32_4_1

Bulletin of the Section of LogicVolume 32/4 (2003), pp. 165–177

George Tourlakis 1

Francisco Kibedi

A MODAL EXTENSION OF FIRST ORDER CLASSICALLOGIC–Part I

Abstract

We formalize a fragment of the metatheory of classical first order logic by adding

a new connective, the modal 2, whose intended interpretation is the (classical)

metalogical predicate “is a theorem” (`). We then illustrate how to employ this

modal extension of classical logic to write equational proofs of classical theorems,

and prove (in Part II) its completeness and soundness with respect to Kripke

models. We also prove that the extension meets our objective: For any classical

formulae A and B, we can prove 2A → 2B modally iff we can prove A ` B

classically.

Keywords: First order logic, modal logic, equational logic, calculationallogic, consistency, completeness, Leibniz rule, derivability conditions, prov-ability predicate, Kripke models.

1. Introduction

First order equational, or calculational, logic was proposed in [4], [6], [7]and was shown to be sound and complete in [13], [14], [15]. As a practi-tioner’s tool, it relies heavily on Leibniz’s principle of “replacing equals byequals” allowing the user to prove assertions in the same manner that oneverifies the equality or inequality of two expressions in high school algebra

1This research was partially supported by NSERC grant No. 8820

166 George Tourlakis and Francisco Kibedi

or trigonometry, namely, by constructing a conjunctional2 chain of equali-ties and inequalities–in the case of logic, equivalences (↔) and implications(→) respectively. A classical formal proof in the equational style ought tolook like

A1annotation◦ A2

annotation◦ . . .annotation◦ An (1)

normally written vertically, where each occurrence of “◦” stands for oneof the formal connectives “→” or “↔”. Such a configuration establishesthe derivability of An if A1 is known to be a theorem. More generally, itestablishes the derivability of A1 → An (A1 ↔ An if all the ◦ are ↔).

In practice, one often needs to break this chain of formal equivalencesand implications and invoke a rule of inference that will spawn a new chain,disjoint from the original. For example,

A1annotation◦ . . .

annotation◦ B • (∀x)Bannotation◦ . . .

annotation◦ An (2)

The “•” above cannot be either of→ or ↔, but is the classical metalogical` (here also the two-sided à works). Thus the configuration (2) proves An

from A1, however it does so using two “connected components” of type (1).It is “disconnected” at the point where one moves into the metatheory andapplies à.

We believe that from a methodological point of view it is desirable tohave a new proof calculus that is sufficiently powerful to simulate classicalequational proofs such as (2) above without having to disconnect the proofby the insertion of the metalogical connectives ` or à, as the latter areoften confused with the formal→ and↔ by inexperienced users. We expectthat such a calculus will eliminate this potential confusion by simplifyingand further mechanizing the process of writing equational proofs, as theuser will now exclusively deal with the formal→ and↔. Similar commentswere contained in [8] were it was, essentially, suggested that one formalize` as the modal 2, extending in some natural way the classical logic (suchan extension was restricted to the propositional case in loc. cit.).

Our goal in this paper is to present a simple modal framework withinwhich we can write equational proofs of classical theorems3 in a manner

2E.g., a = b < c = d < e, meaning a = b and b < c and c = d and d < e.3The arbitrary modal equational proof may also be disconnected. However we are

only interested in modal proofs that simulate classical equational proofs. Such simula-tions will be always connected.

A Modal Extension of First Order Classical Logic–Part I 167

that the only connectives used are the formal→ and↔. In this frameworkthe two formal chains in (2) that were disconnected at “•” will be renderedby the connected modal chain

2A1annotation◦ . . .

annotation◦ 2B ↔ 2(∀x)Bannotation◦ . . .

annotation◦ 2An

(3)To this end we devise a particular modal extension of classical first orderlogic that simulates generalization in a natural manner and prove that itmeets our main conservation requirement: For any classical formulae A andB and classical theory T , we have that T ∪ {A} proves B classically iff Tproves 2A→ 2B modally.

