How Much Propositional Logic Suffices for Rosser's Essential Undecidability Theorem?
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The Review of Symbolic LogicV olume 0, Number 0, Month 202X
How Much Propositional Logic Suffices for Rosser’s EssentialUndecidability Theorem?
GUILLERMO BADIA
School of Historical and Philosophical Inquiry, University of QueenslandPETR CINTULA, PETR HÁJEK, and ANDREW TEDDERThe Institute of Computer Science of the Czech Academy of Sciences
Abstract.
In this paper we explore the following question: how weak can a logic be for Rosser’sessential undecidability result to be provable for a weak arithmetical theory? It is well known thatRobinson’s Q is essentially undecidable in intuitionistic logic, and P. Hájek proved it in the fuzzylogic BL for Grzegorczyk’s variant of Q which interprets the arithmetic operations as non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructurallogic and for a much weaker arithmetic theory, a version of Robinson’s R (with arithmetic operationsalso interpreted as mere relations). Our result is based on a structural version of the undecidabilityargument introduced by Kleene and we show that it goes well beyond the scope of the Boolean,intuitionistic, or fuzzy logic. §1. Introduction In Theorem III of Rosser (1936), it was famously established thatPeano Arithmetic was essentially undecidable (a notion only properly named later by Tarski(1949)); that is, no consistent extension of it is decidable (see Tarski et al (1953) for thestandard reference on this topic). After Rosser’s essential undecidability theorem, it wasnatural to ask for weaker theories of arithmetic that would still yield undecidability alongsimilar lines. Robinson (1950) provided the perhaps best known example of such a theory,namely, Robinson’s Arithmetic Q .The most noteworthy other essentially undecidable weakening of Q which will play aspecial role here is R , also due to Robinson (see Tarski et al (1953), p. 53 ), which allows forthe so-called structural essential undecidability argument (to borrow the terminology fromŠvejdar (2008)) due originally to Kleene (1950); see Proposition 15.9 and Theorem 15.19of Monk (1976) for the textbook version of the argument, or below for our rendering. Beloware the standard axioms of R and Q : Received August 2019 This paper is an extension and generalisation of work started by Cintula and Hájek before theunfortunate death of the latter in 2016. Hájek is included among the authors in recognition of hiswork on this topic, and with the blessing of his family, but it should be noted that he was not ableto contribute directly to the final version of this paper. See Visser (2014) for a survey of results involving R and Vaught (1962) for the original referenceregarding undecidability of this theory. Further noteworthy results on R are given by Jones andShepherdson (1983). In the axiomatisation of R , ≤ is often taken to be a defined predicate, which allows for ( R
4) to besimplified to just the left-to-right direction of our biconditional version. Since we shall take ≤ asprimitive, we include both directions. Note also that we use n to refer to a number, and n to referto the associated numeral (this will be properly defined in Section §4.). © 202X Association for Symbolic Logic doi: U064-05-FPR Final 23 June 2020 0:55 Badia, Cintula, Hájek, and Tedder ( R ) m + n = m + n ( R ) m · n = m · n ( R ) m , n for m , n ( R ) x ≤ n ↔ ( x = ∨ x = ∨ · · · ∨ x = n )( R ) x ≤ n ∨ n ≤ x ( Q ) S ( x ) , ( Q ) S ( x ) = S ( y ) → x = y ( Q ) x , → ( ∃ y )( x = S ( y ))( Q ) x + = x ( Q ) x + S ( y ) = S ( x + y )( Q ) x · = ( Q ) x · S ( y ) = ( x · y ) + x ( Q ) x ≤ y ↔ ( ∃ z ) x + z = y Given the proliferation of non-classical logical systems in the literature, the questionnot only of potential weakenings of the arithmetic theory but also of the background propositional logic become salient, and our aim here is to consider how much (or howlittle) propositional logic suffices for something like Kleene’s argument. Our strategy willconsist in a close inspection of the structural essential undecidability argument in order togeneralise it to the non-classical case, with an eye to the question: what logical principlesare actually required for the argument to work and in which non-classical settings are theyavailable?It was already well-known at least since the 1950s (see Kleene (1952)) that the undecid-ability results for Q hold intuitionistically as well as classically. Hájek (2007) showed thatit was also true for a wide variety of fuzzy logics and an arithmetic theory called Q ∽ , avariant of Q − , introduced by Grzegorczyk (2006) (see also Švejdar (2007)) which resultsfrom Q by replacing the addition and multiplication functions by ternary predicates A and M .We shall show that even against the background of a weaker logic than Hájek everconsidered, we can prove essential undecidability for a cousin (in fact, a weakening againsta certain logical background) of Q ∽ that we call R ∽ , which is a natural generalization of R . R ∽ and Q ∽ are axiomatised as follows: ( R ∽ ) A ( m , n , x ) ↔ m + n = x ( R ∽ ) M ( m , n , x ) ↔ m · n = x ( R ∽ ) m , n for m , n ( R ∽ ) x ≤ n ↔ ( x = ∨ x = ∨ · · · ∨ x = n )( R ∽ ) x ≤ n ∨ n ≤ x ( R ∽ ) x ≤ n ∨ ¬( x ≤ n )( Q ∽ ) x = y ∨ x , y ( Q ∽ ) S ( x ) , ( Q ∽ ) S ( x ) = S ( y ) → x = y ( Q ∽ ) x , → ( ∃ y )( x = S ( y ))( Q ∽ ) A ( x , , y ) ↔ x = y ( Q ∽ ) A ( x , S ( y ) , z ) ↔ ( ∃ u )( A ( x , y , u ) ∧ z = S ( u ))( Q ∽ ) M ( x , , y ) ↔ y = ( Q ∽ a ) M ( x , S ( y ) , z ) → ( ∃ u )( M ( x , y , u ) ∧ A ( u , x , z ))( Q ∽ b ) M ( m , n , u ) → ( A ( u , n , x ) → M ( m , n + , x ))( Q ∽ ) x ≤ y ↔ ( ∃ z ) A ( z , x , y ) Remark
A brief comment is in order concerning concerning these axioms. Firstnote that our additional axiom ( R ∽
6) is an instance of excluded middle. We include ( R ∽ R ∽ axioms in our stronger logic (along U064-05-FPR Final 23 June 2020 0:55
How Much Propositional Logic Suffices for Essential Undecidability? with the assumption of excluded middle for equalities)). Furthermore, it should be notedthat adding the assumption of functionality and totality of A , M to R ∽ or Q ∽ results in R , Q respectively, in classical logic. Finally, in Section §6. we comment further on the relationof our axiomatisation Q ∽ with the system studied by Hájek. In Section §2. we present the necessary basic definitions in an abstract setting andoutline the aforementioned structural essential undecidability argument. We shall see thatthe crucial ingredient of the proof is the existence of formulas separating two disjointrecursively enumerable sets: call such a formula one which strongly separates the sets inquestion. In Section §3. we present our weak logic and prove the existence of stronglyseparating formulae for R ∽ against the background of this logic in Section §4. (afterestablishing its completeness with respect to the standard model of arithmetic for Σ -formulas) and finally in Section §6. we show that, in a slightly stronger logical setting(presented in Section §5.), a variant of Hájek’s Q ∽ strengthens our R ∽ , and thus our resultsindeed generalize those of Hájek (2007). §2. The Structural Proof of Essential Undecidability The first ingredients we needare the formulas.
