Completeness and Decidability of General First-Order Logic (with a Detour Through the Guarded Fragment)

Abstract

This paper investigates the “general” semantics for first-order logic introduced to Antonelli (Review of Symbolic Logic 6(4), 637–58, 2013): a sound and complete axiom system is given, and the satisfiability problem for the general semantics is reduced to the satisfiability of formulas in the Guarded Fragment of Andréka et al. (Journal of Philosophical Logic 27(3):217–274, 1998), thereby showing the former decidable. A truth-tree method is presented in the Appendix.

Appendix: Soundness and Completeness of Truth Trees

In this section we introduce an appropriately modified truth-tree method for gfol, which is shown to be sound and complete (it is not, alas, terminating). Our truth tree method will build upon standard rules such as can be found, e.g., in Smullyan [9] or Jeffrey and Burgess [5]. The main change is guided by the fact that there are now two ways for an existential statement ∃x φ(x) to fail in a model: either the extension of φ in the model is empty, or such an extension is not a member of the second-order domain. In order to represent this latter possibility, we introduce a special notation in truth-trees that runs parallel to our semantic definition: we mark a (possibly complex) predicate’s extension with a + or a − according as the extension is or is not a member of the second-order domain. A further complication is given by the fact that, at the same time, we need to carry out an extensionality check: whenever we enter a notation $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[_{x}^{+}$ on a branch (to indicate that — intuitively — the extension of φ relative to x is to be included in the second-order domain) we also need to check that no formula ψ such that $[{\kern -2.3pt}[ \psi [{\kern -2.3pt}[_{y}^{-}$ is also on the branch, is equivalent to φ (relative to the other formulas on the branch).

Definition 36

The modified truth tree method for gfol comprises the following rules:

• Standard one-sided truth-tree rules for Boolean operators and identity, including rules for closing a branch.

• If ∃x φ(x) occurs on a branch, apply a checkmark, choose a new constant a, and continue the branch with $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{+}_{x}$ and φ(a); furthermore, instantiate all starred formulas ψ(y)^∗ on the same branch (an any that might appear later) with ψ(a). Next, perform an extensionality check on $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{+}_{x}$ (explained below).

• If ¬∃x φ(x) occurs on a branch, apply a checkmark, perform a binary split on the branch and enter $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{-}_{x}$ on the left branch; further, the following formulas are entered on the right branch: ¬φ(x)^∗ as well as ¬φ(a), ¬φ(b) …, where a, b, … are all the constants on that right-hand branch, including those that might appear at a later stage (if no constants appear on that branch, or select a a new constant). Next, perform an extensionality check on $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{-}_{x}$ (explained below).

• An extensionality check is performed on a branch each time a “signed extension” $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[_{x}^{+}$ or $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[_{x}^{-}$ is entered on a branch: the signed extension is boxed, and the branch is split as follows (in case of a negative sign):

  

  where a is a new constant, and $[{\kern -2.3pt}[ \psi _{1}(y) [{\kern -2.3pt}[^{+}_{y}$, $[{\kern -2.3pt}[ \psi _{2}(z) [{\kern -2.3pt}[^{+}_{z}$, …are signed extensions occurring on the branch. Extensionality checks carry special conditions for closure, highlighted by the boxed formula: the branch of $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{-}_{x}$ closes if any one of the children closes. (the symmetric rule holds when the signs are reversed).

This completes the specification of the modified truth-tree rules. The rule for the negated existential reflects the two ways in which a counter-model can be obtained: either by keeping the extension of the existentially quantified formulas out of the second-order domain, or by making sure that such an extension is empty. Formulas for signed extensions are boxed in an extensionality check as a reminder that their branches have non-standard closure conditions.