Fulfillment of this requirement allows one to replace any, possibly dis-connected, classical equational proof by a connected modal equationalproof: First, replace all classical → and ↔ by the classical metalogical` and `a respectively (a valid step by modus ponens). Then replace everyA ` B (respectively, A `a B) by 2A → 2B (respectively, 2A ↔ 2B).It is immediate by the conservation requirement that, conversely, a con-nected modal chain of type (3) can be replaced by a classical, possiblydisconnected, chain of type (2).

Modal extensions of propositional ([2], [8], [9], [10], [12], [17]) and predi-cate ([1], [3], [5], [9], [10], [11]) classical logic are not new, however, previousones differ in at least two significant ways from our approach. One, they donot address, but we do, the needs of the “user”, investigating instead gen-eral or specific philosophical and metamathematical issues. Two, their “2”either denotes (intuitively) a form of “necessity”, or it denotes provabilityin the narrow sense of Godel’s provability predicate for Peano arithmetic(or for ZF). As far as we know, our approach is the first that uses “2” tosimulate classical first order provability in general. This entails some depar-tures from the norm: Firstly, the semantics of Section 5 (in Part II) leadsto models that are similar to but not identical to varying domain models.Secondly, we require 2A to be a sentence for all formulae A (contrast, e.g.,with [1], [3], [5]), a choice that we justify later on.

There are many variants of modal logic to choose from as a startingpoint. We build upon a first order version of logic K4, adding just oneaxiom schema ((M3) of Section 2) that simulates classical generalizationA ` (∀x)A. The conservation requirement determines the form of (M3):


2A → 2(∀x)A.4 This axiom plays a role in making Godel completenesswork (cf. Lemma 6.3 in Part II).

Motivation for keeping the two modal axioms of K4 is straightforward:Axiom (M1) in Section 3 simulates classical modus ponens (cf. 4.10). Ax-iom (M2), of less obvious intuitive value, is technically expedient, for exam-ple, towards the proof of weak necessitation and inner Leibniz rule. Thesetwo axioms are the counterparts of Lob’s derivability conditions DC2 andDC3 respectively that one encounters in proofs of Godel’s second incom-pleteness theorem ([2], [12], [16]), this observation providing immediatepeace of mind with respect to their relative consistency with the classicalaxioms. By the way, our axiom (M3) too is “true” when 2 is interpreted asGodel’s provability predicate.5 This observation, once more, puts to restany consistency worry.

We do not need the reflection principle–that is, axiom schema 2A→ Aof logic S4–and we do not include it. Indeed, it is a trivial exercise to showthat it is underivable in our logic, for one can easily build a countermodel(using an irreflexive frame relation) by the techniques of Section 5 of Part II.We also prefer to simulate the inference “if A, then 2A” (corresponding toLob’s DC1) by hiding it inside the axioms, adopting only classical primitiverules of inference.

The paper is split into two parts. Part I (Sections 2–4) deals withsyntactic issues. Part II (Sections 5–6) introduces semantics, proves thesoundness and strong completeness of the proposed logic with respect toKripke structures, and proves that the main conservation requirement ismet.

2. The Language of Modal Logic

Terms are built from the object variables x, y, z, x′, y′′, z51, . . ., and what-ever nonlogical symbols may be available in any particular theory of in-terest, such as constants a, b, c, a′′, c13, . . ., and functions f, g, h, f ′, h′′

101 . . .,exactly as in classical first order logic.

4Since 2A is closed for all A, one may view (M3) as a special case of the Barcanformula.

5“True” abuses language here, and it means: If P (x) is Godel’s provability predicatefor Peano arithmetic, and if A denotes the (formal) Godel number of the formula A,then one can prove P ( A ) → P ( (∀x)A ) in some appropriate conservative extensionof Peano arithmetic ([16]).


Formulae (well-formed modal formulae–wfmf) are built, via the stan-dard inductive definitions, from atomic formulae–which may involve predi-cate symbols P,Q,R, P ′′′

54, . . .–and the primary logical symbols. The latterare the Boolean variables p, q, p′, p′′, q13, . . ., and the connectives: ¬,∨,>,⊥,2, (, ),=,∀, and the comma.

