# Formalizing Predicate Logic (English to PL)

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• Nov 22nd 2009, 09:38 AM
Knark
Formalizing Predicate Logic (English to PL)
I'm stumped on how to formalize these two sentences properly into predicate logic.

1: There's a city to which all roads lead.
2: All roads lead to a city.

The question says nothing about defining a Universe of Discourse.

I tried using

R = is a road
C = is a city
L = leads to

1: If there exists a city and then every road leads to this city.

(∀y)(∃x)[(Cx & Ry) -> Lyx)]

I'm unsure about my use of the existential quantifier, it seems to me that what I wrote translates more into "If there exists a city and each road then each road leads to this city." I'm very confused at this point because this doesn't seem to make sense to me at all.

2: Everything that is a road leads to a city.

(∀y)(Ry -> Lyx)

With this one I started writing it and I know that x is unqualified and unquantified because I realized that if I were to add something like (∃x)(Cx) then this one would be identical to the first question. Is that right? I'm sure it is not since the first one says that "All roads lead to a specific city" whereas the second question says that "All roads lead to at least one city".
• Nov 22nd 2009, 10:56 AM
Plato
Quote:

Originally Posted by Knark
1: There's a city to which all roads lead.
2: All roads lead to a city.

This is the way I see them.
1) $\left( {\exists x} \right)\left( {\forall y} \right)\left[ {C_x \wedge \left( {R_y \to L_{yx} } \right)} \right]$

2) $\left( {\forall x} \right)\left( {\exists y} \right)\left[ {R_x \to \left( {C_y \wedge L_{xy} } \right)} \right]$
• Nov 22nd 2009, 11:18 AM
Knark
That makes a lot more sense to me than my own answer, thank you very much. Goes to show how big of a difference the placement of brackets makes.