Translate to First Order Logic

I have this two sentences:

1. All men love the woman to whom they are married.
2. No woman loves a man who does not respect all women.

and in the exercise they are translated to First Order Logic as follows:

$\forall{x}\text{ }male(x) \land \exists y [female(y) \land married(x, y)\to loves(x, y)]$ $\lnot\exists{x}\text{ }female(x) \land \forall y\forall z [female(z) \land \lnot{respect(y, z)}\land loves(x, y)]$

but I would translate them in this way:

$\forall{x}\text{ }male(x) \to \exists y [(female(y) \land married(x, y))\to loves(x, y)]$ $\forall{x}\text{ }female(x) \to \forall y\forall z [( female(z) \to(\lnot{respect(y, z)}\to\lnot loves(x, y))]$

Where am I wrong?

• $\forall x \ \text {male} (x)$ means that every human is a male... Thus, $\forall x \ \text {male} (x) \land \ldots$ is TRUE only if all humans are males, and this cannot be true. We need parentheses. Commented Jan 31, 2017 at 15:53
• We must have : $∀x [ \ (\text {male}(x) ∧ ∃y( \text {female}(y) ∧ \text {married}(x,y)) \ ) \to \text {loves}(x,y)]$. Commented Jan 31, 2017 at 15:58
• Try to rewrite "All men love the woman to whom they are married." as : "For every man, [if they are married with a women, then they love that women]." Commented Jan 31, 2017 at 16:01
• I'm confused with the wording of the question, are the first two sentances solutions in your textbook? Commented Jan 31, 2017 at 16:04
• @JeremyBaziw I found an exercise online and the first two senteces are the solutions Commented Jan 31, 2017 at 16:08

Well, the provided answers are certainly not correct.

I see that for both of the you added some parentheses to help disambiguate, which is good. Some books use a 'priority' system to say that something like $P \land Q \to R$ will always mean $(P \land Q) \to R$ (so we can't really 'fault' the book on that one), but I think it is good idea to use explicit parentheses the way you do. Good!

Also, for both answers, your intuition that on several occasions we need to use a $\to$ rather than a $\land$ is correct .... for example, the first provided answer effectively saying that everyone is male! Again: good for you to realize the book's mistake!

But: as Mauro points out, we really need parentheses to indicate the cope of thew quantifier, otherwise the $x$ ends up being free. So, at the very least you want:

$\forall{x}\text{ }\big( male(x) \to \exists y [(female(y) \land married(x, y))\to loves(x, y)]\big)$

OK, but that still isn't right! Indeed, there is a major problem with your translation as well!

To see this, suppose that for $y$ we take something from the domain that is not female (presumably $x$ itself!). Then: it is false that $female(y)$, and hence false that $female(y) \land married(x, y)$, and hence true that $(female(y) \land married(x, y))\to loves(x, y)$.

Hence, of course the whole $\exists y [(female(y) \land married(x, y))\to loves(x, y)]$ is true: just pick something that is not female! ... but that of course says nothing of interest about men loving the woman they are married to! Thus we say that $\exists y [(female(y) \land married(x, y))\to loves(x, y)]$ is 'vacuously true'. ... and it's clearly not what we want.

$\forall{x}\text{ }\big( male(x) \to \forall y [(female(y) \land married(x, y))\to loves(x, y)]\big)$

Well, now at least all men will love whichever woman they are married to ... so that's a whole lot closer to what you want ... and probably acceptable as an answer. ... However ... the statement does not rule out that the man is married to more than one woman. So, to capture that the man loves the woman they are married to (if they are married to a woman at all), we could do this:

$\forall{x}\text{ }\big( male(x) \to \forall y \forall z[(female(y) \land female(z) \land married(x, y) \land married(x, z))\to (y=z \land loves(x, y))]\big)$

For the second one, again we need parentheses for the scope of $x$:

$\forall{x}\text{ }\big( female(x) \to \forall y\forall z [(female(z) \to\lnot{respect(y, z)})\to \neg loves(x, y)]\big)$

But we have a more serious problem ... where is the reference to $y$ being male?!

OK, so at the very least we need something like:

$\forall{x}\text{ }\big ( female(x) \to \forall y\forall z [(male(y) \land (female(z) \to\lnot{respect(y, z))})\to \neg loves(x, y)]\big)$

But this isn't right either! This statement claims something about any $y$ and $z$. OK, so pick for $z$ something that is not female. Then as with the previous sentence, $female(z) \to\lnot respect(y, z)$ becomes vacuously true, and therefore, it would follow that $\neg loves(x, y)$ ... even if the man $y$ does respect all women!

OK, so we need something more like:

$\forall{x}\text{ }\big ( female(x) \to \forall y [(male(y) \land \forall z (female(z) \to\lnot{respect(y, z))})\to \neg loves(x, y)]\big)$

But there is another problem yet! This statement is saying that $y$ is a man who does not respect any women at all .... and I interpret the English sentence as saying that it is about men who don't respect all women (that is, as soon as there is at least one woman that this man does not respect, then $x$ will not love that man). So, we need to change this to:

$\forall{x}\text{ }\big ( female(x) \to \forall y [(male(y) \land \neg \forall z(female(z) \to {respect(y, z))})\to \neg loves(x, y)]\big)$

Another option is what you do in your second attempt:

$\forall x female (x)\to\forall y \forall z[male(y)\to(female(z)\to(\lnot respect(y,z)\to \lnot love(x,z)))]$

Now again, we need to add parentheses for the scope of $x$:

$\forall x \big ( female (x)\to\forall y \forall z[male(y)\to(female(z)\to(\lnot respect(y,z)\to \lnot love(x,z)))]\big)$

And another thing we can do is to make this into the equivalent:

$\forall x \big ( female (x)\to\forall y \forall z[(male(y)\land female(z))\to(\lnot respect(y,z)\to \lnot love(x,z)))]\big)$

.. which looks very much what you had originally, but the change in parentheses makes all the difference!