As we do not yet want to bind ourselves to any particular syntax(propositional or first order), let us only assume that we have a countable set of formulas Fm which contains a special subset of formulas that we will suggestively call the Σ -formulasand that for each Σ -formula ϕ there is a special formula (not necessarily a Σ -formula)which we call the negation of ϕ and suggestively denote ¬ ϕ . The second ingredient is that of a logic L which is identified with a consequence relation ⊢ L over Fm , i.e., ⊢ L ⊆ ℘ ( Fm ) × Fm and for each Γ ∪ ∆ ∪ { ϕ } ⊆ Fm we have: • Γ ∪ { ϕ } ⊢ L ϕ (Reflexivity) • If Γ ⊢ L ϕ and for each γ ∈ Γ we have ∆ ⊢ L γ , then ∆ ⊢ L ϕ (Cut)By theory we understand simply a set of formulas ; for each theory T we define the set ofits consequences in logic L as C L ( T ) = { ϕ | T ⊢ L ϕ } .As the final ingredient we need to define the notion of essential undecidability of a theory.As the underlying logic can vary, we have to be a bit more careful and formalistic in ourdefinitions of (un)decidability, extension and consistency now (so that we can be a bit loosergoing forward). When we speak about the decidability of a theory T , we actually speakabout the decidability of the set C L ( T ) , i.e., questions of decidability depends on the logic inquestion (e.g. all theories are trivially decidable in the inconsistent logic Inc = ℘ ( Fm )× Fm ).Analogously when we say that a theory T strengthens a theory S we do not speak about Note that ¬ ϕ need not be a formula per se, it is merely a notation of the negation of ϕ . Theargument below would work perfectly well if all formulas would be Σ -formulas; we however inprinciple assume it can be a proper subset to cover a wider logical setting. See mainly our notionof consistency below. The abstract framework we develop here can be seen as a still less general version of asimilar framework developed by Smullyan (1961) for abstract reasoning about undecidabilityand incompleteness. We could use his framework of representation systems in order to presentour results by fixing for one of our arithmetic theories U , a representation where S is the set of Σ -formulae in the language of U , T = C L ( U ) , and R = { ϕ | U ⊢ L ¬ ϕ } . This would put ourwork into Smullyan’s context, but we follow our current mode of presentation here because we donot need the additional generality provided by Smullyan’s approach here. Thanks are due to ananonymous referee for pointing out this connection.U064-05-FPR Final 23 June 2020 0:55 Badia, Cintula, Hájek, and Tedder simple subsethood but about the fact that T proves all axioms of S in L , i.e., S ⊆ C L ( T ) .Finally the consistency depends on the logic in question and on the class of Σ -formulas:we say that a theory T is Σ -consistent in L if for no Σ -formula ϕ we have T ⊢ L ϕ and T ⊢ L ¬ ϕ (note the our notion of consistency implies non-triviality, i.e., that there is a ϕ such that T L ϕ , and in sufficiently strong logics the converse is true as well; furthermorein a sufficiently strong arithmetical theory it is equivalent with T L = ).Therefore we should speak about Σ -consistency, decidability, and strengthening in L .To simplify matters, whenever the logic is known from the context we assume that allsubsequent uses of these three notions are parameterized by the logic in question. We alsoomit the prefix Σ , when this is clear from the context. With this convention in place, wecan give the following definition analogously to the classical case: Definition
A theory T is essentially undecidable in L if it is consistent and eachconsistent theory strengthening T is undecidable. The next proposition shows that our notion of essential undecidability is a particularlyrobust version of essential undecidability as it is preserved not only in stronger theoriesbut also in stronger logics. The statement of this proposition is made somewhat intricateto accommodate for the possibility of a stronger logic being defined over a bigger set offormulas.
Proposition
Let L be a logic over a set of formulas Fm and L ′ ⊇ L a logic overa set of formulas Fm ′ ⊇ Fm such that all Σ -formulas of Fm are Σ -formulas of Fm ′ .Furthermore, assume that T is an essentially undecidable theory in L . Then any theory S of L ′ which is consistent and strengthens T (seen as a L ′ -theory) is essentially undecidablein L ′ .Proof. It suffices to show that any theory U which is consistent in L ′ and strengthens S in L ′ is undecidable in L ′ . Define a set V = { χ ∈ Fm | U ⊢ L ′ χ } and observe V = C L ( V ) :indeed if V ⊢ L δ implies V ⊢ L ′ δ and for each χ ∈ V we have U ⊢ L ′ χ and so due to the(cut) rule of L ′ we have U ⊢ L ′ δ , i.e., δ ∈ V .Therefore V is consistent in L (otherwise we would obtain contradiction with theassumption that U is consistent in L ′ ) and strengthens T in L (actually T ⊆ U ) and soby essential undecidability of T in L we know that V is undecidable. Since χ ∈ V iff χ ∈ C L ′ ( U ) , then C L ′ ( U ) is undecidable as well. (cid:3) When we establish that a theory T of a logic L is a Rosser theory, defined below, the proofof the essential undecidability theorem can proceed as in the classical setting. We presentthe proof in some detail so it is obvious that no additional properties of L are needed. Definition
We say that a theory T is Rosser in logic L if foreach pair of disjoint recursively enumerable sets A , B ⊆ N there is a recursive series of Σ -formulas ϕ n which strongly separate A from B , that is: • n ∈ A implies T ⊢ L ϕ n . • n ∈ B implies T ⊢ L ¬ ϕ n . Theorem
Let L be a logic and T an L -consistent Rosser theory.Then T is essentially undecidable in L .Proof. First recall that from recursion theory (see e.g. Theorem 6.24 of Monk (1976)) thatwe know that there are disjoint r.e. sets A , B ⊆ N such that each X ⊇ A such that X ∩ B = ∅ is not recursive. Let ϕ n be the series of Σ -formulas guaranteed strongly separating A and B . U064-05-FPR Final 23 June 2020 0:55
