# Tag Info

Accepted

### Does "either" make an exclusive or?

No, you cannot depend on that. If it were that simple, we wouldn't need clunky phrases like "exclusive or" to make clear when an "or" is exclusive. Linguistically, "either" is simply a marker that ...
Accepted

### Why is "A only if B" equivalent to "(not A) or B"?

"A only if B" means that you can't have A without B, i.e. $\neg(A\wedge(\neg B))$, which simplifies (via de Morgan) to $(\neg A)\vee B$.
Accepted

Accepted

### Expressing "does not imply''

The formalization of a sentence in ordinary discourse claiming that "$A$ implies $B$" or "$A$ does not imply $B$" is outside the languages of propositional or first-order or higher-...
Accepted

### How do we tell whether ‘some’ means Exactly one or At least one?

How do we know whether 'some' means at least one or just one? “For some $k,\, P(k)$ is true”, “there exists at least one $k$ such that $P(k)$ is true” and “there exists one $k$ such that $P(k)$ is ...

### Difference between "for any" and "for all"?

Here's a comparison of the various translations of $$∀x{∈}F\; P(x).$$ “For all elements $x$ in $F,\,P(x)$ holds” sometimes sounds like the property $P$ might belong to $F$ as a whole rather than to ...
Accepted

### Confused between if-then (→) and iff (↔)

Just because the monkey is home, that doesn't automatically mean that Tarzan is happy. That's how "only if" is interpreted in conventional mathematical English. Tarzan can't be happy if the monkey is ...
Accepted

### Writing a statement using logical connectives and determining whether it is a logical implication.

No, the second claim should be $r \to p$: $f$ being integrable $p$ is a necessary condition, rather than a sufficient condition. And when something $p$ is a necessary condition for something else $q$, ...
Accepted

### "Many" quantifier in first order logic

First-order logic does not have a notion of "many" - this, like "most," "almost all," etc. is an example of a generalized quantifier, and handling them takes us to extensions of first-order logic. We ...
Accepted

### Express the definition of even using universal and existential quantifiers

A correct formalization of the phrase "an integer is even iff it equals double some other integer" is the following: \begin{align} \forall x \in \mathbb{Z} \, (\exists y \in \mathbb{Z} \, (x = 2y) \...
### What is the difference between $\neg\exists x$ and $\neg\forall x$ ? Is $\neg\exists x$ ever used?
No. $\neg \forall x$ means 'not every $x$'. So, for example, not every number is even. But that does not mean that there are no even numbers at all. Other examples: There are no unicorns .. so we ...
Let $CanBeStabbed(x)$ mean that $x$ can be stabbed. Let $WillKillForSure(x,y)$ mean that $x$ will kill $y$ for sure. Of course you can use other, simpler names. Then "Someone will kill $y$ for sure"...