# Proof systems for BAT consequence relations

Proof systems for BAT consequence relations Abstract An ongoing debate about the differences between formal provability in an axiomatic system and informal provability of mathematical claims in mathematics as a whole resulted in the construction of various logics whose main purpose is to capture the inferential behaviour of the notion of informal provability, just as multiple logics of formal provability capture the behaviour of the concept of formal provability. Known logics of informal provability, based on classical logic, are unable to incorporate all intuitive principles of informal provability (most notably, reflection, which says that whatever is provable is true). One solution to this problem is to treat informal provability as an operator (Shapiro, 1985, North Holland; Reinhardt, 1986, J. Philos. Log., 15, 427–74; Koellner, 2016, Oxford University Press). Another solution is to weaken some of the intuitively adequate principles (Horsten, 2002, Hansel-Hohenhausen). Recently, in yet another approach to the issue, two three-valued non-deterministic logics of informal provability have been developed (Pawlowski and R. Urbaniak, 2016, Rev. Symbo. Log.) to overcome this difficulty. Alas, the logics have been characterized semantically and no proof systems for them are available. The purpose of this article is to define tree-like proof systems for those logics and to prove the corresponding soundness and completeness theorems. 1 Motivations The main aim of this article is to provide sound and complete proof systems for two modal logics of provability BAT and CABAT developed and discussed in [19]. We will develop tree-like proof systems in the spirit of [5, 6, 20], and prove soundness and strong completeness. These logics were developed to model an informal notion of provability in classical mathematics. Roughly speaking, a mathematical claim is informally provable if and only if it can be proved using widely accepted mathematical techniques.1 It may be the case that commonly accepted method may change through time, but at each point in time there seems to be a certain core set of widely accepted mathematical techniques which give rise to informal provability. On the other hand, we have formal proofs given by means of a proof system of a fully formalized theory. Proofs in this sense are always relative to a given proof system and a given theory. According to the proponents of the standard view2 (see [1] and [21] for an elaborate discussion on this), the relation between formal and informal concepts of provability is straightforward. Informal proofs are incomplete sketches of formal proofs. In principle, they claim, any informal proof can be converted into a proper proof in a relevant axiomatic system—if one thinks there is one system in which all formal proofs can be obtained, say ZFC, one should say the relevant axiomatic system; but this hinges on the particularities of the variant of the view. Some philosophers argue that the above picture is too simplistic and that the relation between formal and informal proofs is different. Antonutti Marfori [1] claims that there is no clear algorithm for converting a given informal proof into a proper proof in a relevant axiomatic system. Tanswell [28] claims that it is not obvious how we can identify different informal proofs with their translations. Rav [21, 22] discusses the epistemological and explanatory superiority of informal proofs over formal ones, arguing that it is not convincingly explained by the proponents of the standard view. Leitgeb [16] observed that these concepts of proofs are different. While in formal proofs the language is precisely defined and divided according to logical order, informal proofs are stated in a natural language expanded with additional mathematical vocabulary. Moreover, the connection between steps in an informal proof has a different nature than in the formal one. The former often employs steps that are supposed to be intuitively seen as truth-preserving, without explicitly following syntactically formulated rules of inference and the latter is based purely on syntactical proof forming rules. For this article the most important difference between formal and informal provability lies in the set of general principles which are sound for both kinds of provability. An important inference pattern for informal provability is the reflection schema.3 It roughly says that whatever is provable, is true. It is a well-known fact that there is no consistent formal theory extending Peano arithmetic in which all instances of the reflection schema for its own formal provability predicate are provable (see notably [17, 18]). So, it seems that the informal notion of provability cannot be formally represented in the standard setting. A couple of interesting approaches to this problem have been developed. Shapiro [25] constructed a theory called Epistemic Arithmetic (EA) where informal provability is formalized as an operator and not as a predicate. On this approach informal provability is governed by a modal logic S4.4 Shapiro defined a theorem-preserving translation $$V$$ from the language of arithmetic based on intuitionistic logic (Heyting arithmetic) to the language of EA by the following: For atomic formulas: $$V(\bar{\varphi})=\Box \varphi$$ $$V(\overline{\varphi \wedge \psi })= \Box(V(\overline{\varphi}) ) \wedge \Box(V(\overline{\psi}))$$ $$V(\overline{\varphi \vee \psi })= \Box(V(\overline{\varphi}) ) \vee \Box(V(\overline{\psi}))$$ $$V(\overline{\varphi \rightarrow \psi })=\Box( \Box(V(\overline{\varphi})) \rightarrow \Box(V(\overline{\psi}) ) )$$ $$V(\overline{\varphi \equiv \psi })=\Box( \Box(V(\overline{\varphi})) \equiv \Box(V(\overline{\psi}) ) )$$ $$V(\overline{\neg \varphi})=\Box(\neg \Box(V(\overline{\varphi})))$$ $$V(\overline{\forall\, {x}\varphi(x) })=\Box(\forall\, {x} V(\overline{\varphi(x)}))$$ $$V(\overline{\exists\, {x}\varphi(x) })=\Box(\exists\, {x} V(\overline{\varphi(x)}))$$ where $$\overline{\varphi}$$ means that $$\varphi$$ is an intuitionistic formula.5 Goodman [9] proved that the translation $$V$$ is faithful. His proof was notably simplified by Friedman [8]. This theory was further developed in three directions: by considering additional principles such as the Epistemic Church thesis (see [8, 10]), by extending the language with a truth predicate Koellner [15], Stern [27] and by a deeper analysis of the informal provability operator [11–13, 24]. The collateral damage of this approach is a serious limitation in expressive power. It is no longer possible to quantify over formulas by means of coding. Moreover, the internal logic of the operator in this theories is quite weak because of the existence of this translation. A somewhat different approach to informal provability was proposed by Horsten [13]. The idea is simple: informal provability remains as a predicate but the set of its intuitive principles is weakened. We split the set of intuitive principles into two, we add some of them to the first arithmetical theory called the basis. Next, we add the rest of the principles to the main theory and we also add all the instances of the bridging schema saying that if something is provable in the basis, it is informally provable in the main theory. This approach seems to be more promising but it has its own problems. First, there is no principled and independently motivated story explaining where the set of intuitive principles should be split. Morover, Stern [27] proved that a lot of similar systems are inconsistent. Thus, the approach seems not to be as promising as initially suspected. Recently, Pawlowski and Urbaniak [19] proposed an alternative way to build a theory of informal provability. The authors developed non-deterministic three-valued logics which can be used for building a formal theory of informal provability. On this approach the move to predicate level seems viable.6 The translations of paradoxical theorems blocking the move to predicate level for theories having classical logic in the background do not hold. The systems developed in [19] have been presented without proof theory, and the goal of this article is to provide proof systems for these logics. 2 BAT and CABAT consequence relations Let $$\mathcal{L}$$ be a modal propositional language with $$\neg, \wedge, \vee, \rightarrow$$ as Boolean connectives and a unary modal operator $$\mathtt{B}$$ whose intended interpretation is informally provable. We will use lower case Latin letters $$p,q,r,s,\ldots$$ as propositional variables. The definitions of atoms and well-formed formulas are standard. Lower case Greek letters $$\varphi,\psi,\chi,\ldots$$ are meta-variables for (possibly complex) formulas. We will use $$\Gamma$$ (possibly with indices) as a variable for finite sets of propositional WFFs. We will treat $$\neg, \vee, B$$, as primitive connectives, since the rest of them can be defined7 in a standard way. BAT-logic has three values: 0,n,1. The intended interpretation of 0 is informally refutable, 1 stands for informally provable and $$n$$ stands for neither. A BAT-assignment is a function which assigns values to all propositional variables. To obtain the possible values of a complex formula we use the following matrices: By $$x/y$$ we mean that the value of a complex formula is not determined by an assignment and it can be either $$x$$ or $$y$$. BAT-assignments do not uniquely determine values for all complex formulas. A BAT-evaluation is a function which extends an assignment, assigns values to all formulas of $$\mathcal{L}$$, according to the tables given above. By $$\Gamma$$$$\varphi$$, we mean that any BAT-evaluation which assigns $$1$$ to all formulas in $$\Gamma$$ assigns $$1$$ to formula $$\varphi$$. The rest of the standard Boolean connectives have the following matrices: The semantics is motivated by a natural division of mathematical claims into provable, refutable