How Much Propositional Logic Suffices for Essential Undecidability? Consider a consistent theory S strengthening T , define X = { n | S ⊢ ϕ n } , and notice that: • n ∈ A implies T ⊢ L ϕ n and so, due to the (cut) rule, S ⊢ ϕ n , which entails X ⊇ A . • n ∈ B implies T ⊢ L ¬ ϕ n and so S ⊢ ¬ ϕ n , which entails X ∩ B = ∅ (otherwise S would not be consistent).Thus X cannot be recursive and as it is clearly recursively reducible to C L ( S ) , S cannot bedecidable in L . (cid:3) This structural argument leaves open the question of which logical principles are neededto prove that some theory T is Rosser in L , which is where our work really begins. §3. A weaker logic The minimal logic in which we prove that R ∽ is Rosser will bedefined as a natural first-order extension of a particular propositional non-classical logic.We take propositional logics to be substitution-invariant consequence relations over a setof propositional formulas given by a propositional language L (a set of connectives witharities). It is well-known that each such logic can be presented by means of a Hilbert styleproof system consisting of axiom and rule schemata (we use the ⊲ symbol to separatepremises of a rule from its conclusion).Our basic propositional language L consists of three binary connectives (implication → , (lattice) conjunction ∧ and (lattice) disjunction ∨ ) and a propositional constant ⊥ . Negation ¬ and equivalence ↔ are defined in the following standard way: ¬ ϕ : = ϕ → ⊥ ϕ ↔ ψ : = ( ϕ → ψ ) ∧ ( ψ → ϕ ) . As usual, we assume that ¬ has the highest binding power, followed by ∧ and ∨ , andfinally → and ↔ have the lowest. Our basic propositional logic in the language L , whichwe denote as L , is given by the following Hilbert style proof system: (identity) ϕ → ϕ (weakening) ϕ ⊲ ψ → ϕ ( ∧ elim) ϕ ∧ ψ → ϕ (MP) ϕ, ϕ → ψ ⊲ ψ ( ∧ elim) ϕ ∧ ψ → ψ (assertion) ϕ ⊲ ( ϕ → ψ ) → ψ ( ∨ intro) ϕ → ϕ ∨ ψ (trans) ϕ → ψ, ψ → χ ⊲ ϕ → χ ( ∨ intro) ψ → ϕ ∨ ψ (morg) ¬ ϕ ∨ ¬ ψ ⊲ ¬( ϕ ∧ ψ ) ( ∧ intro) χ → ϕ, χ → ψ ⊲ χ → ϕ ∧ ψ ( ∨ elim) ϕ → χ, ψ → χ ⊲ ϕ ∨ ψ → χ The following theorems and derived rules are easily shown to be derivable in L : (assoc) ϕ ∨ ( ψ ∨ χ ) ↔ ( ϕ ∨ ψ ) ∨ χ (adj) ϕ, ψ ⊲ ϕ ∧ ψ (dni) ϕ ⊲ ¬¬ ϕ Note that in our weaker logic (axiomatised below) we do not include the axiom ⊥ → ϕ , so thoughwe use this suggestive notation, for now ⊥ is merely a propositional constant (however later inSection §5., we will will work with the logic SL w obeying this additional axiom, in which it willbecome a genuine falsum constant). Note the we use the same symbol for two closely related yet different axioms, we can afford thisslight abuse of language as in any given formal proof it it will be clear which of them we are using.We use the same convention for some other upcoming axiom/theorems/rules. Associativity is a simple consequence of ( ∨ intro), ( ∨ elim) and (trans); adjunction follows using(weakening) for ψ being any theorem and ( ∧ intro) together with (MP); and finally (dni) is aninstance of (assertion).U064-05-FPR Final 23 June 2020 0:55 Badia, Cintula, Hájek, and TedderRemark L extends lattice logic by the rules (morg), (weakening), and (assertion).These additional rules are chosen to fulfill specific tasks in fleshing out the structuralargument for our version of R ∽ . So if L looks somewhat artificial, that’s because it is.However, it should be noted that it is a sublogic of many well known systems, most notablyclassical and intuitionistic logic, as well as Hájek’s BL and (related to the stronger systemwe’ll present later in Section §5.) the non-distributive, non-associative Lambek calculuswith weakening in L , when this is presented with (assertion) as a rule (related systems tothis are discussed, for instance, in Galatos et al (2007)). In fact, it is easy to see that L isa proper sublogic of all these logics. Now we are ready to introduce first-order logics. We present only the syntactical aspects,excepting where we consider a very special model (namely, the natural numbers with aclassical interpretation of the vocabulary). Let us fix a propositional logic L expanding L (thus in particular, L has a propositional language which contains L ). Our notion offirst-order language is standard, i.e., it is given by a set of function and predicate symbolswith a special binary predicate symbol = for equality (we write t , s for ¬( t = s ) ). Then terms; atomic formulas, and formulas are built up as usual. In addition, the notions offree/bounded variable, substitutability, sentence, etc. are defined as usual.The first-order logic QL , for a given predicate language, is axiomatised by the substi-tutional instances of all axioms/rules of L (i.e., formulas resulting by replacing atoms byfirst-order formulas) and the following additional axiom/rule schemata (we assume that t substitutable for x in ϕ and x not free in χ ):( ∀ ins) ( ∀ x ) ϕ ( x ) → ϕ ( t ) ( ∃ intro) ϕ ( t ) → ( ∃ x ) ϕ ( x ) ( ∀ intro) χ → ψ ⊲ χ → ( ∀ x ) ψ ( ∃ elim) ψ → χ ⊲ ( ∃ x ) ψ → χ (id) x = x (com) x = y → y = x (trans) x = y → ( y = z → x = z ) ( = prin) x = y → t ( x ) = t ( y ) ( = prin) x = y → ( ϕ ( x ) → ϕ ( y )) We can easily establish the following auxiliary derived rules: (aux) ϕ ( t ) ⊲ x = t → ϕ ( x ) (gen) ϕ ⊲ ( ∀ x ) ϕ §4. Main result In order to apply the structural argument presented in Section §2. weneed to fix a logic over a set of formulas, identify the Σ -formulas, define a theory, showthat it is consistent and Rosser.We will work with the logic QL over the set of formulas for the predicate languageof Grzegorczyk’s arithmetic, i.e., the language with constant , unary function symbol S ,binary predicate symbols = and ≤ , and ternary predicate symbols A and M . We define the n th numeral n as usual: n = S n . . . S ( ) . To derive (aux), assuming ϕ ( t ) , use (assertion) to obtain ( ϕ ( t ) → ϕ ( x )) → ϕ ( x ) and so ( = prin)and (trans) complete the proof.U064-05-FPR Final 23 June 2020 0:55 How Much Propositional Logic Suffices for Essential Undecidability? Σ -formulas be those of the form ϕ = ( ∃ x ) ψ for some ∆ -formula ψ , where ∆ -formulasare those arithmetical formulas where all quantifiers are bounded, i.e., are of the form: ( ∀ x ≤ y ) ϕ = ( ∀ x )(¬( x ≤ y ) ∨ ϕ )( ∃ x ≤ y ) ϕ = ( ∃ x )( x ≤ y ∧ ϕ ) As expected the negation of a Σ -formula ϕ will be the formula ¬ ϕ . Now we can formallypresent the theory R ∽ which we mentioned in the introduction: ( R ∽ ) A ( m , n , x ) ↔ m + n = x for any n and m ( R ∽ ) M ( m , n , x ) ↔ m · n = x for any n and m ( R ∽ ) m , n for m , n ( R ∽ ) x ≤ n ↔ ( x = ∨ x = ∨ · · · ∨ x = n ) for any n ( R ∽ ) x ≤ n ∨ n ≤ x for any n ( R ∽ ) x ≤ n ∨ ¬( x ≤ n ) for any n Our goal in this section is to establish essential undecidability of R ∽ in QL . Thanks toTheorem 2.5. we know that it suffices to prove that it is QL -consistent and Rosser.By N we denote the set of natural numbers and by N we denote the standard model of thenatural numbers in our arithmetical language, i.e., the structure with constant interpretedas , S interpreted so that S ( k ) = k + , ≤ interpreted by the usual order of natural numbers,and A and M as: A N ( k , l , m ) iff k + l = mM N ( k , l , m ) iff k · l = m . In this model we can interpret all connectives and quantifiers of QL in a fully classicalway (so, for instance, ϕ → ψ should be interpreted as ¬ ϕ ∨ ψ for Boolean ¬ ). For a formula ϕ ( x , . . . , x n ) with free variables x , . . . , x n , we write: • N | = ϕ ( k , . . . , k n ) if ϕ is satisfied in N when variables x i are evaluated as k i • N | = ϕ if N | = ϕ ( k , . . . , k n ) for each k , . . . , k n ∈ N .Note that a numeral n is interpreted in N as the number n and so we have N | = ϕ ( k , . . . , k n ) iff N | = ϕ ( k , . . . , k n ) .It is easy to see that the structure N can be interpreted as a model of R ∽ in QL ; formallyspeaking we can prove the following (as in the rest of this section we work in the logic QL only, we omit it as a subscript of ⊢ ): Proposition
For each formula ϕ , R ∽ ⊢ ϕ implies N | = ϕ . For no Σ -formula ϕ do we have N | = ϕ and N | = ¬ ϕ , and this entails that R ∽ is QL -consistent. As the next step we establish the converse claim for Σ -sentences (known as Σ -completeness), i.e., for each Σ -sentence ϕ we have N | = ϕ only if R ∽ ⊢ ϕ . We prove astronger statement. Let us stress that this definition of bounded quantification is intended for the classical arithmeticalformulas; the bounded quantifiers in non-classical logics may be (and often are) defined usingimplication and strong conjunction; but as we have no need for such quantifiers in the paper, noconfusion should arise. Note that because of (assoc), we may state ( R ∽ ) with no fixed association.U064-05-FPR Final 23 June 2020 0:55 Badia, Cintula, Hájek, and TedderTheorem Σ -completeness) For each Σ -formula ϕ ( x , . . . , x n ) , we have: N | = ϕ ( k , . . . , k n ) iff R ∽ ⊢ ϕ ( k , . . . , k n ) . Proof.
One direction is a consequence of Proposition 4.7.and the fact that N | = ϕ ( k , . . . , k n ) iff N | = ϕ ( k , . . . , k n ) . We prove the converse direction first for ∆ -formulas by inductionon the complexity of ϕ . As we cannot deal with negation directly, the induction step for itwill have to take care of the next principal connective/quantifier. • ϕ is atomic First note that all terms are of the form S . . . S ( x ) for some variable x , so itsuffices to prove the claim for numerals. Observe that (1) if N | = n = m , then R ∽ ⊢ n = m by ( = prin); (2) if N | = A ( n , m , k ) (i.e., n + m = k ), then as before R ∽ ⊢ n + m = k which due to ( R ∽ ) implies R ∽ ⊢ A ( n , m , k ) ; (3) analogously N | = M ( n , m , k ) implies R ∽ ⊢ M ( n , m , k ) using ( R ∽ ) ; and finally (4) from N | = m ≤ n , we know that R ∽ ⊢ m ≤ n thanks to axiom ( R ∽ ) for x = m (as then m = m is one of the disjoints on the right-handside disjunction). • ϕ = ψ ∧ χ From the assumption we obtain N | = ψ and N | = χ . Thus by the inductionassumption R ∽ ⊢ ψ and R ∽ ⊢ χ , and so (adj) completes the proof. • ϕ = ψ ∨ χ From the assumption we obtain N | = ψ or N | = χ . Thus by the inductionassumption R ∽ ⊢ ψ or R ∽ ⊢ χ , and so ( ∨ intro) completes the proof. • ϕ = ( ∃ y ≤ x ) ψ ( y , x , . . . , x n ) From N | = ( ∃ y ≤ k ) ψ ( y , k , . . . , k n ) we obtain an m suchthat N | = m ≤ k and N | = ψ ( m , k , . . . , k n ) . Thus by the induction assumption R ∽ ⊢ m ≤ k and R ∽ ⊢ ψ ( m , k , . . . , k n ) and so (adj)and ( ∃ intro) complete the proof. • ϕ = ( ∀ y ≤ x ) ψ ( y , x , . . . , x n ) From N | = ( ∀ y ≤ k ) ψ ( y , k , . . . , k n ) we know that foreach m ≤ k we have N | = ψ ( m , k , . . . , k n ) . Thus by the induction assumption we obtain, R ∽ ⊢ ψ ( m , k , . . . , k n ) . Next we use (aux) to obtain, for m ≤ k : R ∽ ⊢ y = m → ψ ( y , k , . . . , k n ) . Thus, using ( ∨ elim), we obtain R ∽ ⊢ ( y = ∨ y = ∨ · · · ∨ y = k ) → ψ ( y , k , . . . , k n ) which, using ( R ∽ ) , ( ∨ intro), and (trans), entails: R ∽ ⊢ ( y ≤ k ) → ¬( y ≤ k ) ∨ ψ ( y , k , . . . , k n ) . As clearly using ( ∨ intro) we also have R ∽ ⊢ ¬( y ≤ k ) → ¬( y ≤ k ) ∨ ψ ( y , k , . . . , k n ) . then by ( ∨ elim) and axiom ( R ∽ ) , R ∽ ⊢ ¬( y ≤ k ) ∨ ψ ( y , k , . . . , k n ) . and (gen) complete the proof. U064-05-FPR Final 23 June 2020 0:55
