32
votes
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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 ...
30
votes
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 $(\neg A)\vee B$.
- 42.4k
24
votes
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How to convert numerical claims to first order logic?
Here are some possible ways to make these kinds of numerical claims in general:
'At least n' (Method 1)
"There is at least 1 P" : $\exists x P(x)$
"There are at least 2 P's" : $\exists x \exists y (...
- 102k
18
votes
Accepted
The meaning of an implication with the existential quantifier
Example c was written: $ \exists x (C(x) \to F(x))$
The answer to that example was given as "Someone is a comedian and that means they are funny"
That is an incorrect translation. Imagine a ...
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13
votes
Difference between "for any" and "for all"?
"Any" is ambiguous and it depends on the context. It can refer to "there exists", "for all", or to a third case which I will talk about in the end.
https://en....
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13
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Why is "A only if B" equivalent to "(not A) or B"?
What you have encountered is a variation on vacuous truth. If the premise is false, then the statement is true. For instance, I can say "If I'm from Mars, then it will rain candy tomorrow", and that ...
- 202k
13
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Anyone who knows anything is envied by someone
"Anything" requires care in quantifying, and your confusion about why anything should mean everything, that's good intuition. Because anything doesn't always mean everything.
What we want to say is ...
- 211k
13
votes
Is "William only eats icecream when the sun is shining" a biimplication?
Indeed you are right.
The sun might be shining without William eating an icecream.
12
votes
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Is "Alice loves candies" actually necessary for "Alice loves all sweet foods"?
Your mistake is that for $A$ to be a necessary condition for $B$ it must hold that $\neg A\Rightarrow\neg B$ (if the necessary condition is not met, then the condition for which it is necessary is ...
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11
votes
Does "either" make an exclusive or?
In everyday speech, "or" is usually exclusive even without "either." In mathematics or logic though "or" is inclusive unless explicitly specified otherwise, even with "either."
This is not a ...
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10
votes
How can I express the "uniqueness quantifier" without "$\exists!$"?
An efficient way to express $\exists ! x \ P(x)$ is:
$$\exists x \forall y (P(y) \leftrightarrow y =x)$$
So, if we use $D(x)$ for '$x$ is a dog', and $F(x,y)$ for '$x$ has $y$ as a favorite toy', ...
- 102k
10
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Which one is correct? Don’t we do as necessary as rigorous math?
Assuming we interpret division to mean the usual notion of division in the real numbers, you're correct that the statement $(\exists h \in \mathbb{R})(1/h = 5)$ doesn't quite make sense, for the ...
- 5,154
10
votes
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Learning to translate natural-language phrases to formal logic
"fix arbitrary"
for each
Fix an arbitrary $x$ (or: arbitrarily fix $x$); then $P(x)$ is true.
In other words: $P(a)$ is true; $P(b)$ is true; $P(c)$ is true; etc. $$\forall x\;P(x).$$
...
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10
votes
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Difference between “for some $k$” and “for some arbitrary $k$”
In your first example, I would just write “where $k$ is an integer”. But “some” is okay. The point is, since $n$ is already known, $k$ is completely determined. Writing “where $k$ is some arbitrary ...
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10
votes
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Is "William only eats icecream when the sun is shining" a biimplication?
@b00n Het’s answer is short and sweet and correct; but I’ll expand on it a bit, to back it up and consider the complications raised in @ryang’s answer.
The key issue is that only in natural English is ...
- 9,841
9
votes
The meaning of an implication with the existential quantifier
"Someone is a comedian and that means they are funny" actually means $$\exists x\, C(x)\land \forall x\,\big(C(x)\to F(x)\big),$$ which is a logically stronger assertion than $$∃x\,\big(C(x) ...
- 40.7k
8
votes
Why is "A only if B" equivalent to "(not A) or B"?
Here is a table showing all possible value-combinations for A and B in the first two columns. You can see that columns 3 and 5 are the same, therefore both predicates must be equivalent.
...
- 181
8
votes
Accepted
Writing predicates - is the quantifier necessary?
No, we cannot omit the existential quantifier.
How we read the formula:
$x=2m$ ?
As: "every $x$ is even" ? Or as: "some $x$ is even" ? Or as: "every $x$ is the double of every $m$" ? Or as: "some ...
- 95.3k
7
votes
Accepted
Why is my translation of $\exists{x}\,(C(x) \rightarrow F(x))$ into an English sentence wrong?
Consider a world with no people. Then there is no comedian, so your translation is correct. However, the statement is false because there doesn't exist anyone, in particular no one who satisfies $C(x) ...
- 6,843
7
votes
Accepted
Expressing "does not imply''
The formalization of a sentence in ordinary discourse claiming that "$A$ implies $B$" or "$A$ does not imply $B$" is outside the languages of propositional or first-order or higher-...
- 17.4k
7
votes
Accepted
How do we tell whether ‘some’ means Exactly one or At least one?
How do we know whether 'some' means at least one or just one?
“For some $k,\, P(k)$ is true”, “there exists at least one $k$ such that $P(k)$ is true” and “there exists one $k$ such that $P(k)$ is ...
- 40.7k
6
votes
Difference between "for any" and "for all"?
Here's a comparison of the various translations of $$∀x{∈}F\; P(x).$$
“For all elements $x$ in $F,\,P(x)$ holds” sometimes sounds like the property $P$ might belong to $F$ as a whole rather than to ...
- 40.7k
6
votes
Accepted
How to write "at least n" and "exactly n" in logical notation?
$P$ holds for at least three elements:
$$\exists x\exists y\exists z\ [x\ne y\land x\ne z\land y\ne z\land P(x)\land P(y)\land P(z)]$$
$P$ holds for at most three elements:
$$\forall w\forall x\forall ...
- 79.6k
6
votes
Accepted
What does "only if" mean exactly?
"A if B". When B is true then A must be true (sufficient).
"A only if B". When B is false then A must be false (necessary).
"A if and only if B". When B is true A must be true, When B is false ...
- 4,647
6
votes
Accepted
When to use which quantifier with predicate logic?
The two answers are equivalent.
"$\lnot \forall$" is the same as "$\exists \lnot$".
If not all cats are black, there must be some cat that is not black.
Thus, we have that
$$ ¬∀x \ (Gx \to Lxx) \...
- 95.3k
6
votes
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 ...
- 202k
6
votes
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 $q$, ...
- 102k
6
votes
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 ...
- 251k
6
votes
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) \...
- 17.4k
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