Understanding Predicates Better

Question

With having been studying predicate logic, I have been thinking of the following question and wanted to clear up my doubts.

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First, observe the following definition of a predicate:

A predicate $P$ is a proposition-valued function of some variable(s).

With that said, let $x$ be some variable (the subject of the predicate $P$), then $P(x)$ is $true$ or $false$ dependent on $x$.

My question: if $t$ is such that $P(t)$ is $TRUE$, is it correct to think of $P$ as an assignment of the value $TRUE$ to $t$ (since $P$ is studied as a function)? I believe this to make sense, since for example:

if we have a single-variable predicate $P$ defined as $P(x) :=$ $x$ is greater than $4$, then $P(5) = TRUE$. So, we can say the number $5$ is assigned the value $TRUE$ (since of course the atomic sentence $5$ is greater than $4$ yields $true$).

Any criticism is welcome.

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EDIT: After considering the comments below, I understand the definition of a predicate I have supplied may be circular or misleading. As stated in the comments, here is the definition rewritten appropriately to fit the context of my question.

A predicate $P$ is a {$TRUE, FALSE$}-valued function.

Answer 1

Generally - yes, it makes sense, but there are some subtleties.

Predicates are often understood as linguistic items (like names, e.g. "Pope Francis"), rather than something what linguistic items refer to (i.e. Pope Francis himself). Thus, in various logic textbooks, we have "predicates" (Smith, Sider), "predicate symbols" (Enderton), or "predicate letters" (Boolos and Jeffrey) introduced as a type of non-logical vocabulary of formal languages in question. What are predicates understood as linguistic expressions? Good philosophical question, for a discussion you can see e.g. Alex Oliver's "What is a predicate?" (https://www.phil.cam.ac.uk/people/teaching-research-pages/oliver/what-is-a-predicate.pdf), but that's more of an extra.

Now, we might think about what predicates (or predicate letters, or predicate symbols) refer to. So far, I met two main ways of cashing this out:

1. Predicates refer to (characterstic) functions: one way of presenting it is this:

"For each predicate letter in the language and no others, a characteristic function is [is assigned]. In the case of $n$-place predicate letter $R$, the characteristic function $\phi$ has $n$ argument places. For any objects $o_1, ..., o_n$ in the domain [of interpretation], the value $\phi(o_1, ..., o_n)$ is to be $1$ or $0$ [($TRUE$ or $FALSE$ in your case)] depending on whether or not $R$ is supposed to be true in the interpretation of the sequence $o_1, ..., o_n$ of objects." (Boolos and Jeffrey (3rd ed.), p. 99)

So generalizing your idea, an $n$-place predicate stands for an $n$-place function (in a particular interpretation $I$) that assigns a truth-value $TRUE$ or $FALSE$ to any $n$-tuple objects from the domain of $I$. So, I'd say your intuitions are correct.

But, there is another way to think about what predicates refer to:

2. ($n$-place) predicates refer to sets of $n$-tuples of objects in the domain:

Sometimes textbooks say that ($n$-place) predicates refer to ($n$-ary) relations, and by this they mean sets of $n$-tuples of objects in the domain. Here's Enderton (2nd ed.), p. 81 [notation changed for continuity]:

[Interpretation] $I$ assigns to each $n$-place predicate symbol $P$ an $n$-ary relation $P^I$ ⊆ $Dom(I)^n$; i.e., $P^I$ is a set of $n$-tuples of members of the universe (domain of $I$).

Is it a better way to think about predicates? I don't know - from a mathematical point of view I think it does not matter, but I guess it's good to know both.

Answer 2

My question: if $t$ is such that $P(t)$ is TRUE, is it correct to think of $P$ as an assignment of the value TRUE to $t$ (since $P$ is studied as a function)? I believe this to make sense, since for example:

No, $t$ is not assigned a truth value; $t$ does not change its type or identity.

Rather we say that $t$ is mapped to a truth value by $P$. The function $P$ returns true or false when input with an object such as $t$.

• A predicate is a function that maps its variables to propositional truth-values. The variables of a predicate represent objects in the domain of discussion.

Answer 3

In this example, I will be referring to predicate logic under Tarskian semantics, which is the traditional semantics for predicate logic. It is based on the notion of interpretations of names in terms of objects external to the logic.

Objects: These are the things you are currently talking about. They can be abstract mathematical objects, or objects in the real world.

Names: These are the names given to objects. When I refer to a cat, I do not staple the cat to paper as part of my sentence, I use a name. This same distinction between names and objects is made in predicate logic.

Domain: The current set of objects you are talking about.

Interpretation: A mapping of names to objects ($c$ to my cat at home), functions to predicates ($F(c)$ for my cat is fluffy), and a truth assignment to predicates of names. The truth assignment in this example is $F(c)=True$. It does not mean that $c=True$, since $c$ refers to my cat.

Model: A model is a domain and interpretation $\mathbb{M}=<D,I>$. My model of $F(c)$ consisted of the domain {my cat}, and interpretation: $c$ refers to my cat and my cat is fluffy.

Predicates: A predicate is a property of an object. When we say $F(c)$ is true, we mean that we are reasoning about models where the truth value of $F(c)$ is true. In my example I picked the domain of my cat $c$, the interpretation fluffy $F$, and $F(c)=True$. If $F(c)$ was false, then we would need to reason about other models.

An excellent synopsis of first order logic, which contains predicate logic, can be found in Stanford Universities online course.