Finally, to show that my answer and this second answer are equivalent:

$\forall{x}\text{ }\big ( female(x) \to \forall y [(male(y) \land \neg \forall z(female(z) \to {respect(y, z))})\to \neg loves(x, y)]\big) \Leftrightarrow$

$\forall{x}\text{ }\big ( female(x) \to \forall y [male(y) \to (\neg \forall z(female(z) \to {respect(y, z))}\to \neg loves(x, y))]\big) \Leftrightarrow$

$\forall{x}\text{ }\big ( female(x) \to \forall y [male(y) \to (loves(x, y) \to \forall z(female(z) \to {respect(y, z))})]\big) \Leftrightarrow$

$\forall{x}\text{ }\big ( female(x) \to \forall y [male(y) \to \forall z (loves(x, y) \to (female(z) \to {respect(y, z))})]\big) \Leftrightarrow$

$\forall{x}\text{ }\big ( female(x) \to \forall y \forall z[male(y) \to (loves(x, y) \to (female(z) \to {respect(y, z))})]\big) \Leftrightarrow$

$\forall{x}\text{ }\big ( female(x) \to \forall y \forall z[male(y) \to ((loves(x, y) \land female(z)) \to {respect(y, z)})]\big) \Leftrightarrow$

$\forall{x}\text{ }\big ( female(x) \to \forall y \forall z[male(y) \to (( female(z) \land loves(x, y)) \to {respect(y, z)})]\big) \Leftrightarrow$

$\forall{x}\text{ }\big ( female(x) \to \forall y \forall z[male(y) \to ( female(z) \to (loves(x, y) \to {respect(y, z)}))]\big) \Leftrightarrow$

$\forall{x}\text{ }\big ( female(x) \to \forall y \forall z[male(y) \to ( female(z) \to (\neg {respect(y, z)} \to \neg loves(x, y) ))]\big)$

• Sorry @Bram28, if I annoy you, just another question: the following sentence, $\forall x male(x) \to \exists y[female(y) \land married(x,y) \land loves(x,y)]$ , could be a right version for the first sentence? Commented Jan 31, 2017 at 22:38
• @B.junior You're fine! Eh, your new sentence says that every man is married to a woman that they love. So again, that would rule out any unmarried men (which I don't think is the intent of the English sentence, but it would not rule out any men married to multiple women ... In fact, this sentence opens the possibility for a man to be married to multiple women, only some of which the man loves. Again, I really don't think that is the intent of the original English sentence. Commented Jan 31, 2017 at 23:01
• Understood, the existential quantifier stands for 'more than woman' and not for 'only one woman'. Let's suppose a man could marry more than one woman and we want translate the sentence 'A man loves some women to whom they are married': how could we translate this sentence? Commented Jan 31, 2017 at 23:22
• @B.junior Right ... Technically the existential stands for 'at least one'. And your last sentence comes pretty close to what you want ... Though again it would force every man to be married to at least one woman ... Is that what you want, or would you be lookinf for a sentence that allows some men not to be married to a woman at all? Commented Jan 31, 2017 at 23:25
• No I want to understand how the existential quantifier works if we consider some women to whom the man are married because the implication can't be used as we said before, right? Commented Jan 31, 2017 at 23:36

The phrasing "the woman to whom they are married" seems to assume as a given that every man is married to one unique woman. Expressing that most naturally in first-order logic calls for a function letter rather than a relation, so I would write

$$\forall m.\,\text{man}(m) \to \text{loves}(m,\text{wife}(m))$$

• That's a possible reading, for sure ... but I interpret it as 'if they are married to a woman at all, then they are married to just one woman'. We certainly can use the English sentence in this way in real life without implying that every man is married. Commented Jan 31, 2017 at 18:07

Thank you so much @Bram28. Let's start with the first sentence: I have understood what you said, just one question, Is there a way to realize the sentence using the existential quantifier instead of the universal quantifier?

for the seconde sentence, adding the man reference I've done as follows

$\forall x female (x)\to\forall y \forall z[male(y)\to(female(z)\to(\lnot respect(y,z)\to \lnot love(x,y)))]$

Is that right? Thank you :)

• You can of course replace any $\forall$ into $\neg \exists \neg$ ... but I suppose that's not what you mean. Also, if I say that for any man there does exists a woman that the man is married to, then every man is married, so there can no longer be any unmarried men!) ... so that's not what's meant by the sentence either. So, I would say that no, there is no straightforward way to use an existential here for the first one. Commented Jan 31, 2017 at 18:03
• Your second sentence works, yes! ... assuming you do add those parentheses for the scope of $x$. Commented Jan 31, 2017 at 18:10
• Thank you @Bram28, and I suppose that the second version of the sentence that I provided is equal to your version, for the property $(A \land B) \to C \equiv A \to (B \to C)$ Commented Jan 31, 2017 at 18:13
• Yes! I had just added that to my answer: very good! Commented Jan 31, 2017 at 18:15