# Tagged Questions

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### What does “calculus” mean?

"calculus" and "formal system" From http://en.wikipedia.org/wiki/Propositional_calculus#Terminology a calculus is a formal system that consists of a set of syntactic expressions ...
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### Is there a phrase to describe those objects of $\mathbf{C}$ that can be expressed as quotients of the algebra freely generated by $X$?

Let $\mathbf{C}$ denote the category of models of an algebraic theory in $\mathbf{Set}.$ Now suppose $X$ is an object of $\mathbf{Set}$. Is there a traditional phrase used to describe those objects of ...
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### Does every mathematical principle have a proof?

My question actually narrows down to the meaning of mathematical principle. While I'm looking for some principles, they usually have their proofs, so I thought "principle" has the same meaning as ...
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### Must different constant symbols denote different objects?

In first-order theory with equality and >=2 constant symbols (let's denote two of them by c and d), does it always happen that $\neg(c=d)$ is derivable (possibly stated as an axiom)? In other words, ...
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### What do we call the disjuncts of the conclusion of an argument?

If we have an argument with a single premise of the form $A \wedge B$, then we can refer to $A$ and $B$ collectively as "premises" of the argument without causing any confusion. However if we have an ...
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### Is an anti-symmetric and asymmetric relation the same? Are irreflexive and anti reflexive the same?

I don't understand the difference between an anti symmetric and asymmetric relation. From my understanding, it is asymmetric if there is not any element where: if (x,y) (y,x). But what if you have ...
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### “Logically equivalent formulae express the same _______.” <- What word do logicians use for the blank?

Meaning denotes the truth conditions of a sentence: what would have to be the case for the interpreted formula to be true. Nevertheless, without an interpretation, two logically equivalent formulae ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $i$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $Q$ ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### $\omega$-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 ...
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### 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 ...
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### 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 ...
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### “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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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?
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### 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 ...
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### 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.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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., ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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, ...