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I am a bit confused when translating categorical sentences in language of predicate logic. I am trying to be as exacted I can be. That means capital letters for verbs only, the rest are individual constants.

Example:

All Greeks are men
maybe could this be translated like this:

∀(x)∀(y)[ARE(x) → ARE(y)]
where x= Greek, and y = men

or
∀(x, y)[ARE(x, y)]

Slit
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  • I would write it as $\forall x (\mathrm{Greek}(x) \to \mathrm{Man}(x))$. – md2perpe Aug 27 '17 at 17:09
  • That is like in a book. Are my above solutions valid? – Slit Aug 27 '17 at 17:11
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    No, your solutions are not good. You need to have "Greek" and "Man" inside the formula, not outside. And $ARE(x)$ just says "x is" or, when combined with "$x = \mathrm{Greek}$", "Greek is". Your statement does not say "if $x$ is Greek then $x$ is Man". – md2perpe Aug 27 '17 at 17:30
  • I am confused because in a textbook it is written that capital letter is predicate. Here we have only nouns as predicates. – Slit Aug 27 '17 at 17:43
  • The verb "(to) be" is a copula verb which means it doesn't describe an activity but is used to connect the subject to something more describing like an adjective (e.g. Greek or male). Therefore it's more natural to use the adjective as the predicate in the formula. One could also have written $\forall x ( \mathrm{IS}(x, \mathrm{Greek}) \to \mathrm{IS}(x, \mathrm{Male}) )$ but what is the general definition of $\mathrm{IS}$ and what type of mathematical objects are $\mathrm{Greek}$ and $\mathrm{Male}$? – md2perpe Aug 27 '17 at 18:03
  • Here's a MathJax tutorial :) – Shaun Aug 27 '17 at 18:12
  • That makes sense - when you look on IS as copula. I look at is/are as verb of existence. In a sense All Greeks are (that exist) men. Don't know what math. objects they could be. – Slit Aug 27 '17 at 18:13
  • It has been truly said that existence is not a predicate. – Lubin Aug 27 '17 at 19:31
  • @md2perpe How would you wrote: Some footballer wrote a PhD thesis? The solution that I found is (∃x)(Fx & (∃y)(Wxy & Ty)). I would wrote it without Ty which represents PhD thesis. – Slit Aug 28 '17 at 20:54
  • I would write it as $\exists x \exists y \left( \mathrm{Footballer}(x) \wedge \mathrm{Thesis}(y) \wedge \mathrm{Wrote}(x,y) \right)$. – md2perpe Aug 29 '17 at 05:00
  • @md2perpe what is the correct answer? – Slit Aug 29 '17 at 05:47
  • The "is" in the above statement can be equated to "inclusion" (see set theory): the set of Greeks is included into (is a subset of) the set of Men: $\text {Greeks} \subseteq \text {Men}$. – Mauro ALLEGRANZA Aug 29 '17 at 11:45
  • According to the mathematical definition, the above is defined as follows: $\forall x \ (x \in \text {Greeks} \to x \in \text {Men})$. – Mauro ALLEGRANZA Aug 29 '17 at 11:46
  • We can use instead of names for sets the corersponding predicates, to get: $\forall x \ (\text {Greek}(x) \to \text {Man}(x))$. – Mauro ALLEGRANZA Aug 29 '17 at 11:47
  • This is the correct why to explain where the copula "is" disappeared: first it has been transalted in the relation of (set) inclusion, which in turn is defined in terms of the conditional connecetive: ($\to$ : "if ..., then ..."). – Mauro ALLEGRANZA Aug 29 '17 at 11:49
  • @MauroALLEGRANZA I think i get it. I was confused how to write that statesmen. Now, I am not sure in what detail I should go in expressing predicates. – Slit Aug 29 '17 at 12:29
  • With the predicate expression $\text {Greek}(x)$ we symbolize: "$x$ is Greek", and so on. In set language we have $x \in \text {Greeks}$ where the binary predicate "in" ($\in$) translates the verb to be. – Mauro ALLEGRANZA Aug 29 '17 at 13:58
  • @MauroALLEGRANZA What about Some footballer wrote a PhD thesis? This solution: (∃x)(Fx & (∃y)(Wxy & Ty)) or this one: ∃x∃y(Footballer(x)∧Thesis(y)∧Wrote(x,y)). I am maybe forfirst one although both are valid. I think so.. Right? – Slit Aug 29 '17 at 14:10
  • Yes, the two are equivalent, "modulo" using the same symbols. – Mauro ALLEGRANZA Aug 29 '17 at 14:32
  • @MauroALLEGRANZA that is true although the first solution looks better because explains/has more info. – Slit Aug 29 '17 at 14:40
  • In what sense ? they are equivalent, and thus they have the same "info". – Mauro ALLEGRANZA Aug 29 '17 at 14:43
  • Sorry I tought I defined what kind of thesis that is PhD. Just add & PhD(y) near Thesis(y). – Slit Aug 29 '17 at 14:50
  • @MauroALLEGRANZA that is what I meant in what detail should you express sentences – Slit Aug 29 '17 at 15:19

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