How Much Propositional Logic Suffices for Essential Undecidability? • ϕ = ¬ ψ We have to distinguish the structure of ψ : − ψ is atomic Analogously to the positive case: (1) If N | = n , m , then R ∽ ⊢ ¬( n = m ) by ( R ∽ ) ; (2) if N | = ¬ A ( n , m , k ) (i.e., n + m , k ), then as before R ∽ ⊢ n + m , k whichdue to ( R ∽ ) and (trans) implies R ∽ ⊢ ¬ A ( n , m , k ) ; (3) analogously N | = ¬ M ( n , m , k ) implies R ∽ ⊢ ¬ M ( n , m , k ) using ( R ∽ ) ; and finally (4) from N | = ¬( m ≤ n ) , impliesfor each k ≤ n we have k , m , thus by ( R ∽ ) we obtain R ∽ ⊢ m = k → ⊥ . Thus by ( ∨ elim) we obtain R ∽ ⊢ m = ∨ m = · · · ∨ m = n → ⊥ and so ( R ∽ ) and and (trans) complete the proof. − ψ = α ∧ β From N | = ¬( α ∧ β ) we obtain N | = ¬ α or N | = ¬ β . Thus by the inductionassumption we know that R ∽ ⊢ α → ⊥ or R ∽ ⊢ β → ⊥ and so in both cases ( ∧ elim)and (trans) completes the proof. − ψ = α ∨ β From N | = ¬( α ∨ β ) we obtain N | = ¬ α and N | = ¬ β . Thus by theinduction assumption we know that R ∽ ⊢ α → ⊥ and R ∽ ⊢ β → ⊥ and so ( ∨ elim)completes the proof. − ψ = ¬ χ From N | = ¬¬ χ we obtain N | = χ . Thus by the induction assumption weknow that R ∽ ⊢ χ and so (dni) completes the proof. − ψ = ( ∃ y ≤ x ) χ ( y , x , . . . , x n ) From N | = ¬( ∃ y ≤ k ) χ ( y , k , . . . , k n ) we obtain foreach m ≤ k that: N | = ¬ χ ( m , k , . . . , k n ) . Thus as in the the positive case for ∀ wecould show that R ∽ ⊢ ¬( y ≤ k ) ∨ ¬ χ ( y , k , . . . , k n ) . Thus by (morg) we obtain R ∽ ⊢ y ≤ k ∧ χ ( y , k , . . . , k n ) → ⊥ and so ( ∃ elim) completes the proof. − ψ = ( ∀ y ≤ x ) χ ( y , x , . . . , x n ) From N | = ¬( ∀ y ≤ k ) χ ( y , k , . . . , k n ) we know thatthere is an m such that N | = ¬¬( m ≤ k ) and N | = ¬ χ ( m , k , . . . , k n ) . Thus by the induction assumption we obtain R ∽ ⊢ ¬( m ≤ k ) → ⊥ and R ∽ ⊢ χ ( m , k , . . . , k n ) → ⊥ which using ( ∨ elim) entails R ∽ ⊢ (¬( m ≤ k ) ∨ χ ( m , k , . . . , k n )) → ⊥ and so ( ∀ ins) and (trans) complete the proof.Finally we deal with Σ -formulas. Assume that N | = ( ∃ y ) ψ for some ∆ -formula ψ . Thenthere is some m such that N | = ψ ( m , k , . . . , k n ) . Hence, R ∽ ⊢ ψ ( m , k , . . . , k n ) and so( ∃ intro) completes the proof. (cid:3) U064-05-FPR Final 23 June 2020 0:55 Badia, Cintula, Hájek, and Tedder
Now we have all the ingredients to prove that R ∽ is Rosser in QL ; we will actually showa bit more: there is a single Σ -formula ϕ such that ϕ ( ) , ϕ ( ) , . . . is the series of formulaswitnessing that R ∽ is Rosser. Lemma
For each pair of disjoint r.e. sets A , B ⊆ N , there is a Σ -formula ϕ ( x ) suchthat n ∈ A implies R ∽ ⊢ ϕ ( n ) n ∈ B implies R ∽ ⊢ ¬ ϕ ( n ) Proof.
From the recursion theory (see e.g. Lindström (1997), Fact 1.3(b)) we know that A and B are definable using the classical Σ -formulas, i.e., formulas in the language withfunctions + and · instead of predicates A and M . Let us show that each ‘classical’ ∆ -formula is equivalent to a ∆ -formula in our language (assuming that N interprets all thesesymbols).Let us call the terms and formulas of our language simple . An classical atomic formulais almost-simple if it is simple or of the form x = t ◦ t , where ◦ is either + or · , x is avariable and t and t are simple term. Clearly replacing such a formula by A ( t , t , z ) or M ( t , t , x ) respectively yields an equivalent formula in our language. So it suffices to showthat any ‘classical’ ∆ -formula is equivalent to a ‘classical’ ∆ -formula where all atomicformulas are simple or almost-simple (we omit the adjective ‘classical’ from now on). First,we observe the validity of the following three statements of classical logic for ◦ being each + or · , terms t , t , t and variables x , x , x not occurring in those terms: • N | = t = t ◦ t ↔ ( ∃ x ≤ t )( ∃ x ≤ t )( t = x ◦ x )• N | = t ◦ t ≤ t ↔ ( ∃ x ≤ t )( ∃ x ≤ t )( x ◦ x ≤ t )• N | = t ≤ t ◦ t ↔ ( ∃ x ≤ t )( ∃ x ≤ t )( x ≤ t ∧ x ≤ t ∧ t = x ◦ x ) .Next we note that applying any of these equivalencies to any non-almost-simple atomicsubformula of a given ∆ -formula χ strictly decreases the finite multiset of depths of termsoccurring in the formula according to the standard multiset well-ordering. Thereforeexhaustively applying these equivalencies yields the required equivalent ∆ -formula withonly almost-simple atomic formulas.Thus we can assume that there are ∆ -formulas α ( x , v ) and β ( x , v ) (in our language) suchthat • n ∈ A iff N | = ( ∃ v ) α ( n , v ) . • n ∈ B iff N | = ( ∃ v ) β ( n , v ) .We define the ∆ -formula ψ ( x ) : ψ ( x , v ) = ¬(¬ α ( x , v ) ∨ ( ∃ u ≤ v ) β ( x , u )) and show that the Σ -formula ϕ ( x ) = ( ∃ v ) ψ ( x , v ) has the desired properties. The version of the following lemma concerning extensions of R over classical logic was establishedby Putnam and Smullyan (1960). A finite multiset over a set S is an ordered pair h S , f i , where f is a function f : S → N and { x ∈ S | f ( x ) > } is finite. If ≤ is a well-ordering of S , then: h S , f i ≤ m h S , g i : ⇐⇒ ∀ x ∈ S (cid:0) f ( x ) > g ( x ) = ⇒ ∃ y ∈ S (cid:0) y > x and g ( y ) > f ( y ) (cid:1)(cid:1) is a well-ordering on the set of all finite multisets over S , known as the Dershowitz–Mannaordering, for which see Dershowitz and Manna (1979).U064-05-FPR Final 23 June 2020 0:55 How Much Propositional Logic Suffices for Essential Undecidability? The first case ( n ∈ A ) is easy: observe that this entails that n < B and so we have not only N | = α ( n , m ) for some m but also that N | = ¬ β ( n , k ) for each k and so N | = ¬( ∃ u ≤ m ) β ( n , u ) .Therefore N | = ϕ ( n ) and so by Theorem 4.8. it follows that R ∽ ⊢ ϕ ( n ) .The proof of the second case ( n ∈ B ) is not so direct because ¬ ϕ is not a Σ -formula andso we cannot use Σ -completeness directly. However, because we know that N | = β ( n , m ) for some m and that N | = ¬ α ( n , k ) for each k we can use it to obtain R ∽ ⊢ β ( n , m ) and R ∽ ⊢ ¬ α ( n , k ) . Using these facts we prove that R ∽ ⊢ v ≤ m → ¬ α ( n , v ) (1) R ∽ ⊢ m ≤ v → ( ∃ u ≤ v ) β ( n , u ) . (2)and after establishing these two claims the proof that R ∽ ⊢ ¬ ϕ ( n ) easily follows: indeedusing them together with ( ∨ intro), ( ∨ elim), and ( R ∽ ) we obtain R ∽ ⊢ ¬ α ( n , v ) ∨ ( ∃ u ≤ v ) β ( n , u ) and so using (dni) we get R ∽ ⊢ ψ ( x , v ) → ⊥ and so ( ∃ elim) completes the proof.To prove (1) we start with R ∽ ⊢ ¬ α ( n , k ) and (aux) to obtain for any k ≤ m : R ∽ ⊢ v = k → ¬ α ( n , v ) Therefore the proof is done thanks to ( ∨ elim) and ( R ∽ ) .To prove (2) we use the claim R ∽ ⊢ β ( n , m ) together with (weakening), (identity), and( ∨ intro) to obtain: R ∽ ⊢ m ≤ v → m ≤ v ∧ β ( n , m ) and so ( ∃ intro) completes the proof. (cid:3) All that is left is to apply Theorem 2.5. and Proposition 2.3. to obtain the followingresults.
