Translating English-language sentences into symbolic logic

Question

Here are some statements that I had to translate into symbolic form, then negate and express the negation as a positive statement. In each case the assumed domain is given in parentheses.

a. Everyone loves somebody. (All people)

Let:
$L(x,\;y)$ mean $x$ loves $y$.

Symbolic form: $\forall x \exists y [L(x,\;y)]$

Negation: $\exists x \forall y\;[\lnot L(x,\;y)]$

Negated statement in english: Someone doesn't love everyone.

b. Nobody loves everybody. (All people)

Let:
$L(x,\;y)$ mean $x$ loves $y$.

Symbolic form: $\lnot\exists x \forall y [L(x,\;y)]$

Negation: $\exists x \exists y [\lnot L(x,\;y)]$

Negated statement in english:
"Someone doesn't love someone"

or

"At least one person doesn't love someone"

c. If a man comes, all the women will leave. (All people)

Let:
$M(x)$ mean $x$ is a man.
$C(x)$ mean $x$ comes.
$W(x)$ mean $x$ is a woman.
$L(x)$ mean $x$ is a leaves.

Symbolic form: $\exists x {\big[}M(x) \land C(x){\big]} \Rightarrow \forall x{\big[}W(x) \Rightarrow L(x){\big]}$

Negation:
$\exists x {\big[}M(x) \land C(x){\big]} \nRightarrow \forall x{\big[}W(x) \Rightarrow L(x){\big]}$
$\exists x {\big[}M(x) \land C(x){\big]} \land \exists x{\big[}W(x) \land \lnot L(x){\big]}$

Negated statement in english: At least one man will come at least one woman will stay.

d. Not all precious stones are beautiful. (All stones)

Let:
$P(x)$ mean $x$ is a precious stone.
$B(x)$ mean $x$ is beautiful.

Symbolic form: $\exists x\;[P(x) \land \lnot B(x)]$

Negation
$\exists x\;[P(x) \land \lnot B(x)]$ is equivalent to $\exists x\;[P(x) \nRightarrow B(x)]$ where $(a \land \lnot b) \Leftrightarrow (a \nRightarrow b)$
So the negated version is: $\forall x\;[P(x) \Rightarrow B(x)]$

Negated statement in english: All precious stones are beautiful.

e. Nobody loves me. (All people)

Let:
$L(x,\;y)$ mean $x$ loves $y$.
$m$ mean $me$.

Symbolic form: $\lnot \exists x [L(x,\;m)]$

Negation: $\exists x [\lnot L(x,\;m)]$

Negated statement in english: Somebody doesn't love me.

f. At least one American snake is poisonous. (All snakes)

Let:
$A(x)$ mean $x$ is american.
$P(x)$ mean $x$ is poisonous.

Symbolic form: $\exists x [A(x) \land P(x)]$

Negation:
$\forall x\;[\lnot A(x) \lor\lnot P(x)]$
$\forall x\;[A(x) \Rightarrow\lnot P(x)]$; where $(\lnot a \lor b) \Leftrightarrow (a \Rightarrow b)$.

Negated statement in english: No american snake is poisonous.

g. At least one American snake is poisonous. (All animals)

Let:
$S(x)$ mean $x$ is a snake.
$A(x)$ mean $x$ is american.
$P(x)$ mean $x$ is poisonous.

Symbolic form: $\exists x [S(x) \land A(x) \land P(x)]{\big]}$

Negation
$\forall x\;[\lnot S(x) \lor \lnot A(x) \lor \lnot P(x)]$
Grouping together $A(x)$ and $S(x)$:
$\forall x\;{\big[}\lnot [\lnot S(x) \lor \lnot A(x)] \Rightarrow \lnot P(x){\big]}$; where $(\lnot a \lor b) \Leftrightarrow (a \Rightarrow b)$
$\forall x\;{\big[} [ S(x) \land A(x)] \Rightarrow \lnot P(x){\big]}$

Negated statement in english: No american snake is poisonous.