We illustrate the method with a few examples, given in Figs. 3, 4 and 5. All of these produce a finite tree, but it is easy to check that the tree for, e.g., $\exists x \lnot \exists y x \doteq y$, is infinite. Consider for instance the simple truth tree in Fig. 3, to see how the method works. When searching for a counter-example to ∃x(¬P x∨Q x), the tree branches, allowing for the possibility that the extension of ¬P x∨Q x might not be in the second-order domain (a fact represented by $[{\kern -2.3pt}[ \lnot Px \lor Qx [{\kern -2.3pt}[^{-}_{x}$), or that such an extension might be empty. The expression $[{\kern -2.3pt}[ \lnot Px \lor Qx [{\kern -2.3pt}[^{-}_{x}$ is boxed to indicate that every branch through that node will close provided any of them does: this is because in this case we are trying to keep out of the second-order domain a subset that was already put in D ₂ by an existentially quantified equi-extensional formula. (In this case the boxed expression has only one child, so the non-standard nature of the closure condition is not apparent.)

Fig. 3

$\vDash \exists x(Px \to Qx) \to \exists x(\lnot Px \lor Qx)$, Boolean steps omitted on the left-hand branch

Fig. 4

$\nvDash \lnot \exists x \lnot Px \to \exists y Py$, no extensionality checks are called

Fig. 5

$\nvDash \lnot \exists x \lnot \exists y\, x \doteq y$

Theorem 37

Suppose φ labels a node in a truth tree such that $\mathfrak {M}, s \vDash \psi$ for each ψ on the branch above φ (including φ itself); if a truth tree rule (as given in Definition 36) is applied to φ, at least one of the children so obtained will also be satisfiable in (some appropriate expansion of) $\mathfrak {M}$ . Moreover, if φ has been boxed by a consistency check, all of the children will be satisfiable in some expansion of $\mathfrak {M}$.

Proof

If the rule is one of the standard Boolean rules or an identity rule, then it is immediate that the theorem holds, just as in the classical case. So we consider the two rules for the existential quantifier.

Suppose the rule the rule is applied to ¬∃x φ(x); by hypothesis, $\mathfrak {M}, s \vDash \lnot \exists x\, \varphi (x)$, so that $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x,s}$ is either empty or not a member of D ₂ (the two cases are not exclusive). If the latter, then $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x,s} \notin D_{2}$, so that the left-hand branch with $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{-}_{x}$ is satisfied in the model.

The closure condition for this branch require that if an extensionality check is performed, at least one of its children close, where each of the children starts a sub-tree whose root is labeled by ¬(φ(a) ⇔ ψ(a)), with ψ a formula such that $[{\kern -2.3pt}[ \psi (y) [{\kern -2.3pt}[^{+}_{y}$ occurs in the branch above ¬∃x φ(x) and is introduced by checking off ∃y ψ(y). So consider any one of these formulas ¬(φ(a) ⇔ ψ(a)) with a a new constant. If no expansion of $\mathfrak {M}$ assigning a denotation to a satisfies ¬(φ(a) ⇔ ψ(a)), then $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x,s} = [{\kern -2.3pt}[ \psi [{\kern -2.3pt}[^{\mathfrak {M}}_{y,s}$ , and since the $\mathfrak {M}, s \vDash \exists y \psi (y)$ by hypothesis, it follows that $[{\kern -2.3pt}[ \psi [{\kern -2.3pt}[^{\mathfrak {M}}_{y,s}$ is in D ₂, whence $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x,s}$ would have to be in D ₂, which we have have ruled out in this case. So each of the formulas ¬(φ(a) ⇔ ψ(a)) is satisfied in some expansion of $\mathfrak {M}$ (not necessarily the same for each).

The other case is when $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x,s}$ is empty. Then for each constant a already introduced in the branch, $\mathfrak {M} ,s \vDash \lnot \varphi (a)$, as desired (regardless of whether $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x,s} = \varnothing$ is a member of D ₂ or not).

Now suppose the rule is applied to ∃x φ(x); by hypothesis, $\mathfrak {M}, s \vDash \exists x\, \varphi (x)$ and so by the satisfaction clause, $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x,s}$ is a non-empty member of D ₂. So let d ∈ [[φ[[ _{x, s} and let $\mathfrak {M}^{\prime }$ be the expansion of $\mathfrak {M}$ assigning d as the denotation of a new constant b. Then $\mathfrak {M}, s \vDash \varphi (b)$, as desired. Moreover, if an extensionality check is called, then the closure conditions again require that at least one of the children close, and the argument is just like the one just given for $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{-}_{x}$ to show that each of the formulas ¬(φ(a) ⇔ ψ(a)) is satisfiable in a further expansion of $\mathfrak {M}^{\prime }$.