Theorem
The theory R ∽ is essentially undecidable in QL . Corollary
Let L be a propositional logic expanding L , L a predicate logicexpanding QL , and T a theory strengthening R ∽ in L . If T is L -consistent (i.e. proves ϕ and ¬ ϕ for no Σ -formula ϕ ), then it is essentially undecidable in L . §5. A stronger logic Our stronger propositional logic in the propositional language L , which we denote as SL w —short for “the L -fragment of the non-associative Lambekcalculus with left and right weakening”—is given by the following Hilbert style proof system(note here that SL w , unlike L , is not paraconsistent as we include the axiom ( ⊥ elim)): U064-05-FPR Final 23 June 2020 0:55 Badia, Cintula, Hájek, and Tedder (identity) ϕ → ϕ ( ∧ elim) ( ϕ ∧ ψ ) → ϕ ( ∧ elim) ( ϕ ∧ ψ ) → ψ ( ∧ intro) (( ϕ → ψ ) ∧ ( ϕ → χ )) → ( ϕ → ψ ∧ χ ) ( ∨ intro) ϕ → ( ϕ ∨ ψ ) ( ∨ intro) ψ → ( ϕ ∨ ψ ) ( ∨ elim) ( ϕ → χ ) ∧ ( ψ → χ ) → ( ϕ ∨ ψ → χ ) (weakening) ϕ → ( ψ → ϕ ) ( ⊥ elim) ⊥ → ϕ (MP) ϕ, ϕ → ψ ⊲ ψ (adj) ϕ, ψ ⊲ ϕ ∧ ψ (tone → ) ϕ → ψ, χ → θ ⊲ ( ψ → χ ) → ( ϕ → θ ) (assertion) ϕ ⊲ ( ϕ → ψ ) → ψ Let us first observe that SL w indeed extends L : the latter’s rules of (weakening), ( ∧ intro),and ( ∨ elim) follow from the corresponding axioms of SL w using the rules (MP) and (adj);(trans) follows from (tone → ) by taking ψ = χ using (identity) and (MP); and the rule(morg) follows from the axiomatic form stated below.Let us note that the rule (tone → ) can be equivalently replaced by the following tworules: (suffixing) ϕ → ψ ⊲ ( ψ → χ ) → ( ϕ → χ ) (prefixing) ϕ → ψ ⊲ ( χ → ϕ ) → ( χ → ψ ) The following theorems/rules are derivable in L : (cont) ϕ → ψ ⊲ ¬ ψ → ¬ ϕ .(morg) ¬( ϕ ∨ ψ ) ↔ ¬ ϕ ∧ ¬ ψ (morg) ¬ ϕ ∨ ¬ ψ → ¬( ϕ ∧ ψ ) (weakening) ϕ → ( ψ → ϕ ∧ ψ ) (red) ¬ ϕ → ( ϕ → χ ) (exp) ϕ ∧ ψ → χ ⊲ ϕ → ( ψ → χ ) Remark SL w can be seen as a fragment of the non-associative Lambek calculuswith left and right weakening (in the terminology of Galatos et al (2007) ( ⊥ elim) is left weakening and our (weakening) is their right weakening). It is indeed just a fragment:the language of the full logic SL w also involves fusion (residuated conjunction) and dualimplication and it is well known that they are not definable from our connectives. The non-associativity of SL w refers to the residuated conjunction, but this fact can be expressed,using implication and their statement of residuation, as the failure of formula ( ϕ → ψ ) →(( ψ → χ ) → ( ϕ → χ )) . Therefore SL w is strictly weaker than Hájek’s logic BL . SL w can be seen as the extension of the positive fragment (without distribution) of thebasic relevant logic B , studied, for instance, by Routley et al (1982), by the weakeningaxiom, the assertion rule, and negation defined in terms of ⊥ . For one direction juts consider suitable instances of (tone → ) and the axiom (identity); for conversedirection first use (prefixing) to obtain χ → θ ⊲ ( ψ → χ ) → ( ψ → θ ) and then (suffixing) toobtain ϕ → ψ ⊲ ( ψ → θ ) → ( ϕ → θ ) and the rule (trans) completes the proof. The first rule is a direct consequence of (suffixing); both (morg)s are consequences of ( ∧ intro),( ∧ elim), ( ∨ intro), ( ∨ elim), and (cont); the stronger form of (weakening) follows from applying( ∧ intro) twice on ϕ → ( ψ → ψ ) and ϕ → ( ψ → ϕ ) ; (red) follows from applying (prefixing) on( ⊥ elim); and finally to obtain (exp) apply (prefixing) twice on ϕ ∧ ψ → χ and use (weakening).U064-05-FPR Final 23 June 2020 0:55 How Much Propositional Logic Suffices for Essential Undecidability? Finally we need to prove two important facts about crisp formulae, i.e. formulae ψ where ψ ∨ ¬ ψ is provable. The first claim can be seen as converse of the derived rule (exp). Proposition
Assume that formula ϕ is crisp in T and T ⊢ ϕ → ( ψ → χ ) . Thenalso T ⊢ ϕ ∧ ψ → χ . If furthermore ψ is also a crisp formula in T , then the formula ϕ ∨ ψ is crisp in T as well.Proof. We present formal derivations of both claims • (1) ( ψ → χ ) → ( ϕ ∧ ψ → χ ) ( ∧ elim) and (suffixing)(2) ( ϕ → χ ) → ( ϕ ∧ ψ → χ ) ( ∧ elim) and (suffixing)(3) ϕ → ( ϕ ∧ ψ → χ ) ϕ → ( ψ → χ ) , (1), and (trans)(4) ¬ ϕ → ( ϕ ∧ ψ → χ ) (red), (2), and (trans)(5) ϕ ∧ ψ → χ (3), (4), ( ∨ elim) and crispness of ϕ . U064-05-FPR Final 23 June 2020 0:55 Badia, Cintula, Hájek, and Tedder • Let us denote the formula ( ϕ ∨ ψ ) ∨ ¬( ϕ ∨ ψ ) as χ (1) ϕ → χ ( ∨ intro)(2) ψ → χ ( ∨ intro)(3) ¬ ϕ ∧ ¬ ψ → χ (morg), ( ∨ intro), and (trans)(4) ¬ ϕ → (¬ ψ → χ ) (3) and (exp)(5) ϕ → (¬ ψ → χ ) (1), (weakening), and (trans)(6) ¬ ψ → χ (4), (5), ( ∨ elim), and crispness of ϕ (7) χ (2), (6), ( ∨ elim), and crispness of ψ (cid:3) §6. Q ∽ strengthens R ∽ in QSL w In this section we prove that the arithmetical theory Q ∽ proves all theorems of R ∽ against the background of QSL w . As a reminder, Q ∽ standsto Q as R ∽ stands to R and is axiomatised as follows: ( Q ∽ ) x = y ∨ x , y ( Q ∽ ) S ( x ) , ( Q ∽ ) S ( x ) = S ( y ) → x = y ( Q ∽ ) x , → ( ∃ y )( x = S ( y ))( Q ∽ ) A ( x , , y ) ↔ x = y ( Q ∽ ) A ( x , S ( y ) , z ) ↔ ( ∃ u )( A ( x , y , u ) ∧ z = S ( u ))( Q ∽ ) M ( x , , y ) ↔ y = ( Q ∽ a ) M ( x , S ( y ) , z ) → ( ∃ u )( M ( x , y , u ) ∧ A ( u , x , z ))( Q ∽ b ) M ( m , n , u ) → ( A ( u , n , x ) → M ( m , n + , x ))( Q ∽ ) x ≤ y ↔ ( ∃ z ) A ( z , x , y ) Remark
