Wright’s Strict Finitistic Logic in the Classical Metatheory: The Propositional Case

Crispin Wright in his 1982 paper argues for strict finitism, a constructive standpoint that is more restrictive than intuitionism. In its appendix, he proposes models of strict finitistic arithmetic. They are tree-like structures, formed in his strict finitistic metatheory, of equations between numerals on which concrete arithmetical sentences are evaluated. As a first step towards classical formalisation of strict finitism, we propose their counterparts in the classical metatheory with one additional assumption, and then extract the propositional part of ‘strict finitistic logic’ from it and investigate. We will provide a sound and complete pair of a Kripke-style semantics and a sequent calculus, and compare with other logics. The logic lacks the law of excluded middle and Modus Ponens and is weaker than classical logic, but stronger than any proper intermediate logics in terms of theoremhood. In fact, all the other well-known classical theorems are found to be theorems. Finally, we will make an observation that models of this semantics can be seen as nodes of an intuitionistic model.


Aims
The present paper provides and explores the propositional part of 'strict finitistic logic' according to Wright, as obtained via classical counterparts of his strict finitistic models of arithmetic, under an additional assumption we call the 'atomic prevalence condition'.'Strict finitism' is the view of mathematics according to which an object, or a number in particular, is admitted iff it is constructible in practice, and a statement holds iff it is verifiable in practice.It is constructivist, in that it accepts a number and a statement on grounds of our cognitive capabilities; and more restrictive than intuitionism, since it uses the notion of 'in practice' in place of intuitionism's 'in principle'.Strict finitism is finitistic, because as a consequence, it rejects the idea that there are infinitely many natural numbers.Strict finitistic logic is meant to be the abstract system of logical reasoning concerning actually constructible objects, based on actual verifiability.
Among the literary sources, Wright's in 1982 [7], to us, is the most philosophically inspirational and appears to be in possession of most formal-logical contents.While [7] is a philosophical rebuttal to Dummett's criticism [2]  against strict finitism, its appendix contains a proposal of the models of strict finitistic arithmetic 1 .Wright sketches an 'Outline of a strict finitist semantics for first-order arithmetic' [7, p.167], and describes notions not unlike the models and the forcing conditions of the intuitionistic Kripke semantics.A 'strict finitistic tree' is a model of strict finitistic arithmetic which represents practically possible histories of an ordinary subject's arithmetical development.Each node stands for a stage of construction, and is assigned the numbers (numerals e.g.0 + 0) and the arithmetical truths (equations e.g.0 + 0 = 0) the agent has actually constructed and learnt.The 'verification-conditions' [7, p.169] determine what formulas in the language of first-order arithmetic are forced at a node, based on the assigned truths as atoms.
Our interest lies in what kind of sentence is valid according to Wright's semantics, i.e. forced at every node in every strict finitistic tree.While there has been a certain amount of research that involves elucidation of the structure of the strict finitistic numbers2 , we hope, by our research, we could set foot in a path to understanding strict finitistic reasoning.