Corollary 38

If a formula φ is satisfiable and T is a tree rooted in φ, then T will contain an open branch b.

Proof

Let $\mathfrak {M}, s \vDash \varphi$. Repeated application of Theorem 37 yields that T contains a maximal branch b such that all formulas occuring on it are satisfiable in some appropriate expansion of $\mathfrak {M}$ (using König’s Lemma as appropriate in case T is infinite). In the classical case it is immediate that b has to be open, but in the general case we need to account for the non-standard closure conditions for extensionality checks. But if b closes at all it must do so after some finite number of nodes, and the lowest level at which the branch is closed will contain either a formula 𝜗 and along with its negation ¬𝜗, or a negated self-identity , against the simultaneous satisfiability of the formulas along b.

Theorem 37 and the above Corollary constitute a proof of the soundness of the truth-tree method. We now set out to prove the converse, completeness.

Definition 39

Let b ∈ T be a branch and φ a formula occurring on b. Then b|φ is the set of all unchecked formulas ψ occurring on b strictly above φ, where notations $[{\kern -2.3pt}[ \psi [{\kern -2.3pt}[_{x}^{\pm }$ for signed extensions count among the unchecked formulas.

Definition 40

Given φ ∈ b (in a tree T) we define the extended rank of φ in b, denoted by ${\mathsf {rk}^{*}_{\textit {\textbf {b}}}}(\varphi )$, by recursion on φ:

$$\begin{array}{@{}rcl@{}} {\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(Pt_{1}\ldots t_{k}) & =& 1 \\ {\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\lnot\varphi) & =& {\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\varphi) \\ {\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\varphi \to \psi) &=& \max\{{\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\varphi), {\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\psi)\} +1 \\ {\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\exists x\, \varphi) & =& \max\{{\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\psi) : \psi \in \textit{\textbf{b}}|\exists x\, \varphi\} \cup \{{\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\varphi) \} + 3. \end{array} $$

In order to see that the recursion bottoms out and ${\mathsf {rk}^{*}_{\textit {\textbf {b}}}}$ is well defined, we assign to each formula in φ ∈ b a pair 〈n, r k(φ)〉, where n is the distance (number of nodes) from the root of the tree to φ, and r k(φ) is the ordinary rank of the formula (as measured, say, by the height of φ’s syntactic tree or the number of symbols). Induction on ω ² shows that ${\mathsf {rk}^{*}_{\textit {\textbf {b}}}}(\varphi ) \downarrow$.

The definition of ${\mathsf {rk}^{*}_{\textit {\textbf {b}}}}(\exists x\, \varphi )$ is motivated by the fact that, because of the extensionality checks, ∃x φ not only makes a claim as to the extension of φ, but also about the extensions of all potentially equivalent formulas ψ when ∃y ψ or ¬∃y ψ also occurs on the same branch. (In the last clause of Definition 40 we increase the rank by 3 since when written in primitive notation ¬(φ⇔ψ) has extended rank equal to $\max \{ {\mathsf {rk}^{*}_{\textit {\textbf {b}}}}(\varphi ), {\mathsf {rk}^{*}_{\textit {\textbf {b}}}}(\psi ) \} + 2$.)

Theorem 41

Suppose a truth tree T for a sentence φ contains a maximal open branch b. Then φ is satisfiable. Moreover, if the rule for φ triggers an extensionality check, then each of children ¬(ψ⇔𝜃) also lies on an open branch.

Proof

Assume the truth tree contains a completed (i.e., maximal) open branch b. Using an appropriately modified version of the standard argument we will show that we can extract from b a model $\mathfrak {M} = (D_{1}, D_{2}, I)$ in which φ is true (given that φ is a sentence, and each object in the domain of $\mathfrak {M}$ will be denoted by a constant, we can dispense with variable assignments).

