Abstract
In Pietroski (2018) a simple representation language called SMPL is introduced, construed as a hypothesis about core conceptual structure. The present work is a study of this system from a logical perspective. In addition to establishing a completeness result and a complexity characterization for reasoning in the system, we also pinpoint its expressive limits, in particular showing that the fourth corner in the square of opposition (“Some_not”) eludes expression. We then study a seemingly small extension, called SMPL^{+}, which allows for a minimal predicatebinding operator. Perhaps surprisingly, the resulting system is shown to encode precisely the concepts expressible in firstorder logic. However, unlike the latter class, the class of SMPL^{+} expressions admits a simple procedural (contextfree) characterization. Our contribution brings together research strands in logic—including natural logic, modal logic, description logic, and hybrid logic—with recent advances in semantics and philosophy of language.
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References
Pietroski, P.M. (2018). Conjoining meanings: Semantics without truth values. Oxford University Press.
Fodor, J. (1975). The language of thought. Harvard University Press.
Carey, S. (2009). The origin of concepts. Oxford University Press.
Piantadosi, S.T. (2021). The computational origin of representation. Minds and Machines, 31(1), 1–58.
Montague, R. (1974). Formal philosophy. Yale University Press.
Muskens, R.A. (2007). Higher order modal logic. In P Blackburn, J.F.A.K. van Benthem, & F Wolter (Eds.) Handbook of modal logic (pp. 621–653). Elsevier.
Zylberberg, A, Dehaene, S, Roelfsema, P.R., & Sigman, M. (2011). The human Turing machine: A neural framework for mental programs. Trends in Cognitive Science, 15(7), 293–300.
Partee, B (1975). Montague grammar and transformational grammar. Linguistic Inquiry, 6(2), 203–300.
Keenan, EL, & Stavi, J (1985). A semantic characterization of natural language determiners. Linguistics & Philosophy, 9(3), 253–326.
Lidz, J (2018). The explanatory power of linguistic theory. In N Hornstein, H Lasnik, P PatelGrosz, & C Yang (Eds.) Syntactic structures after 60 years (pp. 85–91). De Gruyter Mouton.
Chomsky, N. (1957). Syntactic structures. Mouton & Co.
Chomsky, N (1959). On certain formal properties of grammars. Information & Control, 2, 137–167.
Pullum, G (1983). Contextfreeness and the computer processing of human languages. In Proceedings of the 21st annual meeting on association for computational linguistics (ACL ’83) (pp. 1–6).
Joshi, A.K., Vijay Shanker, K., & Weir, D. (1991). The convergence of mildly contextsensitive grammar formalisms. In S.M. Shieber, P Sells, & T Wasow (Eds.) Foundational issues in natural language processing (pp. 31–81). MIT Press.
van Benthem, J (1987). Meaning: interpretation and inference. Synthese, 73(3), 451–470.
SánchezValencia, V. (1991). Studies on natural logic and categorial grammar. Universiteit van Amsterdam: PhD thesis.
Keenan, E (2004). Excursions in natural logic. In C Casadio, PJ Scott, & R.A.G Seely (Eds.) Language and grammar: Studies in mathematical linguistics and natural language (pp. 31–52). CSLI: Stanford.
PrattHartmann, I (2004). Fragments of language. Journal of Logic, Language & Information, 13(2), 207–233.
Moss, L.S. (2008). Completeness theorems for syllogistic fragments. In F. Hamm S. Kepser (Eds.) Logics for linguistics structures (pp. 143–173). De Gruyter.
Icard, T.F., & Moss, L.S. (2014). Recent progress on monotonicity. Linguistic Issues in Language Technology, 9(7), 167–194.
Barwise, J, & Cooper, R (1981). Generalized quantifiers and natural language. Linguistics & Philosophy, 4, 159–219.
SteinertThrelkeld, S, & Szymanik, J. (2019). Learnability and semantic universals. Semantics & Pragmatics, 12.
Shepard, R.N., Hovland, C.I., & Jenkins, H.M. (1961). Learning and memorization of classifications. Psychological Monographs, 75(13), 1–42.
Davidson, D (1966). The logical form of action sentences. In N Rescher (Ed.) The logic of decision and action. University of Pittsburgh Press.
Parsons, T. (1990). Events in the semantics of english: A study in subatomic semantics. MIT Press.
Pietroski, P.M. (2005). Events and semantic architecture. Oxford University Press.
Kamp, H (1981). A theory of truth and semantic representation. In J.A.G. Groenendijk, T.M.V. Janssen, & M.B.J. Stokhof (Eds.) Formal methods in the study of language. Mathematical Centre Tracts 135 (pp. 277–322). Amsterdam: Mathematisch Centrum.