We note two slight deviations from the standard definitions: One isthat we add an induction clause “if A is formula, then so is (2A)”. Theother is the inclusion of Boolean variables (traditionally used in equationallogics to facilitate applications of the Leibniz rule) and the Boolean con-stants > (formal “true”) and ⊥ (formal “false”). Every Boolean symbol isan atomic formula. A is the scope of the leftmost 2 in (2A). Every objectvariable in the scope of a 2 is bound; thus, (2A) is closed (a sentence).The motivation regarding object variables is our intended intuitive inter-pretation of 2 as the classical `, and therefore as the classical |= as well.When we say “|= A” classically, we mean that for all structures where weinterpret A, and for all value-assignments to the free object variables ofA, the formula is true. Thus the variables in a statement such as “|= A”are implicitly universally quantified and are unavailable for substitutions.However, Boolean variables in a wfmf are always free.

If a formula does not contain 2, then we say it is a classical formula–or a well-formed formula, or wff. Additional symbols ∧,→,↔ and ∃ areintroduced metalinguistically in the usual manner. The usual metalinguis-tic rules of how to omit brackets apply: highest priority symbols are the“unary” ∀,∃,¬,2; all associativities are right.

We denote substitutions into variables of a wfmf A by the meta ex-pressions A[x := t] (term substituted into all occurrences of the free objectvariable x) and A[p := B] (formula substituted into all occurrences of theBoolean variable p). In the the first type we disallow substitution if pro-ceeding would have some variable of the term t captured by a quantifier.In the second type we proceed regardless. In our notation the symbols“[x := t]” and “[p := B]” have the least scope, thus, e.g., (∀x)A[y := t]–ifallowed–means (∀x)(A[y := t]).

If ∆ is a set of wfmf, then 2∆ denotes {2A : A ∈ ∆}. We say that2A is the “boxed version of A” and 2∆ is the “boxed version of ∆”.


3. Axioms

We shall call our first order modal logic M3, the “3” indicating the presenceof three modal axioms.

Definition 3.1. The set of axioms of M3 is Λ ∪2Λ, where Λ consists ofall instances of the following schemata.

(1) All tautologies(2) (∀x)A→ A[x := t], provided the substitution is allowed(3) A→ (∀x)A, provided x is not free in A(4) (∀x)(A→ B)→ (∀x)A→ (∀x)B(5) x = x(6) s = t → (A[x := s] ↔ A[x := t]), provided the substitutions are

allowed(M1) 2(A→ B)→ 2A→ 2B(M2) 2A→ 22A(M3) 2A→ 2(∀x)AThere are two primary rules of inference. Modus ponens (MP) “if A

and A → B, then infer B”, and generalization (Gen) “if A, then infer(∀x)A, for any object variable x”.

We shall always work within a mathematical theory, generically de-noting its set of nonlogical axioms by T . Examples of T are ZFC, Peanoarithmetic, or something totally wild (including wfmf’s), or ∅. In the lattercase we have a pure theory, i.e., we are doing just logic.

Definition 3.2. [Γ-proofs and Γ-theorems] We say that a formula A is aΓ-theorem of T based on a (possibly empty) set of additional assumptions,Γ–and write Γ `T A–iff there is a Γ-proof of An–from T . By such a proofwe understand a sequence of formulae A1, . . . , An such that A is An andeach Ai satisfies one of the following conditions:

(1) Ai ∈ Λ ∪2Λ ∪ T ∪2T ∪ Γ(2) There are numbers j, k < i such that Ak is Aj → Ai

(3) There is a number j < i such that Ai is (∀x)Aj .The corresponding direct recursive definition of Γ-theorems is to say

that A is a Γ-theorem iff it satisfies (1) (using “A” for “Ai”) or there is aΓ-theorem B, such that B → A is also a Γ-theorem, or A is (∀x)B and Bis a Γ-theorem. We omit writing Γ (or T ) if it is empty.


Having added the boxed versions of all the axioms in T and Λ, it isunnecessary to include “if A, then 2A” as a primary rule, since we canobtain a form of this inference as a derived rule (4.2 below). We also notethat there is a subtle but important difference between writing Γ ` A and`Γ A, namely, `Γ A is the same as Γ ∪2Γ ` A.

