)?

For my own curiosity, I read part of an LSAT study guide yesterday. The "logic games" section comprised questions like, An advertising executive must schedule the advertising during a particular ...
0
votes
0answers
54 views

What do we call functions that are definable by expressions?

Let $X$ denote a model of an algebraic theory $T$. What do we call the functions $f : X^n \rightarrow X$ that are definable by some expression in the language of $T$? e.g. If $S_3$ is the symmetric ...
2
votes
5answers
79 views

Question about definition of binary relation

Wikipedia says: Set Theory begins with a fundamental binary relation between and object $o$ and a set $A$. If $o$ is a member of $A$, write $o \in A $. I thought that a binary relation is a ...
1
vote
2answers
36 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 ...
10
votes
3answers
166 views

What do we call entities (like $\sum$) that bind variables?

In logic, we refer to entities like $\forall$ and $\exists$ as quantifiers, because they bind variables. However, variable-binding doesn't just occur at quantifiers. For example, the symbol $i$ ...
3
votes
2answers
106 views

Validity vs. Tautology and soundness

I see that valid formula (proposition or statement) is the one that is valid under every interpretation. But this is a tautology. Is there any difference between tautology and valid formula? They also ...
2
votes
1answer
29 views

What do we call functions that return axioms / axiom schemata?

Consider the function $\mathrm{Assoc}$ defined by: $$\mathrm{Assoc}(X,*) = (\forall x,y,z \in X)((x*y)*z=x*(y*z))$$ This is a function that accepts symbols $X$ and $*$ and returns the axiom (a ...
2
votes
2answers
124 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
33 views

Quick logic question about $P\leftrightarrow Q$, terminology

I know that if we have $P\rightarrow Q$, $P$ can be called the antecedent and $Q$ the consequent or conclusion. If we have $P \leftrightarrow Q$, are there names for what we would call $P$ and $Q$ ...
6
votes
1answer
202 views

Name of meta-properties

How are properties like "definability" called (in which formulas are involved): A set $X$ is definable when there is a formula $\phi(x)$ such that $X = \lbrace x : \phi(x)\rbrace$. It is not a ...
1
vote
2answers
104 views

Partial functions - where can I learn more about this (heuristic, informal) system of conventions?

Is there a name for the following (heuristic, informal) system of conventions for dealing with partial functions and undefined expressions? I'd like to know whether it has any undesirable quirks that ...
2
votes
2answers
450 views

What is the statement “If not p then q” called?

Let's say I have a statement: if p then q. The converse would be: if q then p. The inverse would be: if not p then not q. The contraposition would be: if not q then not p. What would you call the ...
2
votes
2answers
109 views

The use of any as opposed to every.

This is a really basic question, but it is one I never really thought about until now. Let $\mathscr{G}$ be a tree. Then every pair of vertices in $\mathscr{G}$ is connected by a unique walk. We ...
1
vote
2answers
90 views

About a sentence in logic theory that I don't understand.

Can somebody explain to me the following terms in logic? I have to read a paper in combinatorics that says this, but I don't understand anything in this sentence, where the author speaks about logic. ...
6
votes
1answer
78 views

name of the unit of adjunction between $-\times C$ and $\cdot^C$

Answers to a earlier question about the categorical interpretation of first-order quantification led me to learn more about adjoints. Now, I understand that a category $\mathscr{C}$ with products has ...
3
votes
3answers
159 views

Are statements like “Every time I've done X, Y has happened” (vacuously) true if I've never done X?

I've recently been wondering about vacuous truths. I know a statement like "I've never been beaten in a race" is true if I've never been in a race, but what I'm wondering is if the following ...
5
votes
2answers
219 views

$\omega$-consistency and related terms

We know that a theory $T$ is $\omega$-inconsistent if there is a formula $\psi$ such that $T$ proves $(\exists x)\psi(x)$, and $T$ also proves $\lnot \psi(n)$ separately for each standard natural ...
2
votes
1answer
86 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
3answers
788 views

Claim vs statement vs proposition

In logic, it seems like the words claim, statement and proposition have the same meaning: A sentence which can be true or false (but not both). I'm not sure if this is correct. Isn't there any ...
9
votes
3answers
1k views

“IFF” (if and only if) vs. “TFAE” (the following are equivalent)

If $P$ and $Q$ are statements, $P \iff Q$ and The following are equivalent: $(\text{i}) \ P$ $(\text{ii}) \ Q$ Is there a difference between the two? I ask because formulations ...
4
votes
4answers
167 views

Models vs. Structures

Why are both the terms 'structure' and 'model' used in mathematical logic / model theory? Are they just holdovers from different subjects or is there a principled reason for having both? For ...
3
votes
2answers
162 views

Meaning of “defined”

What are the precise meanings of terms "defined", "well defined" and "undefined", etc.? We can't define what "defined" means since then we would run into circular definitions. (If definitiveness is ...
3
votes
3answers
103 views

What are 'contexts' actually called?

Consider the following argument by contradiction. \begin{array}{|l} \mbox{We wish to deduce A.} \\ {\begin{array}{|l} \mbox{Suppose not A.} \\ \hline \\ \mbox{Then B. Thus C. Therefore, ...
2
votes
1answer
61 views

What's the 'real' way of saying 'cautiously extends'?

Its well known that ZF is equiconsistent with ZFC. Thus we say 'the Axiom of Choice cautiously extends ZF'. Except we don't, because I just made that up. What's the usual way of saying this sort of ...
2
votes
1answer
225 views

What do we call the negation of logical equivalence?

The statement that '$x$, $y$ and $z$ are equivalent' just means all of $x$, $y$ and $z$ are false, or all of $x$, $y$ and $z$ are true. Now suppose its not the case that $x$, $y$ and $z$ are ...
5
votes
3answers
353 views

What's the difference between 'any', 'all' and 'some'?