Heim, I. (1982). The semantics of definite and indefinite noun phrases. PhD thesis, University of Massachusetts Amherst.
Liang, P, Jordan, M.I., & Klein, D. (2013). Learning dependencybased compositional semantics. Computational Linguistics, 39(2), 389–446.
Buch, S, Li, FF, & Goodman, N. (2021). Neural event semantics for grounded language understanding. In Transactions of the association for computational linguistics (TACL), Vol. 9.
Abiteboul, S, Hull, R, & Vianu, V. (1995). Foundations of databases, AddisonWesley.
Tarski, A (1944). The semantic conception of truth. Philosophy & Phenomenological Research, 4, 341–375.
Blackburn, P, & Seligman, J (1995). Hybrid languages. Journal of Logic, Language & Information, 4(3), 251–272.
Goranko, V (1996). Hierarchies of modal and temporal logics with reference pointers. Journal of Logic, Language & Information, 5(1), 1–24.
van Benthem, J (1987). Logical syntax. Theoretical Linguistics, 14(2–3), 119–142.
Marsh, W, & Partee, B.H. How noncontext free is variable binding? WJ Savitch, E Bach, & W.M.G SafranNaveh (Eds.), Springer Studies in Linguistics and Philosophy.
Barwise, J (1977). An introduction to firstorder logic. Studies in Logic and the Foundations of Mathematics, 90, 5–46.
Hemaspaandra, E (1996). The price of universality. Notre Dame Journal of Formal Logic, 37(2), 174–203.
Kurtonina, N, & de Rijke, M (1999). Expressiveness of concept expressions in firstorder description logics. Artificial Intelligence, 107, 303–333.
Malink, M. (2013). Aristotle’s modal syllogistic. Harvard University Press.
Holliday, W.H., & Icard, T.F. (2010). Moorean phenomena in epistemic logic. In L Beklemishev, V Goranko, & V Shehtman (Eds.) Advances in modal logic, (Vol. 8 pp. 178–199). College Publications.
Quine, W.V.O. (1948). On what there is. The Review of Metaphysics, 2(1), 21–38.
Papadimitriou, C.H. (1994). Computational complexity. AddisonWesley.
Ristad, S. (1993). The language complexity game. MIT Press.
Szymanik, J. (2016). Quantifiers and cognition logical and computational perspectives. Springer.
Knowlton, T, Pietroski, P, Halberda, J, & Lidz, J. (2021). The mental representation of universal quantifiers. Linguistics and Philosophy. forthcoming.
PrattHartmann, I, & Moss, L.S. (2009). Logics for the relational syllogistic. Review of Symbolic Logic, 2(4), 647–683.
Horn, L. (1989). A natural history of negation. University of Chicago Press.
Hamilton, W. (1860). Lectures on logic (Vol. 1). Blackwood.
Hoeksema, J (1999). Blocking effects and polarity sensitivity. In J Gerbrandy, M Marx, M de Rijke, & Y Venema (Eds.) JFAK: Essays dedicated to Johan van Benthem on the occasion of his 50th birthday. Vossiuspers. Amsterdam: Amsterdam University.
Sbardolini, G (2021). Assertion, rejection, and semantic universals. In S Ghosh T Icard (Eds.) Logic, rationality, and interaction. LORI 2021, volume 13039 of lecture notes in computer science. Springer.
Moss, L.S. (2010). Natural logic and semantics. In M Aloni, H Bastiaanse, T de Jager, & K Schulz (Eds.) Logic, language and meaning (pp. 84–93). Springer.
Holliday, WH., & Icard, TF (2018). Axiomatization in the meaning sciences. In D Ball B Rabern (Eds.) The science of meaning: Essays on the metatheory of natural language semantics (pp. 73–97). Oxford University Press.
Grätzer, G. (2008). Universal algebra. Springer.
Brandom, R.B. (2000). Articulating reasons: An introduction to inferentialism. Harvard University Press.
Chierchia, G. (1984). Topics in the syntax and semantics of infinitives and gerunds. PhD thesis, University of Massachusetts Amherst.
Pietroski, P.M. (2019). Semantic types: Two is better than too many. In New frontiers in artificial intelligence: JSAIisAI international workshops (pp. 148–163).
van Benthem, J. (1985). Modal logic and classical logic. Bibliopolis.
Purdy, W. C. (1991). A logic for natural language. Notre Dame Journal of Formal Logic, 32, 409–425.
