Predicate problem

Question

I have the sentence: "Every Chef has a existent dish that he prepares deliciously"

I need to translate it to the math language (∃∀,|,:,(),predicate..) I tried to solve it by choosing A-Chef, B- Dish ,C- Prepares, D delicious, but for no success.

My prof said to use predicate to solve it, So I understand the predicate input is P(A,B) but I don't know what the predicate will check, I thought about P(a,b): dish that the chef prepares deliciously, I understand how it can be right for only checking the prepares, but then adding the deliciously it feel to me undefined. Hope you could help me understand and solve this

Thank you

Answer 1

I would like to clarify something in your question. You wrote "the predicate input is $P(A,B)$", and this needs to be defined. The letter $P$, if you use it as a predicate, carries a unique meaning, in the form of a sentence with blanks for each input.

Here's a full example of how I would write out a similar problem:

$B(x):x$ is a bodybuilder

$G(x):x$ is a goat

$L(x,y):x$ lives with $y$

$D(x):x$ is done deliciously

$$\forall x|B(x)\exists y[G(y)\land D(L(x,y))]$$

"For every bodybuilder there is a goat who lives with him deliciously"... or, more literally in terms of the logic, "For every $x$ such that $x$ is a bodybuilder there exists $y$ such that $y$ is a goat and $x$ lives with $y$ deliciously"

The bar notation $\forall x|B(x)$ might not be standard, but in any case do be careful that your statement does not logically mean that "$B(x)$ is true for all $x$", when what you intend is that a certain related statement is true for all $x$ that satisfy $B(x)$.

Answer 2

Translating the statement "Every Chef has a existent dish that he prepares deliciously" will require both monadic and relational predicates.

A monadic predicate assigns a property to an arbitrary subject:

$ \begin{array}{ll} Cx: & \text{$x$ is a chef.} \\ Dx: & \text{$x$ is a dish.} \\ Lx: & \text{$x$ is delicious.} \\ \end{array} $

A relational predicate, also known as a polyadic predicate, defines a relationship between two or more subjects. In particular, we define a dyadic predicate that specifies a relationship between exactly two subjects:

$ \begin{array}{ll} Pxy: & \text{$x$ prepares $y$.} \\ \end{array} $

The phrase "every chef" conveys an assertion about all objects that are chefs, implying the need for the universal quantifier $\forall$. Furthermore, for every chef, it is clear there is at leat one object that is a dish the chef prepares deliciously. This implies the need for the existential quantifier $\exists$.

Given the domain of all things, we may translate as follows:

$$ \forall x [Cx \to \exists y [Dy \wedge Pxy \wedge Ly]] $$

which is logically equivalent to

$$ \forall x \exists y [Cx \to (Dy \wedge Pxy \wedge Ly)] $$

meaning quite literally "For every $x$, there exists at least one $y$ such that, if $x$ is a chef, then $y$ is a dish and $x$ prepares $y$ and $y$ is delicious." Note I interpret the original phrase "he prepares deliciously" to mean the dish the chef prepares is in fact delicious and not as the manner in which he prepares the dish is delicious. Also note the property of the dish being "existent" is already captured by the existential quantifier $\exists$ because usage of this quantifer makes an assertion that the corresponding subject actually exists.