4. Some metatheorems

Metatheorem 4.1. [Tautology Theorem] If A1, . . . , An |=taut B,6 thenA1, . . . , An `T B for any T .

Metatheorem 4.2. [Derived Rule: Weak Necessitation (WN)] If Γ `T A,then Γ `T 2A, provided Γ = 2Γ′ or Γ = Γ′ ∪2Γ′′ for some Γ′′ ⊇ Γ′.

Proof. Induction on Γ-theorems.(1) If A ∈ Λ ∪ T , then 2A ∈ 2(Λ ∪ T ), and we are done. If A is 2B

for some B ∈ Λ ∪ T , then we have `T 2B, but also `T 2B → 22B, by(M2). Thus `T 22B by MP.

(2) If A ∈ Γ, then we proceed as in (1).(3) Let Γ `T A, and also Γ `T B and Γ `T B → A. We have

Γ `T 2B and Γ `T 2(B → A) by induction hypothesis (I.H.). Then wehave Γ `T 2B → 2A by (M1) and MP. Using MP again, we get Γ `T 2A.

(4) Let Γ `T C, and A be (∀x)C. By I.H., Γ `T 2C, hence Γ `T2(∀x)C by (M3) and MP.

Corollary 4.3. If `T A, then `T 2A.

We call an inference rule weak if in order to obtain its conclusion wemust place restrictions on how the premises were derived. Otherwise therule is “strong”. For example, Gen is strong for we place no conditions onthe hypothesis A.

Metatheorem 4.4. [Outer Deduction Theorem] For any formulae A,Band any set of formulae Γ, if Γ + A `T B with a condition, then Γ `TA → B. The condition is that a Γ + A-proof of B exists that contains nogeneralization steps C ` (∀x)C, for any x that is free in A.

6A1, . . . , An |=taut B indicates that A1, . . . , An tautologically imply B. That is thesame as saying that |=taut A1 → . . . → An → B, i.e., that A1 → . . . → An → B is atautology.


N.B. Γ + A is an abbreviation of Γ ∪ {A}.

Proof. By induction on Γ + A-theorems B obtained via Γ + A-proofsthat satisfy the condition.

(1) If B is in one of Λ, 2Λ, T , or 2T , then `T B. Now, B |=taut A→B. So we get `T A→ B by 4.1, from which Γ `T A→ B.

(2) Suppose B is in Γ. Then Γ `T B. Since B |=taut A→ B, as above,we have Γ `T A→ B.

(3) Suppose B is A. Then A → B is the tautology A → A. Hence`T A→ B, and so Γ `T A→ B.

(4) Suppose Γ + A `T C and Γ + A `T C → B. By I.H., Γ `T A→ Cand Γ `T A → (C → B). Since A → C,A → (C → B) |=taut A → B, wehave Γ `T A→ B.

(5) Finally, let Γ+A ` D and (∀x)D be B. By I.H., Γ ` A→ D, henceΓ ` (∀x)(A→ D) by Gen. Axiom (4) now yields

Γ ` (∀x)A→ (∀x)D (i)

via MP. The fact that D ` (∀x)D was employed in the proof of B meansthat x is not free in A. Thus, by axiom (3) and 4.1, (i) yields Γ ` A →(∀x)D.

Metatheorem 4.5. [Inner Generalization] ` 2A↔ 2(∀x)A.

Proof. The← direction is by 2((∀x)A→ A) (boxed axiom (2)), by (M1)and MP. The → direction is (M3).

Remark 4.6. The qualifiers “outer” and “inner” are used with respectto the classical logic that M3 extends. Thus, an inner “rule” is not a ruleof M3 at all. Rather it is a theorem schema of M3 that simulates a ruleof classical logic according to the (soon to be proved) main conservationrequirement. Here “inner generalization” simulates classical generalizationon classical wff A: “A and (∀x)A are mutually derivable”. Note howeverthat 4.5 applies to all wfmf A not only to classical wff A.

The qualifier “outer” applied to a derived rule indicates that, unlike“inner”, this is a bona fide derived rule of M3.

Metatheorem 4.7. `T (∀x)(A↔ B)→((∀x)A↔ (∀x)B

).


