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Every Italian likes to eat pizza.

In predicate logic, are these formalizations syntactically and semantically correct?

  1. $$\forall x \forall y(\text{Italian}(x) \land \text{Pizza}(y) \implies \text{LikeToEat}(x,y))$$ ($∀x∀y \text{ Italian}(x)∧\text{Pizza}(y)⟹\text{LikeToEat}(x,y)$ should also be correct.)
  2. $$\forall x: (\text{Italian}(x) \implies \text{LikeToEatPizza}(x))$$ (Is the colon's placement syntactically correct?)
  3. $$\forall \text{italian} \in \text{Italian}: \text{LikeToEatPizza}(\text{italian})$$
  4. $$\forall i( L(\text{pizza}))$$
  5. $$\forall i: \text{EatingPizza}(x)$$
  6. $$ \forall x: L(x) $$ ($x:$ Italian ; $L:$ likes to eat pizza)
ryang
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user3352632
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  • is too strong: it means every Italian likes to eat any pizza. 2. is better, but eliminate the colon. Also use the \text{} environment inside a math environment to make text non-italic.
    • – Adrian Keister Jul 19 '18 at 15:20
  • is wrongly written and 2. and 3. are "variants".
    • – Mauro ALLEGRANZA Jul 19 '18 at 19:09
  • @AdrianKeister I have a question regarding of removing the parenthesis from 1. and 2.. That would make x and y unbound after AND and the Implikation and would be wrong, right? And about the colon: is this wrong or just not "beauty"? – user3352632 Jul 20 '18 at 11:52
  • @MauroALLEGRANZA 1. and 2. withouth parenthesis around the formula would make both wrong, right? – user3352632 Jul 20 '18 at 11:55
  • Is there a rule to name predicates with capital letters? – user3352632 Jul 20 '18 at 11:58
  • @user3352632: Removing parentheses from 1 and 2 would degrade the answers, for sure, possibly making them syntactically incorrect. It depends a bit on which rules you're following. I would use your number 2, but write it like this: $(\forall,x)(\operatorname{Italian}(x)\implies\operatorname{LikesToEatPizza}(x)).$ – Adrian Keister Jul 20 '18 at 13:06
  • The colon is probably not syntactically correct according to whatever rules you have, but it might be ok. It's not ambiguous, certainly. I can read your number 2 and know unambiguously what it means. It might not fly with a computer, though. Depends on the computer. – Adrian Keister Jul 20 '18 at 13:07
  • Predicates usually are written with CamelCase, such as LikesToEatPizza. – Adrian Keister Jul 20 '18 at 13:08
  • @AdrianKeister thanks for your replies. I think: $\forall x: Italian(x) \implies LikeToEatPizza(x)$ is also correct. The first one written as $\forall x \forall y Italian(x) \land Pizza(y) \implies LikeToEat(x,y)$ should be also correct. x and y are bounded in all occurences, right? – user3352632 Jul 20 '18 at 13:09
  • @MauroALLEGRANZA could you please take a look at the 5 and 6? – user3352632 Jul 24 '18 at 10:26
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    If you want to quantify pizza, use an existential rather than a universal. Because, liking pizza means that there is at least one pizza that you like, but not necessarily all. – Graviton Jun 04 '23 at 03:55
  • Also, I see what you're doing with $:$ meaning "such that", and although this abbreviation is common, it's actually not a formal logic symbol, and is usually only used to save time / space, for verbal clarity, or as a separator. So putting it here is not strictly formal. – Graviton Jun 04 '23 at 04:01
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    @Graviton "Every Italian likes pizza" is rather ambiguous: it is not inconsistent with every Italian liking every pizza, even Hawaiian. A similar issue (but in a bigger sentence) was discussed here. – ryang Jun 04 '23 at 04:01
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    @ryang Haha, agreed. At the least, it is safer to use an existential in a moment of ambiguity, as it will be implied by the stronger universal (assuming a non-empty universe, as Italians definitely exist). – Graviton Jun 04 '23 at 04:07
  • @user3352632 Re: removing the parenthses: No, ∀x Px⇒Qx does not mean ∀x(Px⇒Qx), but instead (∀y Py)⇒Qx. – ryang Jun 04 '23 at 07:55