Rescher, N (1962). Plurality quantification. Journal of Symbolic Logic, 27, 373–374.
Feldman, J (2016). The simplicity principle in perception and cognition. Wiley Interdisciplinary Reviews: Cognitive Science, 7(5), 330–340.
Grädel, E, Kolaitis, PG., Libkin, L, Marx, M, Spencer, J, Vardi, MY, Venema, Y, & Weinstein, S. (2007). Finite model theory and its applications. Springer.
Moss, L.S. (2016). Syllogistic logic with cardinality comparisons. In K Bimbó (Ed.) J. Michael Dunn on information based logics (pp. 391–415). Springer.
van Benthem, J, & Icard, T.F. (2021). Interleaving logic and counting. ILLC Prepublication Series.
Acknowledgements
We would like to thank audiences at Indiana University and Stanford University for useful comments on this material. Thanks especially to Paul Pietroski for many discussions and very helpful comments and contributions on an earlier draft.
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Supported by grant #586136 from the Simons Foundation.
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Appendix
Appendix
In this technical appendix we give detailed proofs of all results in the main text.
1.1 Proof of Theorem 2
To establish Theorem 2 we need to show how any question about satisfiability of a propositional formula α can be reduced to a question about whether tr(α) has a model in which it is nonempty. Let us abbreviate the latter property by saying that tr(α) “has a model.” The first point to notice is that tr(α) is always built up using monadic predicates, ∧, and ⇓; in particular it never includes any subformula ∃R.φ. Such formulas have a special feature:
Lemma 14
Suppose \(\varphi \in {\mathscr{L}}\) is built up using only monadic predicates in \(\mathcal {A}\), ∧, and ⇓. Then if φ has a model, it has a model with just one point.
Proof
Let (M, 〚 〛) be a model such that \([\![\varphi ]\!] \neq \varnothing \). Pick a point a ∈ 〚φ〛 and consider a new model ({a}, 〚 〛_{a}) where 〚A〛_{a} = 〚A〛 ∩{a} for all \(A\in \mathcal {A}\). We then show that 〚φ〛_{a} = {a} by induction on φ. □
To finish the proof of Theorem 2, consider \(\alpha \in {\mathscr{L}}^{prop}\). We know that α is satisfiable iff tr(α) has a model: if α is satisfiable, then by (1) we know tr(α) has a (onepoint) model; and if tr(α) has a model, by Lemma 14 it has a onepoint model, which by (1) again means that α is satisfiable. Because the length of tr(α) involves only a polynomial increase in length, we have achieved a polynomial reduction of propositional satisfiability to the problem whether a formula in \({\mathscr{L}}\) has a model.
Finally, to check whether two concepts are distinct is at least as difficult as checking whether a given concept has a model. In particular, to check that \(\varphi \in {\mathscr{L}}\) has a model it suffices to know that φ≠⊥, i.e., there is a model in which \([\![{\varphi }]\!] \neq [\![\bot ]\!] = \varnothing \).
1.2 Soundness Proof (Theorem 6)
For every model \({\mathscr{M}}\), if \({\mathscr{M}}\) satisfies the hypotheses of one of our rules (the sentence(s) above the line in Fig. 2), then \({\mathscr{M}}\) also satisfies the conclusion (below). For the rules pertaining to ∧, this amounts to basic facts about set intersections. We shall check this soundness fact for the rest of the rules.
Fix the model \({\mathscr{M}}\). We write M for the domain of this model.
First, assume that \({\mathscr{M}}\models \psi \leq \phi \). This means that \([\![\psi ]\!] \subseteq [\![\phi ]\!]\). To see that \([\![{\Downarrow }\phi ]\!] \subseteq [\![{\Downarrow }\psi ]\!] \), we may assume that 〚⇓ ϕ〛≠∅. Thus, there is some x ∈ M such that 〚ϕ〛 = ∅. For this same x (or indeed for any x), 〚ψ〛 = ∅ as well. Thus 〚⇓ψ〛 = M. So we are done.
Second, we check that \([\![\phi ]\!]\subseteq [\![\top ]\!]\cap [\![{\Uparrow }\phi ]\!]\). Again, we may assume that 〚ϕ〛≠∅. In this case, 〚⇑ ϕ〛 = M, and 〚⊤〛 = M always. So 〚⊤〛 ∩ 〚⇑ ϕ〛 = M, and this is a superset of every set.
