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I had just asked a similar question, but realized a few things. I am now looking to see if my understanding of how to define $\mathfrak A \models P$ where $P$ is a predicate of 0-ary (i.e., a propositional symbol) and $\mathfrak A$ is a structure, is correct.


Somewhat informally, an $\mathcal{L}-structure$ for a language $\mathcal{L}$ is a pair ($D, I$) where $D$ is our domain of objects and $I$ maps

  • to each constant symbol an object of $D$
  • to each n-ary function symbol, a function from $D^n$ to $D$
  • to each n-ary predicate symbol $P$ , a relation $I(P) \subseteq D$

A text would then typically talk of denotation and how to "get a hold" of what each term denotes, then talk of variable assignments, and then finally define logical consequence:'$\models$'. Since my question doesn't pertain to variable assignments, for succinctness I won't consider it.

So then for defining when a formula $\varphi$ is a logical consequence of a structure $\mathfrak A$ (written as: $\mathfrak A \models \varphi$), I've seen texts start out with something like:

Given that $t_1 , ... , t_n$ are terms and tha $P$ is a predicate symbol of arity $n$:

$\mathfrak A$ $\models $P( $t_1$,...,$t_n$) iff ($d_1$,...,$d_n$) $\in$ $I(P)$, where $d_i$ = ||$t_i$||

where ||$t_i$|| is the notation used for the denotation of term $t_i$.

So when P is of arity 0, the right hand condition becomes: $ () \in I(P)$, correct?

And according to this post, the empty tuple is equal to the empty set. That is, $( ) = \emptyset$,which would make sense when thinking about it.

It's helpful to note that there is a quote from Bruno Poizat's book on mathematical logic (that I won't copy and paste here to try to minimize the clutter) which deals with this topic and which was put as an answer in this thread about the same thing, but I didn't find it as elucidating as I would have hoped.


So my question is if my following understanding of the situation is accurate.

For a structure $\mathfrak A$ and a 0-ary predicate $P$ we have that:

  • $\mathfrak A$ $\models P$ iff $\emptyset \in I(P)$

And the line of reasoning as to why we associate 'true' with the relation (the set) $\{ \emptyset \}$ and 'false' with $\emptyset$ is as follows

  • An n-ary predicate symbol under an interpretation is mapped to a subset of $D^n$
  • We are dealing with 0-ary predicate symbols, so they should be mapped to $D^0$
  • $D^0 = \{ \emptyset \}$
  • There are two sets which are subsets of $\{ \emptyset \}$: either $\{ \emptyset \}$ itself or $\{\} = \emptyset$
  • Thus, there are two possible relations which the predicate symbol can be mapped to, those being either of the two aforementioned sets.
  • If my understanding is correct, then a 0-ary predicate $P$ is a logical consequence iff $\emptyset \in I(P)$. Depending on our interpretation, $I(P)$ will be either the set $\{ \emptyset \}$ or the set $\emptyset$.
  • If under our interpretation, $P$ is mapped to the set $\emptyset$, then $\mathfrak A$ $\models P$ does not hold as $\emptyset \notin \emptyset$. Hence why we associate false with $\emptyset$.
  • If under our interpretation, $P$ is mapped to the set $\{\emptyset \}$, then $\mathfrak A$ $\models$ $P$ holds. Hence why we associate true with $\{ \emptyset \}$.

  • So we can say $\mathfrak A$ $\models$ $P$ holds for any 0-ary predicate $P$ iff $P$ is mapped to $\{\emptyset \}$ in our interpretation.

Thank you for taking the time to read this.

edit: the very last slide here very briefly discusses nullary relations and they seem to be in line with my understanding. But as one of the comments noted, perhaps it is odd/wrong to consider the empty tuple as the empty set here. So what exactly is a nullary-relation in our context?

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  • 2
    $\begingroup$ That quite a lot to write about something so trivial. $\endgroup$ – Rene Schipperus May 1 '17 at 3:23
  • $\begingroup$ Yeah, I won't argue against that. It stems from a mixture of procrastinating and genuinely being confused about something which I wanted to make sure I figured out (even though it may seem trivial) $\endgroup$ – Parry May 1 '17 at 3:27
  • $\begingroup$ I think identifying empty tuple with the empty set is misguided in this context, just identifying the number $0$ with the empty set is, most of the time (i.e. when not talking about von Neumann ordinals). The empty tuple is just the empty tuple. Otherwise, you end up thinking that empty tuple, a $0\times 0$ matrix, the number $0$, the only function $\emptyset\to {\bf R}^{\bf R}$, and an empty relation on ${\bf N}$ are all the same thing, which is absurd. $\endgroup$ – tomasz May 1 '17 at 3:42
  • $\begingroup$ Hmm, so how would the definition for an n-ary predicate being a logical consequence of a structure (and variable assignment) apply to predicates of 0-arity? What would be the right-hand condition? $\endgroup$ – Parry May 1 '17 at 3:46
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Correct.

See: Dirk van Dalen, Logic and Structure, Springer (5th ed - 2013), page 55:

One also considers the “limiting cases” of relations and functions, i.e. $0$-ary relations and functions. An $0$-ary relation is a subset of $A^{\emptyset}$. Since $A^{\emptyset} = \{ ∅ \}$ there are two such relations: $∅$ and $\{ ∅ \}$ (considered as ordinals: $0$ and $1$). $0$-ary relations can thus be seen as truth values, which makes them play the role of the interpretations of propositions. In practice $0$-ary relations do not appear; e.g. they have no role to play in ordinary algebra. Most of the time the reader can joyfully forget about them, nonetheless we will allow them in our definition because they simplify certain considerations.

In this way, every structure $\mathfrak A$ for a language with $0$-ary relations symbol $R$ must have an "objcet" $R^{\mathfrak A}$ that is a truth value tha interpret $R$.

A typical example is the constant $\bot$ (the falsum), that must be interpeted with $0$.

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  • $\begingroup$ Thank you for the reference. $\endgroup$ – Parry May 2 '17 at 1:04

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