and undecidable. Consider a disjunction of two formulas $$p,q$$. If at least one of them is informally provable, then intuitively so is the whole disjunction. We can refute the disjunction only if we can refute both $$p$$ and $$q$$. But from the undecidability of $$p$$ and $$q$$ we cannot determinately infer the status of their disjunction. For instance, in mathematical practice the status of Continuum Hypothesis (CH) and its negation is regarded as undecided.8 But there is an agreement about the status of their disjunction—it is provable, simply because it is a substitution of excluded middle. For some other undecided sentences it is the case that their disjunction is also undecided.9 For instance, consider the disjunction of the consistency statement of ZFC and the Continuum hypothesis. Similar arguments can be developed for other connectives. Hence, indeterminism is both needed and independently motivated. I do not want to decide whether there are sentences which are absolutely undecidable—merely to allow for such a possibility. It seems that mathematicians behave as if some mathematical claim are undecidable or independent in a certain sense. For instance, some claims can seem independent because they are not provable from the commonly accepted mathematical axioms (the CH) or that currently proofs of them are not known. Thus, this informal division of mathematical claims seems to be justified in mathematical practice. BAT is too weak to be used as a logic of informal provability. For instance, it does not prove that disjunction is symmetric. A rather natural strengthening of BAT logic is obtained by closing its inner logic under classical logic: Closure condition: For every $$\mathcal{L}$$-formulas $$\varphi_1,\varphi_2,\cdots,\varphi_k, \psi$$ such that   $\varphi_1,\varphi_2,\cdots,\varphi_k\models \psi,$ where $$\models$$ is the classical consequence relation in the language with modal operator, for any BAT evaluation $$e$$, if $$e(\mathtt{B} \varphi_i)=1$$ for all $$0<i\leq k$$, then $$e(\mathtt{B} \psi)=1$$. This closure condition does justice to the notion of informal provability, since by doing real proofs we do not question classically correct inferences. By a CABAT evaluation we will mean any BAT evaluation which respects the above closure condition. We will use $$\Gamma$$$$\varphi$$ to denote CABAT consequence relation. 3 Informal provability and Löb’s theorem One of the technical features that distinguish formal and informal provability is the validity of the reflection schema. Roughly speaking, it says that if something is provable, then it is true. In mathematical practice it seems that this principle is presupposed. Usually, the existence of an informal proof (where the connection between two steps may be truth-preservation) is sufficient for taking the claim to be true. Things get interesting when we look at formal provability in a sufficiently strong arithmetical theory. Let $$T$$ be a recursively-axiomatizable theory extending Robinson arithmetic. Let $$Bew_T$$ denote its standard provability predicate. It is well known that we have following Hilbert–Bernays (HB) derivability conditions for it:   \begin{align} T\vdash \varphi \Rightarrow T\vdash Bew_T({\ulcorner {\varphi} \urcorner}) \\ \end{align} (HB1)  \begin{align} T\vdash Bew_T({\ulcorner { \varphi \rightarrow \psi} \urcorner}) \rightarrow(Bew_T ({\ulcorner {\varphi} \urcorner}) \rightarrow Bew_T ({\ulcorner {\psi} \urcorner}) )\\ \end{align} (HB2)  \begin{align} T\vdash Bew_T ({\ulcorner {\varphi} \urcorner})\rightarrow Bew_T({\ulcorner {Bew_T({\ulcorner {\varphi} \urcorner} \urcorner}) }) \end{align} (HB3) The above conditions are sufficient for proving the Löb’s theorem. Suppose that there is an arithmetical formula $$\theta$$ which behaves as if it were a provability predicate (it satisfied the above HB conditions), then we can prove the following: Theorem 1 If $$T \vdash \theta({\ulcorner {\varphi} \urcorner})\rightarrow \varphi$$ then $$T\vdash \varphi$$. The obvious consequence of the above is the fact that it is impossible to have all HB conditions together with the reflection schema. Löb’s theorem informs us how much reflection is allowed for formal provability: we can only have reflection for theorems. There is little to no discussion about the philosophical significance of Löb’s theorem. It seems that it is commonly accepted in virtue of provability obeying HB conditions. Yet, its independent philosophical motivation seems fishy. It seems that there is no reason to accept only those instances of the reflection schema for which the code of a formula used in schema is a code of a theorem. This restriction is completely artificial and stems from the fact that a particular arithmetical theory is capable of proving Löb’s theorem. One way to go about this problem is to drop Löb’s theorem and weaken some of the HB conditions and add more reflection. This is exactly the direction that was taken during the formulation of CABAT logic. Consider the following properties of CABAT: Fact 1 The following hold for CABAT: $$\mathtt{B}(\mathtt{B} \varphi \rightarrow \varphi),\mathtt{B}(\lambda \rightarrow (\mathtt{B}\lambda \rightarrow \varphi) ), \mathtt{B}(\mathtt{B}(\lambda \rightarrow \varphi) \rightarrow \lambda )$$$$\mathtt{B}\varphi$$. $$\mathtt{B}(\mathtt{B}\lambda \rightarrow \lambda), \mathtt{B}( \mathtt{B}(\neg \lambda) \rightarrow \lambda ), \mathtt{B} (\lambda \rightarrow \mathtt{B}(\neg \lambda))$$$$\lambda \wedge \neg \lambda.$$ $$\mathtt{B} (\lambda \rightarrow \neg \mathtt{B} \lambda), \mathtt{B}(\neg \mathtt{B}( \lambda)\rightarrow \lambda), \mathtt{B}(\lambda \rightarrow \mathtt{B} \lambda), \mathtt{B}( \neg \lambda \rightarrow \mathtt{B} \neg \lambda)$$$$\lambda \wedge \neg \lambda.$$ Informally speaking, 1 indicates that a translation of Löb’s theorem does not hold in CABAT. From 2 it follows that it is possible to consistently extend CABAT with all instances of the reflection schema. 3 indicates that we cannot do that for so-called provabilitation (it is a schema roughly saying that if something is true then it is provable).10 This is a good sign. Initially what we wanted for informal provability is the reflection schema. From the beginning provabilitation was an undesirable property, so not having it as a consequence and having this asymmetry is a positive feature. 4 BAT-trees We will construct sound and complete tree-like proof system for BAT-logic. The whole idea behind the proof system is to track the values of formulas which appear on branches by labelling devices called signatures. Some formulas appearing on a tree will be labelled with a letter $$n$$. Intuitively, the letter indicates that under the corresponding evaluation the formula has the value $$n$$. We will say that a formula $$\varphi$$ occurs on a tree with a signature iff it occurs on the tree in the form $$\varphi,n$$. Whenever we write formula $$\varphi$$ occurs on a tree we mean it occurs without a signature and not as a subformula of another formula $$\neg \varphi$$. Definition 1 (Root appropriate for $$\Gamma,\psi$$) Let $$\Gamma= \{\varphi_1, \varphi_2 \ldots \varphi_n \}$$, $$n\in\mathbb{N}$$ be a set of formulas and $$\psi$$ a single formula. By the root appropriate for $$\Gamma,\psi$$ we will mean the following construction: We will use syntactic rules to decompose complex formulas, extending the root appropriate for $$\Gamma, \psi$$: Negation 1: $$\ {\hskip20pt}$$ Negation 2: $$\ {\hskip20pt}$$ Disjunction 1: $$\ {\hskip20pt}$$ Disjunction 2: $$\ {\hskip20pt}$$ Disjunction 3: $$\ {\hskip20pt}$$ Provability 1: $$\ {\hskip20pt}$$ Provability 2: $$\ {\hskip20pt}$$ Provability 3: $$\ {\hskip20pt}$$ Since $$\rightarrow,\wedge,\equiv$$ are defined (in the sense of having the same matrices) as usual, we would not give their specific rules. Definition 2 (Tree appropriate for $$\Gamma,\varphi$$) By a BAT-tree appropriate for $$\Gamma,\varphi$$ we will mean any construction that starts with the root appropriate for $$\Gamma,\varphi$$ and is generated by the set of rules defined above. Definition 3 (Full BAT-tree) We say that a BAT-tree is full if it is not possible to apply any rule to extend the tree further. Definition 4 (Closed branch) Any path from the root is a branch of a BAT-tree. A branch $$b$$ is closed iff for some formula $$\varphi$$, $$\varphi$$ and $$\neg \varphi$$ occur on it, or for some formula $$\varphi$$ it occurs on it both with and without a signature. By the left/right root extension we will mean any path which goes down using the left/right path from the root. Definition 5 (Closed tree) A BAT-tree is closed iff all of its branches are closed. If at least one branch of a tree is open, the tree is open. Note that for $$\Gamma ,\varphi$$ there are many different trees, depending what was the order of the rules that we applied. In this case, trees are finite, so they are order-invariant: either all of them are closed or all of them are open. Fact 2 (Order invariance) If one tree $$t$$ appropriate for $$\Gamma, \varphi$$ is closed (open) so are all of them. Proof. Indirect. Suppose that the theorem does not hold, let $$t_0$$, $$t_1$$ be two trees appropriate for $$\Gamma, \varphi$$ where $$t_0$$ is open and $$t_1$$ is closed. Let $$b$$ be an open branch on $$t_0$$. This branch is constructed by a series of rules. Note, that some subset of these rules must have been applied to a certain branch $$b_1$$ on $$t_1$$. Either it was a proper subset then $$b$$ is an extension of $$b_1$$ and $$b_1$$ is closed, so we have a contradiction, or the both branches are generated by the same set of rules. If this is the case then again, since $$b_1$$ is closed, $$b$$ so must be. ■ Definition 6 (BAT consequence relation) $$\Gamma \vdash_{B} \varphi$$ iff a full BAT-tree appropriate for $$\Gamma$$ and $$\varphi$$ is closed. By $$\Gamma \nvdash_{B} \varphi$$ we mean that BAT-tree appropriate for $$\Gamma$$ and $$\varphi$$ is open. Now, we will prove that the above proof system is sound and complete with respect to BAT-logic. We will start with some definitions and notational conventions. Definition 7 (Faithfulness) We say that an evaluation $$e$$ is faithful to a branch b iff for all formulas $$\varphi$$ occurring on the branch, if $$\varphi$$ occurs without a signature then $$e(\varphi)=1$$ and if $$\varphi$$ occurs with a signature $$e(\varphi)=n$$. Suppose that we have a branch $$b$$ of some BAT-tree and we apply some rule to $$b$$. If the rule generates one extension with formula $$\varphi$$ or $$\varphi,n$$, we will abbreviate it as $$b^c,\varphi$$ or $$b^c, \varphi,n$$. If the rule generates two or three extensions, we will use $$b^{l}$$ (or $$b^{r}$$) to refer to the left extension (or to the right one). In case where we have three extensions, we will use $$b^l, b^c, b^r$$. Lemma 1 Let $$e$$ be a BAT-evaluation and $$b$$ a branch in a BAT-tree. If $$e$$ is faithful to $$b$$, then for any rule that can be applied to $$b$$, there is an extension $$b'$$ of $$b$$ such that $$e$$ is faithful to $$b'$$. Proof. The proof is by cases. Suppose that $$e$$ is a faithful evaluation to a branch $$b$$ up to the point where a formula $$\varphi$$ occurs. It is sufficient to check that after the application of each rule to $$b$$, the rule generates at least one extension of $$b$$ which preserves faithfulness. We will start with rules for negation: Negation 1,2 If $$\varphi=\neg\neg\psi$$, we have just one extension: $$b^c,\psi$$. The assumption implies that $$e(\neg\neg\psi)=1$$. By the matrix of negation, $$e(\psi)=1$$ as desired. If $$\varphi=\neg\psi, n$$ we have just one extension: $$b^c,\psi,n$$. By the assumption $$e(\neg\psi)=n$$ and by the matrix for negation, $$e(\psi)=n$$. Now consider the clauses for disjunction. Disjunction 1 Let $$\varphi= \psi \vee \chi$$. If an evaluation $$e$$ is faithful to $$b$$, then either $$e(\psi)=e(\chi)=n$$, or $$e(\varphi)=1$$, or $$e(\chi)=1$$. If the evaluation $$e$$ fulfills the first condition then the evaluation is faithful to $$b^r$$, if the second one to $$b^l$$, and in the third case to $$b^c$$. Disjunction 2 Suppose $$e(\psi\vee \chi)=n$$ and $$e$$ is faithful to $$b$$. Then, either $$e(\psi)=e(\chi)=n$$, or $$e(\psi)=n$$ and $$e(\chi)=0$$, or $$e(\psi)=0$$ and $$e(\chi)=n$$. In the first case $$e$$ is faithful to $$b^r$$, in the second $$e$$ is faithful to $$b^l$$, and in the third case to $$b^c$$. Disjunction 3 Suppose that $$e(\neg ( \psi \vee \chi) )=1$$ and $$e$$ is faithful to $$b$$. Then, by the matrix for negation and disjunction, $$e(\psi)=0$$ and $$e(\chi)=0$$ which implies that $$e(\neg\psi)=1=e(\neg\chi)=1$$, which shows that $$e$$ is faithful to $$b^c$$. The last set of rules is the set of rules for provability. Provability 1 Let $$\varphi=\mathtt{B}\psi$$. By the assumption we only have one extension of $$b$$ with $$\psi$$. By the assumption and the matrix for $$\mathtt{B}$$ we know that $$e(\psi)=1$$ as desired. Provability 2 Let $$\varphi=\neg \mathtt{B}\psi$$. By the assumption $$e(\neg\mathtt{B}\psi)=1$$. Thus, by the matrix of negation and $$\mathtt{B}$$, $$e(\psi)=0$$ or $$e(\psi)=1$$. In the first case $$b^l$$ is faithful to $$e$$, and in the second case $$b^r$$ preserves faithfulness. Provability 3 Let $$\varphi=\mathtt{B}\psi,n$$. By the assumption we only have one extension of $$b$$ with $$\psi,n$$. By the assumption and the matrix for $$\mathtt{B}$$ we know that $$e(\psi)=n$$ as desired. ■ Theorem 2 (Soundness) If $$\Gamma \vdash_{B} \varphi$$ then $$\Gamma$$$$\varphi$$. Proof. Assume for contradiction that $$\Gamma \vdash_{B} \varphi$$ and $$\Gamma$$$$\varphi$$. Consider a full tree appropriate for $$\Gamma$$ and $$\varphi$$. Suppose $$\varphi \in \Gamma$$. Then we have a contradiction since $$e(\varphi) = 1$$ and $$e(\varphi) \neq 1$$. Let $$\varphi \not\in \Gamma$$, by the assumption we know that there is an evaluation $$e$$ such that $$e(\psi)=1$$ for all $$\psi \in \Gamma$$ and $$e(\varphi)\neq 1$$. Since $$e(\varphi) = 0$$ or $$e(\varphi) = n$$, $$e$$ is faithful to either the left or the right extension of the root. By induction applications of lemma 1, there is a branch $$b$$ extending the root such that $$e$$ is faithful to it. Since $$\Gamma \vdash_{B} \varphi$$, we know that the branch $$b$$ is closed. Thus, for some formula $$\psi$$, either $$\psi$$ and $$\neg \psi$$ occur on it or for some formula $$\psi$$, $$\psi$$ occurs with and without a signature. In the first case $$e (\psi)=1=e(\neg\psi)$$ and in the second case $$e(\psi)=n$$ which leads to inconsistency. ■ Now we will proceed to the strong completeness of the above proof system (for finite sets11 of formulas). Definition 8 (Evaluation induced by $$b$$) Let $$b$$ be an open branch. We will say that an evaluation $$e$$ is induced by $$b$$ iff For all propositional variables $$p$$ occurring without signature $$e(p)=1$$, if $$\neg p$$ occurs on $$b$$ then $$e(p)=0$$, if $$p$$ or $$\neg p$$ occurs on the branch $$b$$ with signature, $$e(\neg p)=e(p)=n$$. Now we will prove the completeness theorem. We will start with the following lemma: Lemma 2 Let a branch $$b$$ be open and complete. Let $$E$$ be the set of evaluations induced by $$b$$. Then there is a BAT-evaluation in $$E$$ such that: if $$\varphi$$ occurs on $$b$$ without a signature then $$e(\varphi)=1$$, if $$\neg \varphi$$ occurs without a signature then $$e(\varphi)=0$$, if $$\varphi$$ or $$\neg \varphi$$ occurs with a signature then $$e(\varphi)=n$$. Proof. By induction on the complexity of $$\varphi$$. If $$\varphi$$ is a propositional variable then we are done by definition of $$e$$ being induced by $$b$$. If $$\varphi$$ is a complex formula it has one of the forms: $$\neg \psi, \psi \vee \chi,\mathtt{B}\psi$$, with or without signature for some formulas $$\psi,\chi$$ for which the lemma already hold. We will divide this proof in two cases depending whether $$\varphi$$ occurs with a signature or without. Case 1: without signature Negation: suppose that $$\varphi=\neg\psi$$. By the induction hypothesis $$e(\psi)=0$$, so by the matrix for negation $$e(\neg \psi)=1$$. If $$\varphi = \neg\neg \psi$$, then the rule for double negation elimination must have been applied to obtain $$\psi$$ on $$b$$. By the induction hypothesis $$e(\psi)=1$$, hence $$e(\neg \psi)=0$$. Disjunction: $$\varphi= \psi \vee \chi$$, by the completeness of the branch, one amongst $$\psi$$, $$\chi$$ or both $$\psi,n$$ and $$\chi, n$$ is on the branch. In the first two cases, by the induction hypothesis $$e(\psi)=1$$ or $$e(\chi)=1$$, both implying that $$e(\psi \vee \chi)=1$$. In the third case, by the matrix for disjunction there is evaluation $$e\in E$$ such that $$e(\psi \vee \chi)=1$$. If $$\varphi= \neg (\psi \vee \chi )$$, by the induction hypothesis and completeness of $$b$$, $$e(\psi)=e_v(\chi)=0$$. Thus, $$e(\varphi)=0$$. Provability: If $$\varphi=\mathtt{B}\psi$$, then by the completeness of the branch we know that $$\psi$$ is on the branch. By the induction hypothesis $$e(\psi)=1$$, so by the matrix for $$\mathtt{B}$$, we have $$e(\mathtt{B}\psi)=1$$. If $$\varphi=\neg\mathtt{B}\psi$$, then by the completeness of the branch we know that either $$\neg\psi$$ or $$\psi,n$$ is on the branch. By the induction hypothesis either $$e(\neg\psi)$$=1 thus $$e(\psi)=0$$, so $$e(\neg\mathtt{B}\psi)=1$$ or $$e(\varphi)=n$$ and by the matrix for $$\mathtt{B}$$, we know that $$e(\neg\mathtt{B}\varphi)=1$$. Case 2: with signature Negation: $$\varphi=\neg \psi,n$$. By the induction hypothesis and the matrix for negation, $$e(\psi)=n$$, which implies that $$e(\neg \psi)=n$$, as required. If $$\varphi=\neg\neg \psi,n$$. Then at some point, since the tree is complete, a rule for double negation must have been applied, so we have both: $$\psi$$ and $$\neg \psi$$ with a signature on the branch. By the induction hypothesis $$e(\neg\psi)=n$$ as required. Disjunction: $$\varphi=\psi \vee \chi,n$$ and since $$b$$ is complete, either $$\psi,n$$ and $$\neg \chi$$ or $$\chi,n$$ and $$\neg\psi$$ or both $$\psi$$ and $$\chi$$ occur on the branch with a signature. By the induction hypothesis either $$e(\psi)=n$$ and $$e(\neg \chi)=1$$, or $$e(\chi)=n$$ and $$e(\neg \psi)=1$$, or $$e(\psi)=n=e(\chi)$$. In the first two cases $$e(\psi \vee \chi)=n$$ as required. In the third, by the matrix for disjunction there is at least one evaluation in $$E$$ such that $$e(\psi\vee \chi)=n$$. If $$\varphi=\neg (\psi \vee \chi),n$$. By completeness of the branch, $$\psi \vee \chi, n$$ occurs on the branch. By the induction hypothesis and matrix for disjunction, $$e(\psi \vee \chi)=n$$, which implies that $$e(\neg( \psi \vee \chi)=n$$. Provability: If $$\varphi= \mathtt{B} \psi,n$$, then by the completeness of the branch $$\psi,n$$ is on the branch. By the induction hypothesis $$e(\psi)=n$$, so we can find an evaluation for which $$e(\mathtt{B}\psi)=n$$. ■ Theorem 3 (Completeness) Let $$\Gamma$$ be a set of propositional formulas, and $$\psi$$ a formula. If $$\Gamma$$$$\psi$$ then $$\Gamma \vdash_{B}\psi$$. Proof. By contraposition. Suppose $$\Gamma \nvdash_{B} \psi$$. By definition, there is a complete open tree appropriate for $$\Gamma$$ and $$\psi$$. Let $$b$$ be the open branch in the tree. By Lemma 2 there is an evaluation $$e$$ induced by $$b$$ such that $$e(\varphi)=1$$ for all $$\varphi \in \Gamma$$ and either $$e(\psi)=n$$, or $$e(\neg \psi)=1$$ and hence $$e(\psi)=0$$, depending on whether $$b$$ starts with the right or the left root. In both cases we have a partial evaluation which shows that $$\Gamma$$$$\psi$$. ■ 5 Filtered trees Suppose that we have a complete BAT-tree appropriate for some $$\Gamma,\varphi$$. We will devise a procedure for eliminating some of the branches in the tree in order to construct a proof system for CABAT logic. Definition 9 (Filtered branch) Let $$b$$ be a complete open branch in a BAT-tree. We will say that $$b$$ is filtered