The first-order domain D ₁ is obtained by selecting an object d _a for each constant a occurring on b, taking care to make d _a = d _b whenever the sentence $a \doteq b$ also occurs on b. Obviously we will have I(a) = d _a . Moreover, the interpretation I(P) of each n-place predicate symbol P will be composed by all those n-tuples $\langle d_{a_{1}},\ldots , d_{a_{n}} \rangle$ such that the sentence P a ₁…a _n occurs on b. Finally, we construct a second-order domain D ₂ as follows: if $[{\kern -2.3pt}[ \psi (x,a_{1},\ldots , a_{n}) [{\kern -2.3pt}[^{+}_{x}$ occurs on b, then we put in D ₂ the set of all those elements d _b such that ψ(b, a ₁,…,a _n ) is also on b. Let $\mathfrak {M}$ be the model so defined.

We will prove that φ is satisfiable in the model by induction on the extended rank ${\mathsf {rk}^{*}_{\textit {\textbf {b}}}}(\varphi )$, giving separate cases according as the main connective in the formula is ¬ or not (to reflect the truth tree rules given in Definition 36):

• (a) Suppose P a ₁…a _n occurs on b; then $\langle d_{a_{1}}, \ldots , d_{a_{n}} \rangle \in I(P)$ and $\frak {M} \vDash Pa_{1}\ldots a_{n}$.

• (b) Suppose ¬P a ₁…a _n occurs on b; then since b is open, P a ₁…a _n is not found on b, so $\mathfrak {M} \vDash \lnot Pa_{1}\ldots a_{n}$.

• (c) Suppose ψ ₁ → ψ ₂ is on b; then since the branch is complete, either ¬ψ ₁ is on b or ψ ₂ is on b; by the inductive hypothesis, either $\frak {M} \vDash \lnot \psi _{1}$ or $\frak {M} \vDash \psi _{2}$ (since ${\mathsf {rk}^{*}_{\textit {\textbf {b}}}}(\psi _{i}) < {\mathsf {rk}^{*}_{\textit {\textbf {b}}}}(\psi _{1} \to \psi _{2})$ for i = 1,2), and in either case $\mathfrak {M} \vDash \psi _{1} \to \psi _{2}$.

• (d) Suppose ¬(ψ ₁ → ψ ₂) is on b. Then since the branch is complete, ψ ₁ and ¬ψ ₂ are both on b, and by inductive hypothesis $\frak {M} \vDash \psi _{1}$ and $\mathfrak {M} \nvDash \psi _{2}$ (since the extended rank of each is less than that of ψ ₁ → ψ ₂), so that $\mathfrak {M} \vDash \lnot (\psi _{1} \to \psi _{2})$.

• (e) Suppose ∃x φ(x) is on b; then $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{+}_{x}$ is on b, along with φ(a) for some constant a. Notice that ${\mathsf {rk}^{*}_{\textit {\textbf {b}}}}(\varphi (a)) < {\mathsf {rk}^{*}_{\textit {\textbf {b}}}}(\exists x\, \varphi )$ , so by inductive hypothesis $\mathfrak {M} \vDash \varphi (a)$ whence $d_{a} \in [{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x} \neq \varnothing$ . Moreover, by construction, $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x}$ is in D ₂, so that $\mathfrak {M} \vDash \exists x\, \varphi (x)$. This step might trigger an extensionality check, in which case all children of the form ¬(φ(b) ⇔ ψ(b)) are also satisfiable in the model, including the one occurring on b (as explained in Part (g) below).