Third, we check that 〚ϕ ∧⇓ϕ〛 is independent of ϕ; indeed, it is always ∅. For if 〚ϕ〛 = ∅, then this is clear. And if 〚ϕ〛≠∅, them 〚⇓ ϕ〛 = ∅. So in this case, 〚ϕ∧⇓ ϕ〛 = 〚ϕ〛 ∩∅ = ∅.
Fourth, we check that 〚⇓⇓⇓ϕ〛 and 〚⇓ϕ〛 are the same set. If 〚ϕ〛 = ∅, then 〚⇓ϕ〛 = M. So 〚⇓⇓ ϕ〛 = ∅ (since M≠∅), and thus 〚⇓⇓⇓ϕ〛 = M. In the other case, 〚ϕ〛≠∅. This time, 〚⇓ϕ〛 = ∅. So 〚⇓⇓ϕ〛 = M, and 〚⇓⇓⇓ϕ〛 = ∅. Either way, we have shown what we want.
In the fifth rule, we assume that \([\![\phi ]\!] \subseteq [\![\psi ]\!]\), and we prove that \([\![\exists R.\phi ]\!] \subseteq [\![\exists R.\psi ]\!]\). For this, let x ∈ 〚∃R.ϕ〛. Then there is some y such that 〈x,y〉∈ 〚R〛 and y ∈ 〚ϕ〛. Since y ∈ 〚ϕ〛, we also have y ∈ 〚ψ〛. And thus y is a witness showing that x ∈ 〚∃R.ψ〛.
In the final rule, suppose that x ∈ 〚∃R.ϕ〛. Let y ∈ 〚ϕ〛 be such that 〈x,y〉∈ 〚R〛. Then y shows that 〚ϕ〛≠∅. Thus, 〚⇑ ϕ〛 = M. And from this, we know that \([\![\exists R.\phi ]\!] \subseteq [\![{\Uparrow }\psi ]\!]\).
At this point, we have checked the soundness of the individual rules. We show by induction on formal proofs that if Γ ⊩ ϕ = ψ, then for all models \({\mathscr{M}}\), such that \({\mathscr{M}}\models {\Gamma }\), 〚ϕ〛 = 〚ψ〛. The base case of the induction is when Γ itself contains ϕ = ψ; this is immediate. We have induction steps for the rules in Fig. 2—these follow from our work just above—and for the (cases) rule. For this, assume that we have a derivation Γ ⊩ ϕ = ψ justified by (cases) at the root. We have Γ ∪{χ = ⊥}⊩ ϕ and also Γ ∪{χ≠⊥}⊩ ϕ = ψ. Let \({\mathscr{M}}\models {\Gamma }\). We show that 〚ϕ〛 = 〚ψ〛 in \({\mathscr{M}}\). There are two cases: 〚χ〛 = ∅, and 〚χ〛≠∅. In the first case, we recall our assumption that Γ ∪{χ = ⊥}⊩ ϕ = ψ by a proof shorter than the given proof. So by induction hypothesis, Γ ∪{χ = ⊥}⊧ϕ = ψ. Our model \({\mathscr{M}}\) satisfies all sentences in Γ ∪{χ = ⊥}, hence it does satisfy ϕ = ψ. In the other case, when 〚χ〛≠∅, we have 〚⇑ χ〛 = M. So \({\mathscr{M}}\) will satisfy all sentences in Γ ∪{χ≠⊥} in this case. And again, the induction hypothesis implies that \({\mathscr{M}}\models \phi = \psi \).
1.3 Completeness Proof (Theorem 6)
We first record some useful facts about the proof calculus.
Lemma 15
The following are all guaranteed by our calculus:

1.
⊥≤ φ

2.
If φ ≤ ψ and ψ ≤ χ, then φ ≤ ψ.

3.
If φ ≤ ψ_{1} and φ ≤ ψ_{2}, then φ ≤ (ψ_{1} ∧ ψ_{2}).

4.
⊤∧ ϕ = ϕ.

5.
If ϕ ≤ ψ, then ⇑ϕ ≤⇑ψ.

6.
⇑⇓ϕ = ⇓⇑ϕ = ⇓⇓⇓ϕ.

7.
⇑⇑ϕ = ⇑ϕ.

8.
⇑⊤ = ⊤.

9.
⇓⊤ = ⊥.

10.
⇑⊥ = ⊥.
Proof
For (1), we use all three rules for ∧, and the rule that allows us to replace ⊥ with ψ ∧⇓ψ for any ψ whatsoever (including φ):
For (2), ϕ ∧ χ = (ϕ ∧ ψ) ∧ χ = ϕ ∧ (ψ ∧ χ) = ϕ ∧ ψ = ϕ.