Metatheorem 4.8. [2-monotonicity] If `T 2(A → B), then `T 2A →2B.

Metatheorem 4.9. [2 over ↔] ` 2(A↔ B)→ (2A↔ 2B).

Proof.2(A↔ B)

→⟨4.8 plus |=taut (A↔ B)→ A→ B

⟩2(A→ B)

→⟨(M1)

⟩2A→ 2B

We similarly prove ` 2(A↔ B)→ (2B → 2A) and are done by 4.1.

Remark 4.10. ` 2(A ↔ B) → (2A → 2B) is the counterpart of theequanimity rule of [6], [13], [14], [15], namely “A↔ B,A ` B”. This followsfrom Boolean manipulation, the modal theorem 2(A ∧ B) ↔ 2A ∧ 2B(exercise) and the main conservation requirement. Similarly, (M1) is innerMP, capturing the classical “A→ B,A ` B”.

Metatheorem 4.11. [Outer ∀-monotonicity] If Γ `T A → B, thenΓ `T (∀x)A→ (∀x)B.

Metatheorem 4.12. [Inner ∀-monotonicity]

` 2(A→ B)→ 2((∀x)A→ (∀x)B

)

Proof.2(A→ B)

→⟨(M3)

⟩2((∀x)(A→ B)

)→

⟨boxed axiom (4), and 2-monotonicity (4.8

⟩2((∀x)A→ (∀x)B

)Metatheorem 4.13.[Inner Leibniz Rule]

`T 2(A↔ B)→ 2(C[p := A]↔ C[p := B]

)


Proof. We prove the claim by induction on the formula C.Basis: If C is one of q (other than p), p, >, ⊥, then the result follows

trivially. If C is P (t1, . . . , tn) for some n-ary predicate symbol P (possiblythe logical “=”) and some terms t1, . . . , tn, then again the result followstrivially. For example, in the latter case we are asked to verify ` 2(A ↔B) → 2

(P (t1, . . . , tn) ↔ P (t1, . . . , tn)

)which follows from 4.1 and axiom

2(P (t1, . . . , tn)↔ P (t1, . . . , tn)

).

Induction steps:(1) If C is ¬D or D ∨G, then the result follows by tautological impli-

cation via the obvious I.H. For example, ` 2(D[p := A] ↔ D[p := B]) →2(¬D[p := A]↔ ¬D[p := B]) by 2-monotonicity.

Hence ` 2(A ↔ B) → 2(¬D[p := A] ↔ ¬D[p := B]) by I.H. andtautological implication.

(2) If C is (∀x)D, then we calculate as follows:2(A↔ B)

→⟨I.H.

⟩2(D[p := A]↔ D[p := B]

)→

⟨(M3)

⟩2

((∀x)

(D[p := A]↔ D[p := B]

))→

⟨4.7 + 2-monotonicity (4.8)

⟩2((∀x)D[p := A]↔ (∀x)D[p := B]

)We are done since (∀x)

(D[p := A]

)is the same as

((∀x)D

)[p := A].

(3) If C is 2D, then we calculate as follows:2(A↔ B)

→⟨I.H.

⟩2(D[p := A]↔ D[p := B]

)→

⟨(M2)

⟩22

(D[p := A]↔ D[p := B]

)→

⟨4.9 + 2-monotonicity (4.8)

⟩2(2D[p := A]↔ 2D[p := B]

)We are done since 2

(D[p := A]

)is the same as (2D)[p := A].

Corollary 4.14. ` 2(A↔ B)→(2C[p := A]↔ 2C[p := B]

).

Metatheorem 4.15. [Inner ∀-Introduction] If A has no free x, then `2(A→ B)→ 2(A→ (∀x)B).


Proof.2(A→ B)

→⟨inner ∀-monotonicity (4.12)

⟩2((∀x)A→ (∀x)B

)↔

⟨Leibniz (4.14): axioms (2, 3) yield 2((∀x)A↔ A)

⟩2(A→ (∀x)B

)Corollary 4.16. [Inner ∃-Introduction] If B has no free x, then

` 2(A→ B)→ 2((∃x)A→ B)

Remark 4.17. Each of the implications in 4.15 and 4.16 is promoted to anequivalence by tautological implication and the fact that the other directionholds. For example, ` 2(A → B) ← 2(A → (∀x)B) by 2-monotonicity(4.8) and the tautological consequence (A → (∀x)B) → (A → B) of aninstance of axiom (2).