There are lots of expressions like, for all x, for any x, for some x, etc. I think 'for some x in R s.t ~' means that there exists at least one point in R s.t ~~. right? However, I can't know the ...
2
votes
1answer
52 views

Elementary theory of an algebraic structures

Could someone elaborate me what the sentence "The elementary theory of finite fields is decidable" means? I'm not sure that for example if I take $x\in \mathbb{F}_4$ and $y\in \mathbb{F}_5$ then can I ...
2
votes
2answers
701 views

Is there a difference between 'inconsistent', 'contrary', and 'contradictory'

Is there a difference between 'inconsistent' 'contrary' and 'contradictory'? As far as I understand, two statements are inconsistent when they can not both be true; two statements are contradictory ...
3
votes
1answer
79 views

Substitution - what's the technical name of the inference rule?

Suppose the following are written down in some context. $$3x^2 < y$$ $$x^2=xy-1$$ Then we may deduce (also within that context) that $$3(xy-1) < y$$ What is the technical name of this ...
5
votes
2answers
121 views

Does this qualify as a statement?

Is this a statement? All positive integers with negative squares are prime. What do we need to qualify as such?
5
votes
1answer
118 views

Inherited topology of logical Stone's spaces.

I'm asking here if the following construction is of any interest. I can not find any reference for that kind of thing, so either the subject is completely trivial, either I just don't have the correct ...
4
votes
0answers
158 views

Difference between elementary submodel and elementary substructure

This is a really "elementary" question, forgive the pun. What is the difference between an elementary submodel and an elementary substructure (in first-order Logic)? Sincere thanks for help.
5
votes
2answers
79 views

What should I call a sentence which must (not) be true, but the provability is still unknown?

For example, let $\phi$ be a sentence in $ZF$ and $ZFC\vdash \neg\phi$. Then, $\phi$ must not be provable in $ZF$, but we still don't know whether $ZF\vdash \neg\phi$. What should i call this sentence ...
9
votes
1answer
161 views

Difference between a Lemma and a Theorem [duplicate]

What essentially is the difference between a lemma and a theorem in mathematics? More specifically, suppose you come across a general result while solving a mathematical problem, what are the ...
2
votes
5answers
112 views

Can any mathematical relation be called an 'operator'?

Mathematics authors agree that $+,-,/,\times$ are basic operators. There are also logical operators like $\text{or, and, xor}$ and the unary negation operator $\neg$. Where there seems to be a ...
4
votes
1answer
164 views

Two forms of Beth's theorem?

The version of Beth's theorem I'm familiar with is that if $\phi$ is a sentence in the language $\Sigma\sqcup \lbrace R\rbrace$ depends only on $\Sigma$ (i.e., ...
1
vote
1answer
123 views

Questions on Basic Terminology in Mathematical Logic

As a beginner, I'm overwhelmed by the usage of terminology , such as theory, model, interpretation, structure et al, which are omnipresent in Mathematical logic. Here's my understanding about them: ...
3
votes
2answers
644 views

What is the difference between an axiom and a postulate?

I here about axioms is set theory and postulates in geometry, but they seem like the same thing. Do the mean the same thing but then are used in different instances or what? Is one word more ...
8
votes
3answers
294 views

Analogy between “One and only one” and “If and only if”

Is there a close analogy between "one and only one", used to mean exactly one (as in "there is one and only one object satisfying that condition" etc.), and "if and only if"? "If" and "Only if" can be ...
2
votes
2answers
1k views

meaning of 'Hypothesis' in simple terms?

could anyone please clarify me the meaning of the term 'hypothesis'? with relation to terms 'reasoning' and 'assumption' ? Many thanks
4
votes
1answer
225 views

Difference between elementary logic and formal logic

In Kelley book on topology, in the appendix on elementary set theory, he says in the second paragraph, that "a working knowledge of elementary logic is assumed, but acquaintance with formal logic is ...
6
votes
2answers
182 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, ...
0
votes
1answer
52 views

Knotted up over “unique”

"Every boy has a unique shirt." Does this mean no two boys share the same shirt, or does it mean no two shirts belong to the same boy? I suppose the former, but then what is the most succinct way ...
1
vote
2answers
165 views

Free variables?

So if I understand correctly, these are examples of free variables: (all occurrences of $x$ are free) $$ x*0 $$ $$ 0+x*0 $$ $$ f: y \mapsto x*0 $$ $$ x*12345*(1-1) $$ $$ x*12345*(5-(10/2)) $$ What is ...
2
votes
1answer
125 views

Terminology for implication of theorems

In Portuguese, the following is considered the accepted terminology for the implication of theorems: $$\text{Theorem:}\\ \text{Hypothesis } \Rightarrow \text{ Thesis}$$ Hypothesis is the antecedent ...
-1
votes
1answer
95 views

Meaning of a Logical Operator

Is it possible to know what those operator mean if they must be involved in this logicical condition? What is all the possible meaning of those two symbol if you don't know the symbol's meaning ...
1
vote
1answer
722 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 ...
3
votes
1answer
132 views

What is a relatively bound variable?

edit: Interestingly, the authors also state at one point that the choice of introduction rule is determined by the structure of the previous goal and the list of introduction rules; but at another ...
2
votes
2answers
323 views

Confused about Wikipedia definition of NP

I've been checking my understanding of the definitions of NP and NP-complete and I am confused by some of the definitions given on Wikipedia; for example, the article about NP-complete describes NP ...
1
vote
3answers
171 views

Are these statements about even numbers called symmetrical statements?

I have these following statements. x is a even number $\Rightarrow$ xy is a even number y is a even number $\Rightarrow$ xy is a even number Can I call them symmetrical statements?