For (3), let φ ∧ ψ_{1} = φ and φ ∧ ψ_{2} = φ. Then φ ∧ (ψ_{1} ∧ ψ_{2}) = (φ ∧ ψ_{1}) ∧ ψ_{2} = φ ∧ ψ_{2} = φ.
For (4), ⊤∧ ϕ ≤ ϕ because (⊤∧ ϕ) ∧ ϕ = ⊤∧ (ϕ ∧ ϕ) = ⊤∧ ϕ. And in the other direction ϕ ≤⊤ and ϕ ≤ ϕ, so ϕ ≤⊤∧ ϕ by part (3).
For (5), assume that ϕ ≤ ψ. We know that ⇓ψ ≤⇓ϕ. So ⇓⇓ϕ ≤⇓⇓ψ. Since ⇓⇓ϕ = ⇑ϕ, and similarly for ψ, we see that ⇑ ϕ ≤⇑ ψ.
(6) is an immediate consequence of the definition of ⇑ϕ as ⇓⇓ϕ.
For (7), ⇑⇑ϕ = ⇓⇓(⇓⇓ϕ) = ⇓(⇓⇓⇓ϕ) = ⇓⇓ϕ = ⇑ϕ.
For (8), use (6) to calculate: ⇑⊤ = ⇑⇓⊥ = ⇓⊥ = ⊤. Another proof: ⊤≤⇑⊤≤⊤.
For (9), first note that ⊤∧⇓⊤ = ⊥, by definition of ⊥ and also using the law ϕ ∧⇓ϕ = ψ ∧⇓ψ. In addition, by (4), ⊤∧⇓⊤ =⇓⊤. Thus ⇓⊤ = ⊤∧⇓⊤ = ⊥.
For (10), ⇑⊥ = ⇓⇓⊥ = ⇓⊤ = ⊥. We used the definition of ⊤ as ⇓⊥ and also part (9). Another proof: ⊥≤⊤, and so ⊤ =⇑⊤≤⇑⊥, using part (8). But then ⇑⊥ =⇑⊥∧⊤ = ⊤, using part (4) and the definition of ≤. □
Proof of Theorem 6 and Corollary 7
At this point we fix a set Γ and an equation ϕ^{∗} = ψ^{∗} such that Γ⊯ϕ^{∗} = ψ^{∗}. We show that there is a model \(\mathcal {N}\) of Γ where 〚ϕ^{∗}〛≠〚ψ^{∗}〛. Going forward, we shall assume that Γ is a finite set, so that we can establish the polynomialtime decidability of the consequence relation. That is, we shall find a model \(\mathcal {N}\) whose size is polynomial in the size of Γ ∪{ϕ^{∗},ψ^{∗}}. The same proof, minus all of the finiteness considerations, proves the completeness of the logic. Since the completeness is easier, we omit the details.
Definition 5
Fix Γ, ϕ^{∗} and ψ^{∗}. We take the set of relevant predicates to be those χ which occur in Γ, ϕ^{∗}, or ψ^{∗}. On the assumption that Γ is finite, there are only finitely many relevant predicates, and indeed the number of them is polynomial in the size of Γ ∪{ϕ^{∗} = ψ^{∗}}. Every subconcept of a relevant predicate is itself relevant. List the relevant predicates as χ_{1},…,χ_{K− 1}.
Expanding Γ
We expand Γ to a larger set of equations which also does not derive ϕ^{∗} = ψ^{∗}, and which has the following additional property: for all relevant predicates χ, either Γ contains χ = ⊥, or Γ contains χ≠⊥. We do this by a stepbystep construction, building sets Γ_{n} for 0 ≤ n ≤ K. We start with Γ_{0} = Γ. For each n, let χ_{n} be the n th relevant predicate. If Γ_{n} ∪{χ_{n} = ⊥}⊯ϕ^{∗} = ψ^{∗}, we set Γ_{n+ 1} to be Γ_{n} ∪{χ_{n} = ⊥}. Otherwise, Γ_{n} ∪{χ_{n} = ⊥}⊩ ϕ^{∗} = ψ^{∗}, and we must have Γ_{n} ∪{χ_{n}≠⊥}⊯ϕ^{∗} = ψ^{∗}. [For if not, Γ_{n} ⊩ ϕ^{∗} = ψ^{∗} by (cases).] We set Γ_{n+ 1} to be Γ_{n} ∪{χ_{n}≠⊥} in this case. This builds sets \({\Gamma } = {\Gamma }_{0} \subseteq {\Gamma }_{1} \subseteq {\cdots } \subseteq {\Gamma }_{K}\).