iff for all formulas $$\varphi,\psi$$ on $$b$$ the following hold: If $$\varphi$$ is a classical tautology, then it doesn’t appear with a signature or in a negated form on $$b$$. If $$\varphi$$ is a classical countertautology, it doesn’t appear with a signature or in an unnegated form on $$b$$ If $$\varphi,\psi$$ are classically equivalent then they appear in the same form: either both with signatures, or both negated, or both in the standard form: $$\varphi,\psi$$. Definition 10 (CABAT-tree) By a CABAT-tree we mean any BAT-tree whose open not filtered branches are deleted. By definition, any closed branch in a BAT-tree is not filtered. Definition 11 (Open CABAT-tree) We say that a CABAT-tree is open iff it contains an open branch $$b$$. Otherwise the tree is closed. We will use symbol $$\Gamma \vdash_c \varphi$$ to denote the fact that any full CABAT-tree appropriate for $$\Gamma,\varphi$$ is closed. In order to prove completeness and soundness of the CABAT-consequence relation with respect to CABAT-trees, we will use an alternative, equivalent formulation of CABAT by means of filtration of a set of evaluations [19]: Definition 12 (CL-filtered evaluations) Let CL be classical propositional logic. We say that a BAT-evaluation $$e$$ is CL-filtered just in case the following conditions hold: For any two formulas $$\varphi,\psi$$, if $$\models \varphi \equiv \psi$$ then $$e(\varphi)= e (\psi)$$, For any CL-tautology $$\varphi$$, $$e (\varphi)=1$$, For any CL-countertautology $$\varphi$$, $$e(\varphi)=0$$. We will use symbol $$\models_{fcl}$$ to denote a consequence relation defined by preservation of $$1$$ in the CL-filtered set of BAT-evaluations. Fact 3 $$\Gamma$$$$\varphi$$ iff $$\Gamma \models_{fcl}\varphi$$ and any CABAT-evaluation is a $$CL$$-filtered evaluation. Proof. The proof can be found in [19]. ■ Theorem 4 For any finite set of formulas $$\Gamma$$ and a formula $$\varphi$$, $$\Gamma \vdash_c \varphi$$ iff $$\Gamma$$$$\varphi$$. Proof. $$\Rightarrow$$: We will argue by contraposition. Suppose that $$\Gamma$$$$\varphi$$. We have to show that $$\Gamma \nvdash_c \varphi$$. By the definition of we know that there is a CABAT-evaluation $$e$$ which assign 1 to all formulas in $$\Gamma$$ and either 0 or n to $$\varphi$$. Note that any CABAT-evaluation is also a BAT-evaluation. By the completeness of the proof system $$\Gamma \nvdash_{B} \varphi$$. So there is an open branch in a BAT-tree that corresponds to $$e$$. We will argue that $$e$$ is filtered, at the same time showing that it is not the case that $$\Gamma \nvdash_c \varphi$$. We know that $$e$$ is a CABAT-evaluation, so by fact 3 it is a filtered BAT-evaluation. It is easy to see that the conditions of a filtered branch correspond to conditions of filtered evaluation and since the branch is generated by filtered evaluation it has to be filtered as well. In other words there is at least one open filtered branch, thus $$\Gamma \nvdash_c \varphi$$. $$\Leftarrow$$: We will argue again by contraposition. Suppose that $$\Gamma \nvdash_c \varphi$$, we will show that $$\Gamma$$$$\varphi$$. By the assumption we know that there is an open filtered branch $$b$$ on a tree. Take an evaluation $$e$$ induced by an open filtered branch $$b$$. We will argue that this BAT-evaluation is also a CABAT-evaluation. Suppose that $$e$$ is not in the set of filtered BAT-evaluation. Then it has to invalidate one of the filtration conditions. We can assume that $$e$$ is a partial evaluation restricted only to the formulas on the tree, since it is a trivial matter to extend from there the partial evaluation into a full one. Suppose that for some classical tautology $$\varphi$$, $$e(\varphi)\neq 1$$. Then either $$\varphi$$ is on the branch with a signature or in a negated form. In both cases it is impossible, since $$b$$ is filtered. Suppose that for some classical countertautology $$\varphi$$, $$e(\varphi)=1$$. Then $$\varphi$$ appears on the branch in unnegated form, but this is impossible since, $$b$$ is filtered. Suppose that for some two classically equivalent formulas $$\varphi,\psi$$, $$e(\varphi)\neq e(\psi)$$. Since $$b$$ is filtered and $$e$$ is induced by $$b$$, we know that the form $$\varphi, \psi$$ that appear on the branch is the same: either both are negated, both are without signature or both are with signature. In all the cases evaluation $$e$$ assigns to these formulas the same values. ■ Funding Research on this paper has been funded by the Research Foundation Flanders (FWO). Acknowledgements The author would like to express his gratitude to all those who commented on the earlier versions of this paper, especially to Rafal Urbaniak, Frederik Van De Putte and anonymous referees. Footnotes 1I do not claim that mathematics is a unified discipline. In some branches of mathematics some additional mathematical techniques are available whereas these additional techniques may lead to incorrect results in the other branches. Nonetheless, it seems that there is a common core of mathematical ways of proving things which is accepted thorough all its sub-disciplines. 2This view is usually shared by mathematicians, for instance Enderton [7, 10–11] It is sometimes said that “mathematics can be embedded in set theory.” This means that mathematical objects (such as numbers and differentiable functions) can be defined to be certain sets. And the theorems of mathematics (such as the fundamental theorem of calculus) then can be viewed as statements about sets. Furthermore, these theorems will be provable from our axioms. Hence our axioms provide a sufficient collection of assumptions for the development of the whole of mathematics — a remarkable fact. (In Chapter 5 we will consider further the procedure for embedding mathematics in set theory.) Also, for a bit more sophisticated version of the standard view, see [26]. 3This schema was thoroughly studied in [2–4] 4Recall $$S4$$ is axiomatized by: $$\vdash \Box\varphi\rightarrow \varphi$$ $$\vdash \Box \varphi\rightarrow \Box\Box \varphi$$ $$\vdash \Box\varphi \equiv \Box \Box\varphi$$ If $$\vdash \varphi$$ then $$\vdash \Box\varphi$$ $$\vdash \Box(\varphi\rightarrow \psi)\rightarrow (\Box\varphi\rightarrow \Box\psi).$$ 5The overline is important because the meaning of functors in intuitionistic logic is different than in classical. 6Note that in these logics informal provability is treated as an operator and not as a predicate. It is possible to develop a first-order version of these logics where informal provability is a predicate. This lies beyond the scope of this article. 7In the sense of having the same matrices. 8Philosophically speaking the question whether the Continuum Hypothesis is really undecidable is a bit more complex. Simply stating that CH and its negation are undecided does not do justice to the range of contemporary opinions on the topic. Arguments have been proposed in favour of the truth of both of these statements. However, the independence of CH and its negation from both the axioms of $$ZFC$$ set theory, and its extensions with large cardinal axioms makes them reasonable candidates for undecidable statements. 9Note that saying that the disjunction of CH and its negation is informally provable in virtue of being a substitution of excluded middle is something different than claiming that every sentence is either informally provable or refutable. We agree with the former not with the latter. 10see [19] for a more elaborate discussion of motivations and properties of BAT and CABAT. 11Since BAT and CABAT are compact, we will be interested only in finite sets of premises. References [1] Antonutti Marfori. M. Informal proofs and mathematical rigour. Studia Logica , 96, 261– 272, 2010. Google Scholar CrossRef Search ADS   [2] Arai. T. Some results on cut-elimination, provable well-orderings, induction and reflection. Annals of Pure and Applied Logic , 95, 93– 184, 1998. Google Scholar CrossRef Search ADS   [3] Beklemishev. L. D. Induction rules, reflection principles, and provably recursive functions. Annals of Pure and Applied Logic , 85, 193– 242, 1997. Google Scholar CrossRef Search ADS   [4] Beklemishev. L. D. Proof-theoretic analysis by iterated reflection. Archive for Mathematical Logic , 42, 515– 552, 2003. Google Scholar CrossRef Search ADS   [5] Beth. E. Semantic entailment and formal derivability. Mededelingen van de Koninklijke Nederlandse Akademie van Wetenschappen , 18, 309– 342, 1955. [6] Carnielli. W. A. Systematization of finite many-valued logics through the method of tableaux. The Journal of Symbolic Logic , 52, 473– 493, 1987. Google Scholar CrossRef Search ADS   [7] Enderton. H. Elements of Set Theory . Academic Press, 1977. [8] Flagg R. C. and Friedman. H. Epistemic and intuitionistic formal systems. Annals of Pure and Applied Logic , 32, 53– 60, 1986. Google Scholar CrossRef Search ADS   [9] Goodman. N. D. Epistemic arithmetic is a conservative extension of intuitionistic arithmetic. Journal of Symbolic Logic , 49, 192– 203, 1984. Google Scholar CrossRef Search ADS   [10] Halbach V. and Horsten. L. Two proof-theoretic remarks on EA + ECT. Mathematical Logic Quarterly , 46, 461– 466, 2000. Google Scholar CrossRef Search ADS   [11] Heylen. J. Modal-epistemic arithmetic and the problem of quantifying in. Synthese , 190, 89– 111, 2013. Google Scholar CrossRef Search ADS   [12] Horsten. L. Modal-epistemic variants of Shapiro’s system of epistemic arithmetic. Notre Dame Journal of Formal Logic , 35, 284– 291, 1994. Google Scholar CrossRef Search ADS   [13] Horsten. L. Provability in principle and controversial constructivistic principles. Journal of Philosophical Logic , 26, 635– 660, 1997. Google Scholar CrossRef Search ADS   [14] Horsten. L. An axiomatic investigation of provability as a primitive predicate. In Principles of Truth , pp. 203– 220. Hansel-Hohenhausen, 2002. [15] Koellner. P. Godel’s disjunction. In Godel’s Disjunction: The Scope and Limits of Mathematical