• (f) Suppose ¬∃x φ(x) is on b; then there are two subcases, according as b is right-branching or left-branching. If b is right-branching, then for each constant a occurring on b, the formula ¬φ(a) is also on b, so that by inductive hypothesis $\frak {M} \vDash \lnot \varphi (a)$. Now, and since D ₁ = {d _a : a is on b}, also $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x} = \varnothing$ , and $\mathfrak {M} \vDash \lnot \exists x\, \varphi (x)$. The second subcase is when b is left-branching, so that $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{-}_{x}$ is also on b. We need to verify that $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{s}$ is not in D ₂. If it is in D ₂ it must be because there is a formula ∃y ψ(y) ∈ b such that $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x} = [{\kern -2.3pt}[ \psi [{\kern -2.3pt}[^{\mathfrak {M}}_{y}$. We distinguish two subcases, according as the formula ∃y ψ(y) occurs above or below ¬∃x φ(x):

  • (i) Suppose it occurs above: then an extensionality check is triggered, and one of the children will have the form ¬(φ(a) ⇔ ψ(a)) for some constant a. By Part (g) below this formula lies on an open branch b ^′, and it is immediate that:

    $${\mathsf{rk}^{*}_{\textit{\textbf{b}}^{\prime}}}(\lnot (\varphi(a) \leftrightarrow \psi(a))) < {\mathsf{rk}^{*}_{\textit{\textbf{b}}^{\prime}}}(\lnot\exists x\, \varphi) = {\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\lnot\exists x\, \varphi) $$

    (since b and b ^′ split at ¬∃x φ). So by inductive hypothesis ¬(φ(a) ⇔ ψ(a)) is satisfiable in some expansion of $\mathfrak {M}$, against the fact that $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\mathfrak {M}}_{x} = [{\kern -2.3pt}[ \psi [{\kern -2.3pt}[^{\mathfrak {M}}_{y}$.

  • (ii) Suppose ∃y ψ(y) occurs below ¬∃x φ(x). Then again an extensionality check is triggered, with one of the children labeled by ¬(ψ(a) ⇔ φ(a)) for some constant a. Notice that ∃y ψ(y) must come occur as a subformula of some formula above ¬∃x φ(x) or as a subformula of φ itself, or as a substitution instance of these. In each case,

    $${\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\lnot (\psi(a) \leftrightarrow \varphi(a))) \le {\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\exists y \psi(y)) < {\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\lnot \exists x\, \varphi(x)). $$

    It follows by the inductive hypothesis that $\mathfrak {M} \vDash \lnot (\psi (a) \leftrightarrow \varphi (a)$, against the hypothesis that $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{\frak {M}}_{x} = [{\kern -2.3pt}[ \psi [{\kern -2.3pt}[^{\frak {M}}_{y}$.

  To conclude this case we observe if an extensionality check is triggered, then all children of the form ¬(φ(b) ⇔ ψ(b)), are also satisfiable in the model, including the one occurring on b (see Part (g) immediately below).

• (g) Suppose the application of a rule for ∃x φ or ¬∃x φ triggers an extensionality check. Suppose the former is the case (the other case is perfectly analogous). Then the branch b will contain a notation for a signed extension $[{\kern -2.3pt}[ \varphi [{\kern -2.3pt}[^{+}_{x}$, and the children of ∃x φ(x) will be labeled by formulas ¬(φ(b) ⇔ ψ(b)), where $[{\kern -2.3pt}[ \psi [{\kern -2.3pt}[^{-}_{x}$ occurs higher up on b; one of these will occur on b and the others will occur on branches b ^′,b ^″,… etc., all going through ∃x φ(x) and they are all open, otherwise b itself would be closed given the closure condition for boxed formulas. Now pick any of these formulas, occurring on a branch b ^′ (possibly b = b ^′). Then, like before, we have:

  $${\mathsf{rk}^{*}_{\textit{\textbf{b}}^{\prime}}}(\lnot (\varphi(b) \leftrightarrow \psi(b))) < {\mathsf{rk}^{*}_{\textit{\textbf{b}}^{\prime}}}(\exists x\, \varphi(x)) = {\mathsf{rk}^{*}_{\textit{\textbf{b}}}}(\exists x\, \varphi(x)), $$

  So by inductive hypothesis $\mathfrak {M} \vDash \lnot (\varphi (b) \leftrightarrow \psi (b))$, as desired.