Replace Γ by Γ_{K}
Note that Γ and Γ_{K} have the same predicates. So the notion of a relevant predicate is the same if we expand Γ to Γ_{K}. Moreover, Γ_{K}⊯ϕ^{∗} = ψ^{∗}. The rest of this proof produces a model \(\mathcal {N}\) of Γ_{K} where 〚ϕ^{∗}〛≠〚ψ^{∗}〛, and such a model \(\mathcal {N}\) is a model of Γ as well. To save on notation, we simply replace our original set Γ with Γ_{K}. In other words, we assume that the original set Γ came to us expanded as in the last paragraph.
A Point on on Γ
We claim that Γ⊯⊤≤⊥. For if Γ ⊩⊤≤⊥, then for all χ, Γ ⊩⊥≤ χ ≤⊤≤⊥, and so Γ ⊩ χ = ⊥. In particular, for the concepts ϕ^{∗} and ψ^{∗} which we fixed above, we would have Γ ⊩ ϕ^{∗} = ⊥ = ψ^{∗}, contrary to what we assumed at the outset.
The Model \(\mathcal {N}\)
Let
We interpret our language on this set N as follows: For a monadic predicate A,
For each binary R, we set
This turns our set N into a model \(\mathcal {N}\). Please note that we need not have ∃R.ψ in N in order to have 〈ϕ,ψ〉∈ 〚R〛 in the model.
Lemma 16 (Truth Lemma for \(\mathcal {N}\))
For all relevant ψ (not necessarily in N),
Proof
By induction on ψ. For ψ a basic predicate A, the statement in (8) is the definition in (10).
Consider a conjunction ψ, say ψ_{1} ∧ ψ_{2}, where ψ ∈ N and therefore ψ_{1} and ψ_{2} also belong to N. We reason as follows:
where the last line follows from Lemma 15, part (3).
The next induction step is for a relevant predicate ⇓ψ. Assume (8) for ψ. Since ⇓ψ is relevant, so is ψ. This time we prove that
The two cases are: Γ ⊩ ψ = ⊥, and Γ ⊩ ψ≠⊥. (Notice that we used the fact that Γ contains either ψ = ⊥ or ψ≠⊥. We arranged this by expanding Γ.)
In the first case, Γ ⊩ ψ = ⊥. We claim that 〚ψ〛 = ∅. [Here is the proof: If χ ∈ 〚ψ〛, then we use the induction hypothesis to see that Γ ⊩⊤≤⇑χ ≤⇑ψ ≤⇑⊥ = ⊥. And this contradicts our earlier assumption that Γ⊯⊤≤⊥.] Since 〚ψ〛 = ∅, 〚⇓ψ〛 = N. We evaluate the set on the right in (9). Let ϕ ∈ N. Then Γ ⊩ ϕ ≤⊤ = ⇓⊥ = ⇓ψ. This shows that the set on the right in (9) is all of N, as desired.
In the second case, ψ ∈ N, and indeed ψ ∈ 〚ψ〛 by induction hypothesis. And so 〚⇓ψ〛 = ∅. We claim that the set on the right in (9) is empty. For assume not, and let ϕ ∈ N have Γ ⊩ ϕ ≤⇓ψ. Since ϕ ∈ N, Γ ⊩⊤≤⇑ϕ. We also have Γ ⊩⊤≤⇑ψ, and therefore ⇓⇑ψ ≤⇓⊤ = ⊥. Suppose towards a contradiction that Γ ⊩ ϕ ≤⇓ψ. Then
But recall that we are assuming about Γ that Γ⊯⊤≤⊥. This contradiction confirms that the set on the right in (9) is empty in this case.
The last induction step is for ∃R. Assume that ∃R.ψ is relevant. So ψ also is relevant. We have (8) for ψ ∈ N, and prove that
First, let ϕ ∈ 〚∃R.ψ〛. Then there is some χ ∈ 〚ψ〛 such that 〈ϕ,χ〉∈ 〚R〛. Then χ ∈ N and Γ ⊩ ϕ ≤∃R.χ. By induction hypothesis on ψ, Γ ⊩ χ ≤ ψ. Hence by the logic, Γ ⊩∃R.χ ≤∃R.ψ. Thus, Γ ⊩ ϕ ≤∃R.ψ. This is half of (10).
