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Translating:

All humans have a father

Let $Hx$ be x is human and $Fx$ be x has a father. Then should it be $$\forall x(HxFx)$$? or $$\forall x(Hx\rightarrow Fx)$$?

Xetrez
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    There's more than one way to do it, depending on how the "father" relation is to be defined. My guess is that the intention is for there to be a binary predicate $F(x,y)$ defined as "$x$ is the father of $y$". Any idea how you would write it that way? – Ian Feb 18 '21 at 21:39
  • hmm, I mean I thought the second one only worked for "if-then" sentences so I'm guessing the first one is the most correct one, can you confirm? Also what you mean by how the father relation is to be defined? – Xetrez Feb 18 '21 at 21:41
  • oh then if we assume it's the binary predicate, could it maybe be: $$\forall x(F(Hx,x))$$? – Xetrez Feb 18 '21 at 21:46
  • I don't really know what the first one was even intended to mean; what is the connection between the propositions $Hx$ and $Fx$? Is it "and"? Same for your last one in the comment...what person is $Hx$? The second one is alright; it expands to "For all $x$, if $x$ is human then $x$ has a father", which is semantically what you were asked to express. But I doubt this is what they wanted you to say. – Ian Feb 18 '21 at 21:49
  • hmm, yeah the first one doesn't really make sense, noted. Also I don't know how to make that one you asked me, maybe $\forall x(F(H,x))$? or should I maybe try doing it with another variable maybe $y$ to express this with your binary predicate? – Xetrez Feb 18 '21 at 22:00
  • You should be careful about the types of objects. If $H$ is a unary predicate (i.e. a predicate that takes one argument) then $F(H,x)$ generally makes no sense (it would make $F$ a higher order predicate which is a thing but is irrelevant relevant to your interests here). – Ian Feb 18 '21 at 22:01
  • true. maybe $\forall x\forall y(Hy\rightarrow F(y,x))$? Other than this I don't know what to do – Xetrez Feb 18 '21 at 22:07
  • Well, part of this is specifying the domain. Is something in the domain automatically a human, or are we also talking about non-human things? If it is automatically a human then you don't need any $H(x) \Rightarrow \dots$. How about this: can you tell me why, even if the domain is just humans, that $\forall x , \forall y , F(x,y)$ is wrong? – Ian Feb 18 '21 at 22:21
  • i'm clueless here, also when you refer to domain do you mean universal set? or by domain you mean the $\forall x\forall y$ part? – Xetrez Feb 18 '21 at 22:27
  • In predicate logic there's no "set", there's just a domain, also called a universe of discourse. It's the thing that the quantifiers range over: when you say $\forall x , P(x)$, which things are you saying satisfy $P$, exactly? If you're talking about humans only then no implication for human-ness is required. If not, then you need "For all $x$, if $x$ is a human, then ..." – Ian Feb 18 '21 at 22:41

1 Answers1

2

The second formula is ok.

The first one isn’t. It isn’t even a well formed first order logic formula.

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