It is noteworthy that our Q ∽ differs slightly from Hájek’s. There are twoways in which this is the case. First, we include the additional axiom ( Q ∽
0) stating thatidentities are crisp. Hájek includes this as an assumption of the first order logic, whereaswe build it directly into the theory.In addition, his system includes only one axiom ( Q ∽ )—the biconditional version ofour ( Q ∽ a ) —and furthermore in his ( Q ∽ ) and ( Q ∽ BL (that of which the conditional is residual). Note that all of our Q ∽ axioms are provable from Hájek’s version of the theory in BL . First, ( Q ∽ a ) and ( Q ∽ b ) are consequences of his version stated with strong conjunction. In addition, one can proveour ( Q ∽ ) from his arithmetic theory in BL . First, since ϕ → ( ψ → ϕ ∧ ψ ) is provable inboth our systems, our left-to-right direction of ( Q ∽ ) is an immediate consequence of his(this is the same reason as that for why our ( Q ∽ a ) is a consequence of his axiom). Second, ( ∃ u )( A ( x , y , u ) ∧ z = S ( u )) → A ( x , S ( y ) , z ) is provable in his system given the crispness ofidentity, as he shows that in first order BL (which is actually strictly stronger then QBL by using the additional axiom of constants domains), whenever ϕ is crisp, then any weakconjunction of ϕ with some other formula is equivalent to their strong conjunction; seeHájek (2007), remark 2.1(1). So, since z = S ( u ) is crisp, the result follows.Hence, our results do indeed generalise Hájek’s, despite our using a variant on his Q ∽ . Note that in Q ∽ , thanks to ( Q ∽ = prin) for formulas by its equivalent formulation:( = prin) x = y ∧ ϕ ( x ) → ϕ ( y ) U064-05-FPR Final 23 June 2020 0:55
How Much Propositional Logic Suffices for Essential Undecidability? Theorem Q ∽ strengthens R ∽ in Q SL w .Proof. Before we start proving the axioms of R ∽ let us prove two useful preliminaries: Claim 1 Q ∽ ⊢ x ≤ y ↔ S ( x ) ≤ S ( y )) . Claim 2
For any formula ϕ ( x ) such that Q ∽ ⊢ ϕ ( ) and Q ∽ ⊢ ϕ ( S ( y )) we have Q ∽ ⊢ ϕ ( x ) .To prove the first claim it clearly suffices to prove A ( x , y , z ) ↔ A ( x , S ( y ) , S ( z )) anduse ( ∃ intro), ( ∃ elim) and ( Q ∽ ) to complete the proof. The proof of the left-to-rightimplication is easy: clearly A ( x , y , z ) → A ( x , y , z ) ∧ S ( z ) = S ( z ) . Thus, by ( ∃ intro) and(trans), A ( x , y , z ) → ( ∃ u )( A ( x , y , u ) ∧ S ( z ) = S ( u )) and so ( Q ∽ ) completes the proof. Theconverse implication is a bit more complex:(1) S ( z ) = S ( u ) → z = u ( Q ∽ )(2) A ( x , y , u ) ∧ S ( z ) = S ( u ) → A ( x , y , u ) ∧ z = u (1), ( ∧ intro), ( ∧ elim)(3) A ( x , y , u ) ∧ S ( z ) = S ( u ) → A ( x , y , z ) (2), ( = prin), and (trans)(4) ( ∃ u )( A ( x , y , u ) ∧ S ( z ) = S ( u )) → A ( x , y , z ) (3) and ( ∃ elim)(5) A ( x , S ( y ) , S ( z )) → A ( x , y , z ) ( Q ∽ ), (4), (trans)To prove the second claim let us use (aux) for both premises to obtain Q ∽ ⊢ x = → ϕ ( ) and Q ∽ ⊢ x = S ( y ) → ϕ ( x ) . Using ( ∃ elim) and ( Q ∽ ) we obtain Q ∽ ⊢ x , → ϕ ( x ) andthus ( ∨ elim) and ( Q ∽ ) complete the proof. ( R ∽ ) : First we prove A ( m , n , x ) → x = m + n by metainduction on n . The case n = follows from ( Q ∽ ). The inductive case:(1) A ( m , n , u ) → u = m + n by IH(2) A ( m , n , u ) ∧ x = S ( u ) → u = m + n ∧ x = S ( u ) (1), ( ∧ intro), ( ∧ elim)(3) A ( m , n , u ) ∧ x = S ( u ) → x = m + n + (2), ( = prin), (trans)(4) ( ∃ u )( A ( m , n , u ) ∧ x = S ( u )) → x = m + n + (3), ( ∃ elim)(5) A ( m , n + , u ) → x = m + n + (4), ( Q ∽ ), (trans)To prove the converse direction we show, again by metainduction on n , that A ( m , n , m + n ) and use (aux) to complete the proof. Again, the case n = follows from ( Q ∽ ) andthe inductive case follows immediately from the induction assumption and the fact that A ( x , y , z ) ↔ A ( x , S ( y ) , S ( z )) which we established in the proof of Claim 1. ( R ∽ ) : The proof is similar; first establish M ( m , n , x ) → x = m · n by metainduction on n . The case n = follows from ( Q ∽ ) . The inductive case:(1) M ( m , n , u ) → u = m · n by IH(2) M ( m , n , u ) ∧ A ( u , m , x ) → u = m · n ∧ A ( u , m , x ) (1), ( ∧ intro), ( ∧ elim)(3) M ( m , n , u ) ∧ A ( u , m , x ) → A ( m · n , m , x ) (2), ( = prin), (trans)(4) M ( m , n , u ) ∧ A ( u , m , x ) → x = m · ( n + ) (3), ( R ∽ ) , (trans)(5) ( ∃ u )( M ( m , n , u ) ∧ A ( u , m , x )) → x = m · ( n + ) (4), ( ∃ elim)(6) M ( m , n + , u ) → x = m · ( n + ) (5), ( Q ∽ a ), (trans)To prove the converse direction we show, again by metainduction on n , that M ( m , n , m · n ) and then (aux) completes the proof. Again, the case n = follows from ( Q ∽ ) . The inductivecase:(1) M ( m , n , m · n ) IH(2) A ( m · n , n , m · ( n + )) ( R ∽ ) (3) M ( m , n , m · n ) → ( A ( m · n , n , m · ( n + )) → M ( m , n + , m · ( n + ))) ( Q ∽ b )(4) M ( m , n + , m · ( n + )) (3) and MP twice U064-05-FPR Final 23 June 2020 0:55 Badia, Cintula, Hájek, and Tedder ( R ∽ ) : It suffices to establish the case where n < m . Observe that m = n → m − n = by repeated use of ( Q ∽ ) . As m − n , we know that m = n → S ( m − n − ) = and as S ( m − n − ) = → ⊥ due to ( Q ∽ ), the claim follows. ( R ∽ ) : To prove the right-to-left direction observe that for k ≤ n we have A ( n − k , k , n ) due to ( R ∽ ) and so k ≤ n using ( ∃ intro) and ( Q ∽ ) and thus x = k → x ≤ n by (aux).Repeated use of ∨ elim) then completes the proof of this direction.To prove the converse direction set ϕ n = x ≤ n → x = ∨ x = ∨ · · · ∨ x = n and weprove ϕ n by metainduction over n .For the base case, x ≤ → x = , we employ Claim 2. First note that ≤ → = follows from (weakening). Next, ( Q ∽ ) gives us that , S ( u ) and so by ( ∧ elim) and(cont), ¬( ∃ u )( A ( z , y , u ) ∧ = S ( u )) holds. By ( Q ∽ ) , it follows that ¬ A ( z , S ( y ) , ) , and thus( ⊥ elim), (trans), ( ∃ elim), and ( Q ∽ ) entail that S ( y ) ≤ → S ( y ) = . So Claim 2 deliversthe desired result.Next, observe that thanks to ( ∨ intro) and (weakening) we have ϕ n + ( ) and so if we prove ϕ n + ( S ( y )) the claim follows using Claim 2.(1) S ( y ) ≤ n + → y ≤ n Claim 1(2) S ( y ) ≤ n + → y = ∨ y = ∨ . . . y = n IH, (2), and (trans)(3) y = ∨ y = ∨ . . . y = n → S ( y ) = ∨ S ( y ) = ∨ . . . S ( y ) = n + repeated use of ( = prin), ( ∨ elim), and ( ∨ intro).(4) ϕ n + ( S ( y )) (2), (3), and (trans) ( R ∽ ) : Let us set ϕ n = x ≤ n ∨ n ≤ x and we prove ϕ n by metainduction over n . Clearlyfrom ( Q ∽ ) and ( Q ∽ ) we get that ≤ x and so by ( ∨ intro) we obtain both the base caseand also ϕ n + ( ) . Thus again proving ϕ n + ( S ( y )) completes the prof due to Claim 2.(1) y ≤ n → S ( y ) ≤ n + Claim 1(2) n ≤ y → n + ≤ S ( y ) Claim 1(3) y ≤ n ∨ n ≤ y → S ( y ) ≤ n + ∨ n + ≤ S ( y ) (2), (3), ( ∨ elim), and ( ∨ intro)(4) ϕ n + ( S ( y )) (3), IH, and (trans) ( R ∽ ) : Thanks to ( R ∽ ) we know that x ≤ n is equivalent to a disjunction of crispformulas and so it is crisp as a result of Proposition 5.13.. (cid:3) As before, it is obvious that the structure N can be interpreted as a model of Q ∽ in Q SL w ,hence Q ∽ is consistent in QSL w . Therefore the previous theorem and Corollary 4.11., allowsus to prove the following theorem, which can indeed be seen as a generalisation of Hájek’sresult in first order BL. Corollary Q ∽ is essentially undecidable in Q SL w . §7. Concluding remarks We have shown that the weak arithmetic theory R ∽ isessentially undecidable against the background of the weak propositional logic L extendedby minimal first-order axioms. The first upshot of this is that the cost of entry for essentialundecidability is very low indeed – one needs only a fairly weak arithmetic theory anda fairly weak logic. Furthermore, we can show that R ∽ is a weaker theory than even thevery weak Q ∽ in the context of a slightly stronger (but still quite weak) logic. This extendsand strengthens Hájek’s result and suggests avenues of further investigation, perhaps usingSmullyan’s representation systems, into the limits of undecidability in mathematical theoriesagainst the background provided by weak logics. U064-05-FPR Final 23 June 2020 0:55
How Much Propositional Logic Suffices for Essential Undecidability? §8. Acknowledgments P. Cintula was supported by the project GA17-04630S of theCzech Science Foundation (GAČR) and by RVO 67985807. A. Tedder was supportedby the GAČR project 18-19162Y. Thanks are due to an anonymous referee for helpfulcomments. This paper was presented at the Melbourne Logic Seminar, the conference LogicColloquium in Prague, and the conference Services to Logic: 50 Years of the Logicians’Liberation League in Mexico City. We are grateful to the audiences in all these venues.Finally, Albert Visser provided some useful comments on an earlier version of this work.BIBLIOGRAPHYL. Běhounek, P. Cintula, and P. Hájek. Introduction to Mathematical Fuzzy Logic.
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