In the other direction, let ϕ ∈ N have Γ ⊩ ϕ ≤∃R.ψ. Since ϕ ∈ N, Γ ⊩⊤≤⇑ϕ. By the monotonicity law in the logic, Γ ⊩⇑ϕ ≤⇑∃R.ψ. Using the logic, Γ ⊩⇑∃R.ψ ≤⇑ψ. Putting these together, Γ ⊩⊤≤⇑ψ. Since ψ is relevant, we now have ψ ∈ N. We have Γ ⊩ ψ ≤ ψ, and so by induction hypothesis, ψ ∈ 〚ψ〛. Since ϕ and ψ belong to N and 〚R〛 is given by (7), we have 〈ϕ,ψ〉∈ 〚R〛. Thus, ϕ ∈ 〚∃R.ψ〛, as desired.
This completes the proof. □
Lemma 17
\(\mathcal {N}\) satisfies every equation in γ = δ in Γ. That is, \(\mathcal {N}\models {\Gamma }\).
Proof
Fix such an equation, and note that γ and δ are relevant by Definition 5. Let ϕ ∈ 〚γ〛. Then ϕ ∈ N, and by the Truth Lemma for \(\mathcal {N}\), Γ ⊩ ϕ ≤ γ. But also Γ ⊩ γ ≤ δ, and so Γ ⊩ ϕ ≤ δ. Since ϕ ∈ N, we use the Truth Lemma for \(\mathcal {N}\) again, this time to see that ϕ ∈ 〚δ〛. This for all ϕ shows that \([\![\gamma ]\!] \subseteq [\![{\delta }]\!]\). The converse is similar, and we conclude that \(\mathcal {N}\models \gamma = \delta \). □
Lemma 18
For all predicates γ, if Γ ⊩ γ = ⊥, then 〚γ〛 = ∅. For all relevant predicates γ, if Γ ⊩ γ≠⊥, then 〚γ〛≠∅.
Proof
The first assertion just comes from the fact that 〚⊥〛 = ∅ in every model of Γ. And since we know from Lemma 17 that \(\mathcal {N}\) is a model of Γ, we see from soundness that 〚γ〛 = 〚⊥〛 = ∅.
For the second assertion, let γ be relevant with Γ ⊩ γ≠⊥. Then γ ∈ N. By the Truth Lemma for \(\mathcal {N}\) and the fact that Γ ⊩ γ ≤ γ, we see that γ ∈ 〚γ〛. In particular, 〚γ〛≠∅. □
In the next lemma, recall that our standing assumption in this proof is that Γ⊯ϕ^{∗} = ψ^{∗}.
Lemma 19
\(\mathcal {N}\not \models \phi ^{*}= \psi ^{*}\).
Proof
Before we start, let us recall the definition of N in (5) and also Definition 5. Note that both ϕ^{∗} and ψ^{∗} are relevant. Recall our construction began by arranging that for relevant χ, either Γ contains (and thus derives) χ = ⊥ or else Γ ⊩ χ≠⊥. We thus have four cases, depending on whether Γ ⊩ ϕ^{∗} = ⊥ or Γ ⊩ ϕ^{∗}≠⊥, and similarly for ψ. In two of these cases, we shall show that \(\mathcal {N}\not \models \phi ^{*}= \psi ^{*}\), and in the other two we derive a contradiction to the standing assumption in this completeness theorem that Γ ⊩ ϕ^{∗}≠ψ^{∗}.
First, if Γ ⊩ ϕ^{∗} = ⊥ and also Γ ⊩ ψ^{∗} = ⊥, then we easily have our contradiction Γ ⊩ ϕ^{∗} = ψ^{∗}.
Second, suppose that Γ ⊩ ϕ^{∗} = ⊥ but that Γ ⊩ ψ^{∗}≠⊥. In this case, Lemma 18 shows that in \(\mathcal {N}\), 〚ϕ^{∗}〛 = ∅ and 〚ψ^{∗}〛≠∅. This tells us that \(\mathcal {N}\models \phi ^{*}\neq \psi ^{*}\), as desired.
Mutatis mutandis, we obtain the same conclusion \(\mathcal {N}\models \phi ^{*}\neq \psi ^{*}\) in the case that Γ ⊩ ϕ^{∗}≠⊥ but Γ ⊩ ψ^{∗} = ⊥.