Due to lack of space we present here only two examples of use of M3

as a tool for writing equational proofs of classical theorems.

Example 4.18. [∀∀-swap] To prove the classical “(∀x)(∀y)A and (∀y)(∀x)Aare mutually derivable” we prove instead ` 2(∀x)(∀y)A↔ 2(∀y)(∀x)A.

2(∀x)(∀y)A↔

⟨gen (4.5)

⟩2(∀y)A

↔⟨gen

⟩2A

↔⟨gen

⟩2(∀x)A

↔⟨gen

⟩2(∀y)(∀x)A

The classical equational proof of the above is totally disconnected since `cannot be replaced by → in the ∀-introduction direction.

Example 4.19. What if we want the classical ` (∀x)(∀y)A ↔ (∀y)(∀x)Ainstead? We can do this by proving → and ← directions separately, fol-lowed by tautological implication. E.g., for the → direction we verify


` 2((∀x)(∀y)A→ (∀y)(∀x)A

):

2((∀y)A→ A

)→

⟨inner ∀-mon.

⟩2((∀x)(∀y)A→ (∀x)A

)→

⟨inner ∀-intro.

⟩2((∀x)(∀y)A→ (∀y)(∀x)A

)References

[1] Sergei Artemov and Franco Montagna, On First-Order Theorieswith Provability Operator, J. of Symb. Logic 59/4 (1994), pp. 1139–1153.

[2] George Boolos, The unprovability of Consistency; An essayin modal logic, Cambridge University Press, Cambridge, 1979.

[3] George Boolos and Vann McGee, The degree of the set of sentencesof predicate provability logic that are true under every interpretation, J. ofSymb. Logic 52/1 (1987), pp. 165–171.

[4] Edsger W. Dijkstra and Carel S. Scholten, Predicate Calculusand Program Semantics, Springer-Verlag, New York, 1990.

[5] Melvin Fitting and Richard L. Mendelsohn, First-Order ModalLogic, Kluwer Academic Publishers, Dordrecht, 1998.

[6] David Gries and Fred B. Schneider, A Logical Approach toDiscrete Math, Springer-Verlag, New York, 1994.

[7] David Gries and Fred B. Schneider, Equational propositional logic,Information Processing Letters 53 (1995), pp. 145–152.

[8] David Gries and Fred B. Schneider, Adding the Everywhere Operatorto Propositional Logic, J. Logic Computat. 8/1 (1998), pp. 119–129.

[9] G. E. Hughes and M. J. Cresswell, An Introduction to ModalLogic, Methuen and Co. Ltd., 1968, London.

[10] G. E. Hughes and M. J. Cresswell, A New Introduction toModal Logic, Routledge, 1996, London.

[11] Franco Montagna, The predicate modal logic of provability, NotreDame J. of Formal Logic 25 (1984), pp. 179–189.

[12] C. Smorynski, Self-Reference and Modal Logic, Springer-Verlag, New York, 1985.

[13] G. Tourlakis, A Basic Formal Equational Predicate Logic-Part I,Bulletin of the Section of Logic 29:1/2 (2000), pp. 43-56.


[14] G. Tourlakis, A Basic Formal Equational Predicate Logic-Part II,Bulletin of the Section of Logic 29:3 (2000), pp 75–88.

[15] G. Tourlakis, On the soundness and Completeness of EquationalPredicate Logics, J. Logic Computat. 11/4 (2001), pp. 623–653.

[16] G. Tourlakis, Lectures in Logic and Set Theory, Volume 1,Mathematical Logic, (2003), Cambridge University Press, Cambridge.

[17] Johan van Benthem, Modal Logic and Classical Logic, Bib-liopolis, Napoli, 1983.

Department of Computer ScienceYork UniversityToronto, OntarioM3J [email protected]

Department of Mathematics and StatisticsYork UniversityToronto, OntarioM3J 1P3Canadafrancisco [email protected]

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