Knowledge , Horsten L. and Welch, P. eds. Oxford University Press, 2016. Google Scholar CrossRef Search ADS   [16] Leitgeb. H. On formal and informal provability. In New Waves in Philosophy of Mathematics , pp. 263– 299. Palgrave Macmillan, 2009. Google Scholar CrossRef Search ADS   [17] Montague. R. Syntactical treatments of modality, with corollaries on reflexion principles and finite axiomatizability. Acta philosophica Fennica , 153– 167, 1963. [18] Myhill. J. Some remarks on the notion of proof. Journal of Philosophy , 57, 461– 471, 1960. Google Scholar CrossRef Search ADS   [19] Pawlowski P. and Urbaniak. R. Many-valued logic of informal provability: a non-deterministic strategy. The Review of Symbolic Logic , 2016. Forthcoming. [20] Priest. G. In Contradiction . Oxford University Press, 2006. Google Scholar CrossRef Search ADS   [21] Rav. Y. Why do we prove theorems? Philosophia Mathematica , 7, 5– 41, 1999. Google Scholar CrossRef Search ADS   [22] Rav. Y. A critique of a formalist-mechanist version of the justification of arguments in mathematicians’ proof practices. Philosophia Mathematica , 15, 291– 320, 2007. Google Scholar CrossRef Search ADS   [23] Reinhardt. W. N. Epistemic theories and the interpretation of Gödel’s incompleteness theorems. Journal of Philosophical Logic , 15, 427– 74, 1986. [24] Rin B. G. and Walsh. S. Realizability semantics for quantified modal logic: generalizing Flagg’s 1985 construction. The Review of Symbolic Logic , 9, 752– 809, 2016. Google Scholar CrossRef Search ADS   [25] Shapiro, S. ed. Epistemic and intuitionistic arithemtic. In Intensional Mathematics . North Holland, 1985. [26] Sjögren. J. A note on the relation between formal and informal proof. Acta Analytica , 25, 447– 458, 2010. Google Scholar CrossRef Search ADS   [27] Stern. J. Toward Predicate Approaches to Modality . Trends in Logic. Springer, 2015. [28] Tanswell. F. A problem with the dependence of informal proofs on formal proofs. Philosophia Mathematica , 23, 295– 310, 2015. Google Scholar CrossRef Search ADS   © The Author 2017. Published by Oxford University Press. All rights reserved. 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# Proof systems for BAT consequence relations

, Volume 26 (1) – Feb 1, 2018
13 pages

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Oxford University Press
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1367-0751
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1368-9894
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10.1093/jigpal/jzx055
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### Abstract

Abstract An ongoing debate about the differences between formal provability in an axiomatic system and informal provability of mathematical claims in mathematics as a whole resulted in the construction of various logics whose main purpose is to capture the inferential behaviour of the notion of informal provability, just as multiple logics of formal provability capture the behaviour of the concept of formal provability. Known logics of informal provability, based on classical logic, are unable to incorporate all intuitive principles of informal provability (most notably, reflection, which says that whatever is provable is true). One solution to this problem is to treat informal provability as an operator (Shapiro, 1985, North Holland; Reinhardt, 1986, J. Philos. Log., 15, 427–74; Koellner, 2016, Oxford University Press). Another solution is to weaken some of the intuitively adequate principles (Horsten, 2002, Hansel-Hohenhausen). Recently, in yet another approach to the issue, two three-valued non-deterministic logics of informal provability have been developed (Pawlowski and R. Urbaniak, 2016, Rev. Symbo. Log.) to overcome this difficulty. Alas, the logics have been characterized semantically and no proof systems for them are available. The purpose of this article is to define tree-like proof systems for those logics and to prove the corresponding soundness and completeness theorems. 1 Motivations The main aim of this article is to provide sound and complete proof systems for two modal logics of provability BAT and CABAT developed and discussed in [19]. We will develop tree-like proof systems in the spirit of [5, 6, 20], and prove soundness and strong completeness. These logics were developed to model an informal notion of provability in classical mathematics. Roughly speaking, a mathematical claim is informally provable if and only if it can be proved using widely accepted mathematical techniques.1 It may be the case that commonly accepted method may change through time, but at each point in time there seems to be a certain core set of widely accepted mathematical techniques which give rise to informal provability. On the other hand, we have formal proofs given by means of a proof system of a fully formalized theory. Proofs in this sense are always relative to a given proof system and a given theory. According to the proponents of the standard view2 (see [1] and [21] for an elaborate discussion on this), the relation between formal and informal concepts of provability is straightforward. Informal proofs are incomplete sketches of formal proofs. In principle, they claim, any informal proof can be converted into a proper proof in a relevant axiomatic system—if one thinks there is one system in which all formal proofs can be obtained, say ZFC, one should say the relevant axiomatic system; but this hinges on the particularities of the variant of the view. Some philosophers argue that the above picture is too simplistic and that the relation between formal and informal proofs is different. Antonutti Marfori [1] claims that there is no clear algorithm for converting a given informal proof into a proper proof in a relevant axiomatic system. Tanswell [28] claims that it is not obvious how we can identify different informal proofs with their translations. Rav [21, 22] discusses the epistemological and explanatory superiority of informal proofs over formal ones, arguing that it is not convincingly explained by the proponents of the standard view. Leitgeb [16] observed that these concepts of proofs are different. While in formal proofs the language is precisely defined and divided according to logical order, informal proofs are stated in a natural language expanded with additional mathematical vocabulary. Moreover, the connection between steps in an informal proof has a different nature than in the formal one. The former often employs steps that are supposed to be intuitively seen as truth-preserving, without explicitly following syntactically formulated rules of inference and the latter is based purely on syntactical proof forming rules. For this article the most important difference between formal and informal provability lies in the set of general principles which are sound for both kinds of provability. An important inference pattern for informal provability is the reflection schema.3 It roughly says that whatever is provable, is true. It is a well-known fact that there is no consistent formal theory extending Peano arithmetic in which all instances of the reflection schema for its own formal provability predicate are provable (see notably [17, 18]). So, it seems that the informal notion of provability cannot be formally represented in the standard setting. A couple of interesting approaches to this problem have been developed. Shapiro [25] constructed a theory called Epistemic Arithmetic (EA) where informal provability is formalized as an operator and not as a predicate. On this approach informal provability is governed by a modal logic S4.4 Shapiro defined a theorem-preserving translation $$V$$ from the language of arithmetic based on intuitionistic logic (Heyting arithmetic) to the language of EA by the following: For atomic formulas: $$V(\bar{\varphi})=\Box \varphi$$ $$V(\overline{\varphi \wedge \psi })= \Box(V(\overline{\varphi}) ) \wedge \Box(V(\overline{\psi}))$$ $$V(\overline{\varphi \vee \psi })= \Box(V(\overline{\varphi}) ) \vee \Box(V(\overline{\psi}))$$ $$V(\overline{\varphi \rightarrow \psi })=\Box( \Box(V(\overline{\varphi})) \rightarrow \Box(V(\overline{\psi}) ) )$$ $$V(\overline{\varphi \equiv \psi })=\Box( \Box(V(\overline{\varphi})) \equiv \Box(V(\overline{\psi}) ) )$$ $$V(\overline{\neg \varphi})=\Box(\neg \Box(V(\overline{\varphi})))$$ $$V(\overline{\forall\, {x}\varphi(x) })=\Box(\forall\, {x} V(\overline{\varphi(x)}))$$ $$V(\overline{\exists\, {x}\varphi(x) })=\Box(\exists\, {x} V(\overline{\varphi(x)}))$$ where $$\overline{\varphi}$$ means that $$\varphi$$ is an intuitionistic formula.5 Goodman [9] proved that the translation $$V$$ is faithful. His proof was notably simplified by Friedman [8]. This theory was further developed in three directions: by considering additional principles such as the Epistemic Church thesis (see [8, 10]), by extending the language with a truth predicate Koellner [15], Stern [27] and by a deeper analysis of the informal provability operator [11–13, 24]. The collateral damage of this approach is a serious limitation in expressive power. It is no longer possible to quantify over formulas by means of coding. Moreover, the internal logic of the operator in this theories is quite weak because of the existence of this translation. A somewhat different approach to informal provability was proposed by Horsten [13]. The idea is simple: informal provability remains as a predicate but the set of its intuitive principles is weakened. We split the set of intuitive principles into two, we add some of them to the first arithmetical theory called the basis. Next, we add the rest of the principles to the main theory and we also add all the instances of the bridging schema saying that if something is provable in the basis, it is informally provable in the main theory. This approach seems to be more promising but it has its own problems. First, there is no principled and independently motivated story explaining where the set of intuitive principles should be split. Morover, Stern [27] proved that a lot of similar systems are inconsistent. Thus, the approach seems not to be as promising as initially suspected. Recently, Pawlowski and Urbaniak [19] proposed an alternative way to build a theory of informal provability. The authors developed non-deterministic three-valued logics which can be used for building a formal theory of informal provability. On this approach the move to predicate level seems viable.6 The translations of paradoxical theorems blocking the move to predicate level for theories having classical logic in the background do not hold. The systems developed in [19] have been presented without proof theory, and the goal of this article is to provide proof systems for these logics. 