Finally, suppose that Γ ⊩ ϕ^{∗}≠⊥ and also Γ ⊩ ψ^{∗}≠⊥. Thus, both ϕ^{∗} and ψ^{∗} belong to the set N which underlies our model. Since Γ ⊩ ϕ^{∗}≤ ϕ^{∗}, the Truth Lemma implies that ϕ^{∗}∈ 〚ϕ^{∗}〛. Similarly ψ^{∗}∈ 〚ψ^{∗}〛. Suppose towards a contradiction that \(\mathcal {N} \models \phi ^{*}= \psi ^{*}\). Since 〚ϕ^{∗}〛 = 〚ψ^{∗}〛, ϕ^{∗}∈ 〚ψ^{∗}〛 and ψ^{∗}∈ 〚ϕ^{∗}〛. By the Truth Lemma again, Γ ⊩ ϕ^{∗}≤ ψ^{∗}≤ ϕ^{∗}. This contradicts Γ⊯ϕ^{∗} = ψ^{∗}. □
Concluding the Proof of Completeness
We have shown that if Γ⊯ϕ^{∗} = ψ^{∗}, then there is a model of Γ where ϕ^{∗} = ψ^{∗} is false. This is the completeness of the proof system.
On Complexity
Our foregoing work also shows that if Γ has any model whatsoever in which ϕ^{∗} = ψ^{∗} fails, then it has such a model whose universe is a subset of S, where S is the set of predicates that appear in Γ ∪{ϕ^{∗} = ψ^{∗}}. And the size of S is polynomially related to the size of Γ ∪{ϕ^{∗} = ψ^{∗}}. This is behind the complexity assertion that the relation “Γ⊯ϕ^{∗} = ψ^{∗}” is in the class np of problems decidable in nondeterministic polynomial time.
1.4 Proof of Theorem 10
Recall that we assume in our language \({\mathscr{L}}^{+}\) that we have a predicate variable X for every firstorder variable \(x \in {\mathscr{L}}^{FO}\). We use this correspondence freely in what follows.
For the translations τ_{x} as defined in Section 4.2, we show Theorem 10 by induction on the structure of φ. For the first base case we simply have \({\mathscr{M}},f \models \tau _{x}(A)\) iff f(x) ∈ 〚A〛. For the second base case we have \({\mathscr{M}},f \models \tau _{x}(Y)\) iff \({\mathscr{M}},f \models x=y\) iff f(x) = f(y) iff f(x) ∈{g_{f}(Y )} iff \(f(x) \in [\![Y]\!]_{g_{f}}\). The case of conjunction is straightforward, as are the next three cases. For the final case:
The third equivalence is by the induction hypothesis, and fourth is by the fact that the assignment to Z is not included in g_{f}[Y ↦f(x)] since the variable z was chosen fresh. That is, Z did not occur in φ, and so the assignment \([\![{\varphi }]\!]_{g_{f[z\mapsto a,y\mapsto f(x)]}} = [\![{\varphi }]\!]_{g_{f}[Z \mapsto a,Y\mapsto f(x)]}\) is the same as \([\![{\varphi }]\!]_{g_{f}[Y\mapsto f(x)]}\).
1.5 Proof of Theorem 11
We show both simultaneously by induction on formulas in \({\mathscr{L}}^{FO}\). Consider the second base case. We have \({\mathscr{M}},f \models R(x_{i},x_{j})\) implies 〈f(x_{i}),f(x_{j})〉∈ 〚R〛, which in turn implies \([\![{X_{i}}]\!]_{g_{f}} \cap [\![{\exists R.X_{j}}]\!]_{g_{f}} \neq \varnothing \), and thus \([\![{\Uparrow (X_{i} \wedge \exists R. X_{j})}]\!]_{g_{f}} = M\). For (8) we have \({\mathscr{M}},f \ \neq R(x_{i},x_{j})\) implies 〈f(x_{i}),f(x_{j})〉∉〚R〛, which gives \([\![{X_{i}}]\!]_{g_{f}} \cap [\![{\exists R.X_{j}}]\!]_{g_{f}} = \varnothing \), and thus \([\![{\Uparrow (X_{i} \wedge \exists R. X_{j})}]\!]_{g_{f}} = \varnothing \). The other two base cases are analogous.
Conjunction is straightforward, so consider negation. For (2) we have:
where the last implication holds because Y is chosen fresh and thus is not in dom(g_{f}): if \([\![\widehat {\alpha }]\!]_{g_{f}} = \varnothing \) then every point a is such that \(\{a\} \cap [\![{\widehat {\alpha }}]\!]_{g_{f}} = \varnothing \). And for (3) we have:
where the reasoning in the last step is as in the previous case.
Consider (2) for existential quantification:
And for (3) we have:
This completes the proof.
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Icard, T.F., Moss, L.S. A Simple Logic of Concepts. J Philos Logic 52, 705–730 (2023). https://doi.org/10.1007/s10992022096851
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DOI: https://doi.org/10.1007/s10992022096851