Answer 1

a. Checks out symbolically, but I would recommend the wording "somebody doesn't love anyone."

b. Your negation is a bit suspicious here, you can just cancel out the leading negation. You "distribute" the negation inwards as if you applied DeMorgan's without actually using DeMorgan's. (this of course trickles down to the written negation)

c. Checks out from what I can tell, the wording is a bit awkward but I can't think of a better alternative.

d. Good

e. Again, for the negation you can just get rid of the leading negation.

f. Good

g. Good, except for a minor typo where you put $y$ where an $x$ should be

Overall, I think you've done well but there are a few errors which I think could be avoided if you write out more of your intermediate steps and focus on the logic which moves you from one step to the next. Once you've become more accustomed to this then I think it'll feel a bit more natural and you can skip steps more safely.

Also, I recommend using your written statements to check your answers: starting from the first statement and not considering the symbolic logic, how would you negate the statement if you were speaking? If it disagrees with the answer you end up with, you should double-check your symbolic argument.

Hope this helps!

Answer 2

a. Everyone loves somebody. (All people)

Symbolic form: $\forall x \exists y [L(x,\;y)]$

Negation: $\exists x \forall y\;[\lnot L(x,\;y)]$

Negated statement in english: Someone doesn't love everyone.

Correction: "Someone doesn't love anybody". (Your symbolic negation is correct though.)

If everybody loves Raymond and Raymond loves only himself, then your suggested statement is True while my correction statement is False.

An orthogonal point worth raising: the given sentence is technically ambiguous, because it could have been alternatively mechanically translated as $\exists x \forall y [L(y,\;x)]$ due to the last word being a hanging quantifier.

b. Nobody loves everybody. (All people)

Symbolic form: $\lnot\exists x \forall y [L(x,\;y)]$

Negation: $\exists x \exists y [\lnot L(x,\;y)]$

Negated statement in english: "Someone doesn't love someone" or "At least one person doesn't love someone"

The negation (just drop that negation symbol!) ought to be $$\exists x \forall y [L(x,\;y)]$$ "Somebody loves everybody" instead.

c. If a man comes, all the women will leave. (All people)

Negated statement in english: At least one man will come at least one woman will stay.

No need to change the tense: "A man comes and some woman will stay".

d. Not all precious stones are beautiful. (All stones)

Symbolic form: $\exists x\;[P(x) \land \lnot B(x)]$

Isn't the symbolic form merely $$\lnot \forall x\;[P(x) \Rightarrow B(x)]?$$

Negation $\forall x\;[P(x) \Rightarrow B(x)]$

Negated statement in english: All precious stones are beautiful.

Correct.

e. Nobody loves me. (All people)

Symbolic form: $\lnot \exists x [L(x,\;m)]$

Negation: $\exists x [\lnot L(x,\;m)]$

Negated statement in english: Somebody doesn't love me.

The negation (just drop that negation symbol!) ought to be $$\exists x [L(x,\;m)]$$ "Somebody loves me".

f and g are both correct.

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Addendum

OP (emphasis mine): The problem I was pointing out is the order of precedence that ¬ has. I used it as to negate just the first quantifier, whereas it seems that you take it to negate the whole expression, even if it's not made explicit by means of parentheses.

Sentences (e.g., ‘the rose is red’, ‘the number of marbles is $7$’, ‘$2x-7=0$’) can be negated. Expressions (e.g., ‘red’, ‘the number of marbles’, ‘$2x-7$’) and quantifiers (e.g., $∀y$) can neither be true nor false, so how can their truth value be flipped, in other words, how can they be negated?

Facts (inserting parentheses to demonstrate that order of precedence is irrelevant): \begin{align}(¬∀x)\:P(x)\;&\equiv\;∃x\:¬P(x)\\&\boldsymbol{\not\equiv}\;∃x\:P(x)\;;\\(¬∀x)∃y\:P(x,y)\;&\equiv\;∃x∀y\:¬P(x,y)\\&\boldsymbol{\not\equiv}\;∃x∃y\:P(x,y).\end{align}

After all, the negation of $$\text{not all xylophones are pink}$$ is $$\text{some xylophone is not pink},$$ rather than $$\text{some xylophone is pink}.$$

If this addendum has revised your understanding, then my above responses to Parts b & d, and my first comment below, will now make more sense.