2 BAT and CABAT consequence relations Let $$\mathcal{L}$$ be a modal propositional language with $$\neg, \wedge, \vee, \rightarrow$$ as Boolean connectives and a unary modal operator $$\mathtt{B}$$ whose intended interpretation is informally provable. We will use lower case Latin letters $$p,q,r,s,\ldots$$ as propositional variables. The definitions of atoms and well-formed formulas are standard. Lower case Greek letters $$\varphi,\psi,\chi,\ldots$$ are meta-variables for (possibly complex) formulas. We will use $$\Gamma$$ (possibly with indices) as a variable for finite sets of propositional WFFs. We will treat $$\neg, \vee, B$$, as primitive connectives, since the rest of them can be defined7 in a standard way. BAT-logic has three values: 0,n,1. The intended interpretation of 0 is informally refutable, 1 stands for informally provable and $$n$$ stands for neither. A BAT-assignment is a function which assigns values to all propositional variables. To obtain the possible values of a complex formula we use the following matrices: By $$x/y$$ we mean that the value of a complex formula is not determined by an assignment and it can be either $$x$$ or $$y$$. BAT-assignments do not uniquely determine values for all complex formulas. A BAT-evaluation is a function which extends an assignment, assigns values to all formulas of $$\mathcal{L}$$, according to the tables given above. By $$\Gamma$$$$\varphi$$, we mean that any BAT-evaluation which assigns $$1$$ to all formulas in $$\Gamma$$ assigns $$1$$ to formula $$\varphi$$. The rest of the standard Boolean connectives have the following matrices: The semantics is motivated by a natural division of mathematical claims into provable, refutable and undecidable. Consider a disjunction of two formulas $$p,q$$. If at least one of them is informally provable, then intuitively so is the whole disjunction. We can refute the disjunction only if we can refute both $$p$$ and $$q$$. But from the undecidability of $$p$$ and $$q$$ we cannot determinately infer the status of their disjunction. For instance, in mathematical practice the status of Continuum Hypothesis (CH) and its negation is regarded as undecided.8 But there is an agreement about the status of their disjunction—it is provable, simply because it is a substitution of excluded middle. For some other undecided sentences it is the case that their disjunction is also undecided.9 For instance, consider the disjunction of the consistency statement of ZFC and the Continuum hypothesis. Similar arguments can be developed for other connectives. Hence, indeterminism is both needed and independently motivated. I do not want to decide whether there are sentences which are absolutely undecidable—merely to allow for such a possibility. It seems that mathematicians behave as if some mathematical claim are undecidable or independent in a certain sense. For instance, some claims can seem independent because they are not provable from the commonly accepted mathematical axioms (the CH) or that currently proofs of them are not known. Thus, this informal division of mathematical claims seems to be justified in mathematical practice. BAT is too weak to be used as a logic of informal provability. For instance, it does not prove that disjunction is symmetric. A rather natural strengthening of BAT logic is obtained by closing its inner logic under classical logic: Closure condition: For every $$\mathcal{L}$$-formulas $$\varphi_1,\varphi_2,\cdots,\varphi_k, \psi$$ such that   $\varphi_1,\varphi_2,\cdots,\varphi_k\models \psi,$ where $$\models$$ is the classical consequence relation in the language with modal operator, for any BAT evaluation $$e$$, if $$e(\mathtt{B} \varphi_i)=1$$ for all $$0<i\leq k$$, then $$e(\mathtt{B} \psi)=1$$. This closure condition does justice to the notion of informal provability, since by doing real proofs we do not question classically correct inferences. By a CABAT evaluation we will mean any BAT evaluation which respects the above closure condition. We will use $$\Gamma$$$$\varphi$$ to denote CABAT consequence relation. 3 Informal provability and Löb’s theorem One of the technical features that distinguish formal and informal provability is the validity of the reflection schema. Roughly speaking, it says that if something is provable, then it is true. In mathematical practice it seems that this principle is presupposed. Usually, the existence of an informal proof (where the connection between two steps may be truth-preservation) is sufficient for taking the claim to be true. Things get interesting when we look at formal provability in a sufficiently strong arithmetical theory. Let $$T$$ be a recursively-axiomatizable theory extending Robinson arithmetic. Let $$Bew_T$$ denote its standard provability predicate. It is well known that we have following Hilbert–Bernays (HB) derivability conditions for it:   \begin{align} T\vdash \varphi \Rightarrow T\vdash Bew_T({\ulcorner {\varphi} \urcorner}) \\ \end{align} (HB1)  \begin{align} T\vdash Bew_T({\ulcorner { \varphi \rightarrow \psi} \urcorner}) \rightarrow(Bew_T ({\ulcorner {\varphi} \urcorner}) \rightarrow Bew_T ({\ulcorner {\psi} \urcorner}) )\\ \end{align} (HB2)  \begin{align} T\vdash Bew_T ({\ulcorner {\varphi} \urcorner})\rightarrow Bew_T({\ulcorner {Bew_T({\ulcorner {\varphi} \urcorner} \urcorner}) }) \end{align} (HB3) The above conditions are sufficient for proving the Löb’s theorem. Suppose that there is an arithmetical formula $$\theta$$ which behaves as if it were a provability predicate (it satisfied the above HB conditions), then we can prove the following: Theorem 1 If $$T \vdash \theta({\ulcorner {\varphi} \urcorner})\rightarrow \varphi$$ then $$T\vdash \varphi$$. The obvious consequence of the above is the fact that it is impossible to have all HB conditions together with the reflection schema. Löb’s theorem informs us how much reflection is allowed for formal provability: we can only have reflection for theorems. There is little to no discussion about the philosophical significance of Löb’s theorem. It seems that it is commonly accepted in virtue of provability obeying HB conditions. Yet, its independent philosophical motivation seems fishy. It seems that there is no reason to accept only those instances of the reflection schema for which the code of a formula used in schema is a code of a theorem. This restriction is completely artificial and stems from the fact that a particular arithmetical theory is capable of proving Löb’s theorem. One way to go about this problem is to drop Löb’s theorem and weaken some of the HB conditions and add more reflection. This is exactly the direction that was taken during the formulation of CABAT logic. Consider the following properties of CABAT: Fact 1 The following hold for CABAT: $$\mathtt{B}(\mathtt{B} \varphi \rightarrow \varphi),\mathtt{B}(\lambda \rightarrow (\mathtt{B}\lambda \rightarrow \varphi) ), \mathtt{B}(\mathtt{B}(\lambda \rightarrow \varphi) \rightarrow \lambda )$$$$\mathtt{B}\varphi$$. $$\mathtt{B}(\mathtt{B}\lambda \rightarrow \lambda), \mathtt{B}( \mathtt{B}(\neg \lambda) \rightarrow \lambda ), \mathtt{B} (\lambda \rightarrow \mathtt{B}(\neg \lambda))$$$$\lambda \wedge \neg \lambda.$$ $$\mathtt{B} (\lambda \rightarrow \neg \mathtt{B} \lambda), \mathtt{B}(\neg \mathtt{B}( \lambda)\rightarrow \lambda), \mathtt{B}(\lambda \rightarrow \mathtt{B} \lambda), \mathtt{B}( \neg \lambda \rightarrow \mathtt{B} \neg \lambda)$$$$\lambda \wedge \neg \lambda.$$ Informally speaking, 1 indicates that a translation of Löb’s theorem does not hold in CABAT. From 2 it follows that it is possible to consistently extend CABAT with all instances of the reflection schema. 3 indicates that we cannot do that for so-called provabilitation (it is a schema roughly saying that if something is true then it is provable).10 This is a good sign. Initially what we wanted for informal provability is the reflection schema. From the beginning provabilitation was an undesirable property, so not having it as a consequence and having this asymmetry is a positive feature. 4 BAT-trees We will construct sound and complete tree-like proof system for BAT-logic. The whole idea behind the proof system is to track the values of formulas which appear on branches by labelling devices called signatures. Some formulas appearing on a tree will be labelled with a letter $$n$$. Intuitively, the letter indicates that under the corresponding evaluation the formula has the value $$n$$. We will say that a formula $$\varphi$$ occurs on a tree with a signature iff it occurs on the tree in the form $$\varphi,n$$. Whenever we write formula $$\varphi$$ occurs on a tree we mean it occurs without a signature and not as a subformula of another formula $$\neg \varphi$$. Definition 1 (Root appropriate for $$\Gamma,\psi$$) Let $$\Gamma= \{\varphi_1, \varphi_2 \ldots \varphi_n \}$$, $$n\in\mathbb{N}$$ be a set of formulas and $$\psi$$ a single formula. By the root appropriate for $$\Gamma,\psi$$ we will mean the following construction: We will use syntactic rules to decompose complex formulas, extending the root appropriate for $$\Gamma, \psi$$: Negation 1: $$\ {\hskip20pt}$$ Negation 2: $$\ {\hskip20pt}$$ Disjunction 1: $$\ {\hskip20pt}$$ Disjunction 2: $$\ {\hskip20pt}$$ Disjunction 3: $$\ {\hskip20pt}$$ Provability 1: $$\ {\hskip20pt}$$ Provability 2: $$\ {\hskip20pt}$$ Provability 3: $$\ {\hskip20pt}$$ Since $$\rightarrow,\wedge,\equiv$$ are defined (in the sense of having the same matrices) as usual, we would not give their specific rules. Definition 2 (Tree appropriate for $$\Gamma,\varphi$$) By a BAT-tree appropriate for $$\Gamma,\varphi$$ we will mean any construction that starts with the root appropriate for $$\Gamma,\varphi$$ and is generated by the set of rules defined above. Definition 3 (Full BAT-tree) We say that a BAT-tree is full if it is not possible to apply any rule to extend the tree further. Definition 4 (Closed branch) Any path from the root is a branch of a BAT-tree. A branch $$b$$ is closed iff for some formula $$\varphi$$, $$\varphi$$ and $$\neg \varphi$$ occur on it, or for some formula $$\varphi$$ it occurs on it both with and without a signature. By the left/right root extension we will mean any path which goes down using the left/right path from the root. Definition 5 (Closed tree) A BAT-tree is closed iff all of its branches are closed. If at least one branch of a tree is open, the tree is open. Note that for $$\Gamma ,\varphi$$ there are many different trees, depending what was the order of the rules that we applied. In this case, trees are finite, so they are order-invariant: either all of them are closed or all of them are open. Fact 2 (Order invariance) If one tree $$t$$ appropriate for $$\Gamma, \varphi$$ is closed (open) so are all of them. Proof. Indirect. Suppose that the theorem does not hold, let $$t_0$$, $$t_1$$ be two trees appropriate for $$\Gamma, \varphi$$ where $$t_0$$ is open and $$t_1$$ is closed. Let $$b$$ be an open branch on $$t_0$$. This branch is constructed by a series of rules. Note, that some subset of these rules must have been applied to a certain branch $$b_1$$ on $$t_1$$. Either it was a proper subset then $$b$$ is an extension of $$b_1$$ and $$b_1$$ is closed, so we have a contradiction, or the both branches are generated by the same set of rules. If this is the case then again, since $$b_1$$ is closed, $$b$$ so must be. ■ Definition 6 (BAT consequence relation) $$\Gamma \vdash_{B} \varphi$$ iff a full BAT-tree appropriate for $$\Gamma$$ and $$\varphi$$ is closed. By $$\Gamma \nvdash_{B} \varphi$$ we mean that BAT-tree appropriate for $$\Gamma$$ and $$\varphi$$ is open. Now, we will prove that the above proof system is sound and complete with respect to BAT-logic. We will start with some definitions and notational conventions. Definition 7 (Faithfulness) We say that an evaluation $$e$$ is faithful to a branch b iff for all formulas $$\varphi$$ occurring on the branch, if $$\varphi$$ occurs without a signature then $$e(\varphi)=1$$ and if $$\varphi$$ occurs with a signature $$e(\varphi)=n$$. Suppose that we have a branch $$b$$ of some BAT-tree and we apply some rule to $$b$$. If the rule generates one extension with formula $$\varphi$$ or $$\varphi,n$$, we will abbreviate it as $$b^c,\varphi$$ or $$b^c, \varphi,n$$. If the rule generates two or three extensions, we will use $$b^{l}$$ (or $$b^{r}$$) to refer to the left extension (or to the right one). In case where we have three extensions, we will use $$b^l, b^c, b^r$$. Lemma 1 Let $$e$$ be a BAT-evaluation and $$b$$ a branch in a BAT-tree. If $$e$$ is faithful to $$b$$, then for any rule that can be applied to $$b$$, there is an extension $$b'$$ of $$b$$ such that $$e$$ is faithful to $$b'$$. Proof. The proof is by cases. Suppose that $$e$$ is a faithful evaluation to a branch $$b$$ up to the point where a formula $$\varphi$$ occurs. It is sufficient to check that after the application of each rule to $$b$$, the rule generates at least one extension of $$b$$ which preserves faithfulness. We will start with rules for negation: Negation 1,2 If $$\varphi=\neg\neg\psi$$, we have just one extension: $$b^c,\psi$$. The assumption implies that $$e(\neg\neg\psi)=1$$. By the matrix of negation, $$e(\psi)=1$$ as desired. If $$\varphi=\neg\psi, n$$ we have just one extension: $$b^c,\psi,n$$. By the assumption $$e(\neg\psi)=n$$ and by the matrix for negation, $$e(\psi)=n$$. Now consider the clauses for disjunction. Disjunction 1 Let $$\varphi= \psi \vee \chi$$. If an evaluation $$e$$ is faithful to $$b$$, then either $$e(\psi)=e(\chi)=n$$, or $$e(\varphi)=1$$, or $$e(\chi)=1$$. If the evaluation $$e$$ fulfills the first condition then the evaluation is faithful to $$b^r$$, if the second one to $$b^l$$, and in the third case to $$b^c$$. Disjunction 2 Suppose $$e(\psi\vee \chi)=n$$ and $$e$$ is faithful to $$b$$. Then, either $$e(\psi)=e(\chi)=n$$, or $$e(\psi)=n$$ and $$e(\chi)=0$$, or $$e(\psi)=0$$ and $$e(\chi)=n$$. In the first case $$e$$ is faithful to $$b^r$$, in the second $$e$$ is faithful to $$b^l$$, and in the third case to $$b^c$$. Disjunction 3 Suppose that $$e(\neg ( \psi \vee \chi) )=1$$ and $$e$$ is faithful to $$b$$. Then, by the matrix for negation and disjunction, $$e(\psi)=0$$ and $$e(\chi)=0$$ which implies that $$e(\neg\psi)=1=e(\neg\chi)=1$$, which shows that $$e$$ is faithful to $$b^c$$. The last set of rules is the set of rules for provability. Provability 1 Let $$\varphi=\mathtt{B}\psi$$. By the assumption we only have one extension of $$b$$ with $$\psi$$. By the assumption and the matrix for $$\mathtt{B}$$ we know that $$e(\psi)=1$$ as desired. Provability 2 Let $$\varphi=\neg \mathtt{B}\psi$$. By the assumption $$e(\neg\mathtt{B}\psi)=1$$. Thus, by the matrix of negation and $$\mathtt{B}$$, $$e(\psi)=0$$ or $$e(\psi)=1$$. In the first case $$b^l$$ is faithful to $$e$$, and in the second case $$b^r$$ preserves faithfulness. Provability 3 Let $$\varphi=\mathtt{B}\psi,n$$. By the assumption we only have one extension of $$b$$ with $$\psi,n$$. By the assumption and the matrix for $$\mathtt{B}$$ we know that $$e(\psi)=n$$ as desired. ■ Theorem 2 (Soundness) If $$\Gamma \vdash_{B} \varphi$$ then $$\Gamma$$$$\varphi$$. Proof. Assume for contradiction that $$\Gamma \vdash_{B} \varphi$$ and $$\Gamma$$$$\varphi$$. Consider a full tree appropriate for $$\Gamma$$ and $$\varphi$$. Suppose $$\varphi \in \Gamma$$. Then we have a contradiction since $$e(\varphi) = 1$$ and $$e(\varphi) \neq 1$$. Let $$\varphi \not\in \Gamma$$, by the assumption we know that there is an evaluation $$e$$ such that $$e(\psi)=1$$ for all $$\psi \in \Gamma$$ and $$e(\varphi)\neq 1$$. Since $$e(\varphi) = 0$$ or $$e(\varphi) = n$$, $$e$$ is faithful to either the left or the right extension of the root. By induction applications of lemma 1, there is a branch $$b$$ extending the root such that $$e$$ is faithful to it. Since $$\Gamma \vdash_{B} \varphi$$, we know that the branch $$b$$ is closed. Thus, for some formula $$\psi$$, either $$\psi$$ and $$\neg \psi$$ occur on it or for some formula $$\psi$$, $$\psi$$ occurs with and without a signature. In the first case $$e (\psi)=1=e(\neg\psi)$$ and in the second case $$e(\psi)=n$$ which leads to inconsistency. ■ Now we will proceed to the strong completeness of the above proof system (for finite sets11 of formulas). Definition 8 (Evaluation induced by $$b$$) Let $$b$$ be an open branch. We will say that an evaluation $$e$$ is induced by $$b$$ iff For all propositional variables $$p$$ occurring without signature $$e(p)=1$$, if $$\neg p$$ occurs on $$b$$ then $$e(p)=0$$, if $$p$$ or $$\neg p$$ occurs on the branch $$b$$ with signature, $$e(\neg p)=e(p)=n$$. Now we will prove the completeness theorem. We will start with the following lemma: Lemma 2 Let a branch $$b$$ be open and complete. Let $$E$$ be the set of evaluations induced by $$b$$. Then there is a BAT-evaluation in $$E$$ such that: if $$\varphi$$ occurs on $$b$$ without a signature then $$e(\varphi)=1$$, if $$\neg \varphi$$ occurs without a signature then $$e(\varphi)=0$$, if $$\varphi$$ or $$\neg \varphi$$ occurs with a signature then $$e(\varphi)=n$$. Proof. By induction on the complexity of $$\varphi$$. If $$\varphi$$ is a propositional variable then we are done by definition of $$e$$ being induced by $$b$$. If $$\varphi$$ is a complex formula it has one of the forms: $$\neg \psi, \psi \vee \chi,\mathtt{B}\psi$$, with or without signature for some formulas $$\psi,\chi$$ for which the lemma already hold. We will divide this proof in two cases depending whether $$\varphi$$ occurs with a signature or without. Case 1: without signature Negation: suppose that $$\varphi=\neg\psi$$. By the induction hypothesis $$e(\psi)=0$$, so by the matrix for negation $$e(\neg \psi)=1$$. If $$\varphi = \neg\neg \psi$$, then the rule for double negation elimination must have been applied to obtain $$\psi$$ on $$b$$. By the induction hypothesis $$e(\psi)=1$$, hence $$e(\neg \psi)=0$$. Disjunction: $$\varphi= \psi \vee \chi$$, by the completeness of the branch, one amongst $$\psi$$, $$\chi$$ or both $$\psi,n$$ and $$\chi, n$$ is on the branch. In the first two cases, by the induction hypothesis $$e(\psi)=1$$ or $$e(\chi)=1$$, both implying that $$e(\psi \vee \chi)=1$$. In the third case, by the matrix for disjunction there is evaluation $$e\in E$$ such that $$e(\psi \vee \chi)=1$$. If $$\varphi= \neg (\psi \vee \chi )$$, by the induction hypothesis and completeness of $$b$$, $$e(\psi)=e_v(\chi)=0$$. Thus, $$e(\varphi)=0$$. Provability: If $$\varphi=\mathtt{B}\psi$$, then by the completeness of the branch we know that $$\psi$$ is on the branch. By the induction hypothesis $$e(\psi)=1$$, so by the matrix for $$\mathtt{B}$$, we have $$e(\mathtt{B}\psi)=1$$. If $$\varphi=\neg\mathtt{B}\psi$$, then by the completeness of the branch we know that either $$\neg\psi$$ or $$\psi,n$$ is on the branch. By the induction hypothesis either $$e(\neg\psi)$$=1 thus $$e(\psi)=0$$, so $$e(\neg\mathtt{B}\psi)=1$$ or $$e(\varphi)=n$$ and by the matrix for $$\mathtt{B}$$, we know that $$e(\neg\mathtt{B}\varphi)=1$$. Case 2: with signature Negation: $$\varphi=\neg \psi,n$$. By the induction hypothesis and the matrix for negation, $$e(\psi)=n$$, which implies that $$e(\neg \psi)=n$$, as required. If $$\varphi=\neg\neg \psi,n$$. Then at some point, since the tree is complete, a rule for double negation must have been applied, so we have both: $$\psi$$ and $$\neg \psi$$ with a signature on the branch. By the induction hypothesis $$e(\neg\psi)=n$$ as required. Disjunction: $$\varphi=\psi \vee \chi,n$$ and since $$b$$ is complete, either $$\psi,n$$ and $$\neg \chi$$ or $$\chi,n$$ and $$\neg\psi$$ or both $$\psi$$ and $$\chi$$ occur on the branch with a signature. By the induction hypothesis either $$e(\psi)=n$$ and $$e(\neg \chi)=1$$, or $$e(\chi)=n$$ and $$e(\neg \psi)=1$$, or $$e(\psi)=n=e(\chi)$$. In the first two cases $$e(\psi \vee \chi)=n$$ as required. In the third, by the matrix for disjunction there is at least one evaluation in $$E$$ such that $$e(\psi\vee \chi)=n$$. If $$\varphi=\neg (\psi \vee \chi),n$$. By completeness of the branch, $$\psi \vee \chi, n$$ occurs on the branch. By the induction hypothesis and matrix for disjunction, $$e(\psi \vee \chi)=n$$, which implies that $$e(\neg( \psi \vee \chi)=n$$. Provability: If $$\varphi= \mathtt{B} \psi,n$$, then by the completeness of the branch $$\psi,n$$ is on the branch. By the induction hypothesis $$e(\psi)=n$$, so we can find an evaluation for which $$e(\mathtt{B}\psi)=n$$. ■ Theorem 3 (Completeness) Let $$\Gamma$$ be a set of propositional formulas, and $$\psi$$ a formula. If $$\Gamma$$$$\psi$$ then $$\Gamma \vdash_{B}\psi$$. Proof. By contraposition. Suppose $$\Gamma \nvdash_{B} \psi$$. By definition, there is a complete open tree appropriate for $$\Gamma$$ and $$\psi$$. Let $$b$$ be the open branch in the tree. By Lemma 2 there is an evaluation $$e$$ induced by $$b$$ such that $$e(\varphi)=1$$ for all $$\varphi \in \Gamma$$ and either $$e(\psi)=n$$, or $$e(\neg \psi)=1$$ and hence $$e(\psi)=0$$, depending on whether $$b$$ starts with the right or the left root. In both cases we have a partial evaluation which shows that $$\Gamma$$$$\psi$$. ■ 5 Filtered trees Suppose that we have a complete BAT-tree appropriate for some $$\Gamma,\varphi$$. We will devise a procedure for eliminating some of the branches in the tree in order to construct a proof system for CABAT logic. Definition 9 (Filtered branch) Let $$b$$ be a complete open branch in a BAT-tree. We will say that $$b$$ is filtered iff for all formulas $$\varphi,\psi$$ on $$b$$ the following hold: If $$\varphi$$ is a classical tautology, then it doesn’t appear with a signature or in a negated form on $$b$$. If $$\varphi$$ is a classical countertautology, it doesn’t appear with a signature or in an unnegated form on $$b$$ If $$\varphi,\psi$$ are classically equivalent then they appear in the same form: either both with signatures, or both negated, or both in the standard form: $$\varphi,\psi$$. Definition 10 (CABAT-tree) By a CABAT-tree we mean any BAT-tree whose open not filtered branches are deleted. By definition, any closed branch in a BAT-tree is not filtered. Definition 11 (Open CABAT-tree) We say that a CABAT-tree is open iff it contains an open branch $$b$$. Otherwise the tree is closed. We will use symbol $$\Gamma \vdash_c \varphi$$ to denote the fact that any full CABAT-tree appropriate for $$\Gamma,\varphi$$ is closed. In order to prove completeness and soundness of the CABAT-consequence relation with respect to CABAT-trees, we will use an alternative, equivalent formulation of CABAT by means of filtration of a set of evaluations [19]: Definition 12 (CL-filtered evaluations) Let CL be classical propositional logic. We say that a BAT-evaluation $$e$$ is CL-filtered just in case the following conditions hold: For any two formulas $$\varphi,\psi$$, if $$\models \varphi \equiv \psi$$ then $$e(\varphi)= e (\psi)$$, For any CL-tautology $$\varphi$$, $$e (\varphi)=1$$, For any CL-countertautology $$\varphi$$, $$e(\varphi)=0$$. We will use symbol $$\models_{fcl}$$ to denote a consequence relation defined by preservation of $$1$$ in the CL-filtered set of BAT-evaluations. Fact 3 $$\Gamma$$$$\varphi$$ iff $$\Gamma \models_{fcl}\varphi$$ and any CABAT-evaluation is a $$CL$$-filtered evaluation. Proof. The proof can be found in [19]. ■ Theorem 4 For any finite set of formulas $$\Gamma$$ and a formula $$\varphi$$, $$\Gamma \vdash_c \varphi$$ iff $$\Gamma$$$$\varphi$$. Proof. $$\Rightarrow$$: We will argue by contraposition. Suppose that $$\Gamma$$$$\varphi$$. We have to show that $$\Gamma \nvdash_c \varphi$$. By the definition of we know that there is a CABAT-evaluation $$e$$ which assign 1 to all formulas in $$\Gamma$$ and either 0 or n to $$\varphi$$. Note that any CABAT-evaluation is also a BAT-evaluation. By the completeness of the proof system $$\Gamma \nvdash_{B} \varphi$$. So there is an open branch in a BAT-tree that corresponds to $$e$$. We will argue that $$e$$ is filtered, at the same time showing that it is not the case that $$\Gamma \nvdash_c \varphi$$. We know that $$e$$ is a CABAT-evaluation, so by fact 3 it is a filtered BAT-evaluation. It is easy to see that the conditions of a filtered branch correspond to conditions of filtered evaluation and since the branch is generated by filtered evaluation it has to be filtered as well. In other words there is at least one open filtered branch, thus $$\Gamma \nvdash_c \varphi$$. $$\Leftarrow$$: We will argue again by contraposition. Suppose that $$\Gamma \nvdash_c \varphi$$, we will show that $$\Gamma$$$$\varphi$$. By the assumption we know that there is an open filtered branch $$b$$ on a tree. Take an evaluation $$e$$ induced by an open filtered branch $$b$$. We will argue that this BAT-evaluation is also a CABAT-evaluation. Suppose that $$e$$ is not in the set of filtered BAT-evaluation. Then it has to invalidate one of the filtration conditions. We can assume that $$e$$ is a partial evaluation restricted only to the formulas on the tree, since it is a trivial matter to extend from there the partial evaluation into a full one. Suppose that for some classical tautology $$\varphi$$, $$e(\varphi)\neq 1$$. Then either $$\varphi$$ is on the branch with a signature or in a negated form. In both cases it is impossible, since $$b$$ is filtered. Suppose that for some classical countertautology $$\varphi$$, $$e(\varphi)=1$$. Then $$\varphi$$ appears on the branch in unnegated form, but this is impossible since, $$b$$ is filtered. Suppose that for some two classically equivalent formulas $$\varphi,\psi$$, $$e(\varphi)\neq e(\psi)$$. Since $$b$$ is filtered and $$e$$ is induced by $$b$$, we know that the form $$\varphi, \psi$$ that appear on the branch is the same: either both are negated, both are without signature or both are with signature. In all the cases evaluation $$e$$ assigns to these formulas the same values. ■ Funding Research on this paper has been funded by the Research Foundation Flanders (FWO). Acknowledgements The author would like to express his gratitude to all those who commented on the earlier versions of this paper, especially to Rafal Urbaniak, Frederik Van De Putte and anonymous referees. Footnotes 1I do not claim that mathematics is a unified discipline. In some branches of mathematics some additional mathematical techniques are available whereas these additional techniques may lead to incorrect results in the other branches. Nonetheless, it seems that there is a common core of mathematical ways of proving things which is accepted thorough all its sub-disciplines. 2This view is usually shared by mathematicians, for instance Enderton [7, 10–11] It is sometimes said that “mathematics can be embedded in set theory.” This means that mathematical objects (such as numbers and differentiable functions) can be defined to be certain sets. And the theorems of mathematics (such as the fundamental theorem of calculus) then can be viewed as statements about sets. Furthermore, these theorems will be provable from our axioms. Hence our axioms provide a sufficient collection of assumptions for the development of the whole of mathematics — a remarkable fact. (In Chapter 5 we will consider further the procedure for embedding mathematics in set theory.) Also, for a bit more sophisticated version of the standard view, see [26]. 3This schema was thoroughly studied in [2–4] 4Recall $$S4$$ is axiomatized by: $$\vdash \Box\varphi\rightarrow \varphi$$ $$\vdash \Box \varphi\rightarrow \Box\Box \varphi$$ $$\vdash \Box\varphi \equiv \Box \Box\varphi$$ If $$\vdash \varphi$$ then $$\vdash \Box\varphi$$ $$\vdash \Box(\varphi\rightarrow \psi)\rightarrow (\Box\varphi\rightarrow \Box\psi).$$ 5The overline is important because the meaning of functors in intuitionistic logic is different than in classical. 6Note that in these logics informal provability is treated as an operator and not as a predicate. It is possible to develop a first-order version of these logics where informal provability is a predicate. This lies beyond the scope of this article. 7In the sense of having the same matrices. 8Philosophically speaking the question whether the Continuum Hypothesis is really undecidable is a bit more complex. Simply stating that CH and its negation are undecided does not do justice to the range of contemporary opinions on the topic. Arguments have been proposed in favour of the truth of both of these statements. However, the independence of CH and its negation from both the axioms of $$ZFC$$ set theory, and its extensions with large cardinal axioms makes them reasonable candidates for undecidable statements. 9Note that saying that the disjunction of CH and its negation is informally provable in virtue of being a substitution of excluded middle is something different than claiming that every sentence is either informally provable or refutable. 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