Methods
Wright introduced the trees and the conditions in his strict finitistic metatheory.This causes some problems.Firstly, further mathematical investigations are hindered.Strict finitism rejects mathematical induction and the law of excluded middle, the metalevel principles allowed in the classical metatheory.Secondly, it is not clear how to interpret the strict finitistic trees.They are structures 'small enough to practically construct'.But this notion can be classically inconsistent.At least the following are supposed to hold 3 .
(i) 0 is a practically constructible numeral.(ii) If a numeral is practically constructible, then so are all of its direct successors.(iii) There is a numeral that is not practically constructible.
So some path of a strict finitistic tree cannot be regarded as isomorphic to an initial segment of N, since these conditions would entail a contradiction.Indeed, these are common assumptions of strict finitism and the main reason why formalising it has been a challenge.
With these considerations, we will adopt the classical metatheory to investigate in, and consider classical structures with some constraints as our counterparts of the strict finitistic trees, rather than try to render them.Our 'models of arithmetic' will be intuitionistic Kripke frames with an assignment of arithmetical sentences, which do not involve variables.After investigating them, we define and explore an abstract, Kripke-style 'strict finitistic semantics' to capture the validity in them schematically.The models in this system, the 'strict finitistic models', deal with formulas built from variables which can stand for concrete sentences via substitution.The same kinds of constraint will be applied, and all semantic properties are then carried over.We will then define a sequent calculus, and prove it to be sound and complete with regard to the strict finitistic semantics.
We will maintain Wright's forcing conditions.His condition of negation (and that of implication, section 1.3) will then affect how we interpret the forcing relation.Let us write k |= A for a sentence or formula A holding at a node k of a tree-like structure we consider.The condition is k |= ¬A iff l |= A for any node l.Thus strict finitistic negation stands for practical unverifiability.Also, it is 'global' in that if ¬A holds somewhere in the structure, it holds everywhere.Therefore A of k |= ¬A may involve an expression the agent has not constructed, and as a result k |= ¬A may not represent the agent's judgement at step k.Rather a safer interpretation would be that k |= A represents the agent's knowledge at k if A is atomic; otherwise (or in all cases), it only displays our judgement as the classically-thinking model-maker. 4e assume the following 'finitistic' constraints on our models of arithmetic.
(i) The frame is finite. 3We will touch upon a fourth condition and see how it is realised in our system, after proposition 2.5. 4 We thank an anonymous referee for suggesting that therefore, some might say that this article's logic is the logic of our judgement about practical constructibility, not the logic of strict finitistic reasoning itself.It would indeed be a sensible reaction, and we agree that it might be the case.The genuine 'logic of strict finitistic reasoning' may be the one that captures an ordinary subject's judgement about their own practical constructibility.It may be Wright's original semantics in the strict finitistic metatheory.It is a model-theoretic approach to understanding an ordinary subject's assertion, e.g., that a statement is unverifiable to themselves.The models there are strict finitistic themselves, since they need be e.g.'small enough'.We hope that this article contributes to making the semantics understandable to us, classical thinkers, by providing its classical counterpart.
(ii) Each node forces at most finitely many atomic sentences (the finite verification condition).
Each model stands for one collection of the entire possibilities of an agent's actual construction and verification.Our models will be finite by (i), but indeed include ones that are not 'small enough'.We accept them as part of our classical idealisation, in hope that the totality of the valid sentences is not affected when seen schematically.We will assume an auxiliary condition.We say a sentence A is 'assertible' in a model of arithmetic if k |= A for some k.A sentence B is 'prevalent' if where ≤ is the partial order on the structure 5 .
Assertibility is practical verifiability in a weaker sense, while prevalence is stronger, since A is verified in a case and B is eventually verified in any case.
A model of arithmetic has the 'atomic prevalence property' if all assertible, atomic sentences are prevalent.Throughout this article, we will only consider ones with this property.This is a strong assumption that collapses the two notions, and indeed, Wright noted that we cannot generally assume it [7,  pp.168-9].There is no guarantee that a truth found in one history is found in every other history, because an agent's construction power would reduce as the construction continues.A truth may require so tremendous an amount of power, that they cannot discover it after obtaining some others.However, to make our study feasible, we will restrict the scope of discussion.We will touch upon this issue in the ending remarks.We will also use, only as a tool, a structure S ∞ with ω-sequences as its branches.Our models of arithmetic will be defined to be S ∞ 's finite parts.Certainly, it does not satisfy the 'finitistic' constraints.But it would appear to be a natural extension of the models, has the atomic prevalence property, and does not affect the totality of the valid sentences (proposition 2.10).

Characteristics (i): Strict finitistic implication
Strict finitistic implication A → B means that B holds after A sooner or later.We translate Wright's condition as that This is a version of intuitionistic implication that allows for a time-gap between A and B. As we see it, Wright did not restrict the gap, since every strict finitistic tree is supposed to be small enough.As part of our classical idealisation, we accept any finite gap (cf. the ending remarks). 6ur implication generalises the atomic prevalence property to all complex sentences (proposition 2.4).Under prevalence, an assertible, atomic sentence describes a fact in the realm of the foreseeable future, which figuratively is a 'candy bin' of a jumble of facts, regardless of the order of their verification.Implication's condition matches this idea: A → B means that if A holds in the future, so does B, regardless of which comes first.As a result, assertibility and prevalence are equivalent in general, and stand for practical verifiability equally.
This idea of 'order-less truths' may suggest classical logic.In fact, assertibility can be calculated just as the classical truth-values (proposition 2.5), and assertibility in all strict finitistic models coincides with provability in classical propositional logic CPC (corollary 2.24) 7 .Accordingly, most of the classical tautologies (and rules) are found valid (propositions 2.5, 2.26).Exceptions are the law of the excluded middle and Modus Ponens (proposition 2.6).
With generalised prevalence, we can aptly treat ¬A as an abbreviation of A → ⊥, where ⊥ stands for a statement practically unverifiable.Formally, the prevalence property implies that k |= A → ⊥ iff l |= A for all l (cf.proposition 2.4).Conceptually, too, A → ⊥ and ¬A coincide.A → ⊥ means that if A eventually holds, then so does ⊥.Since the consequent is impossible, it means that A does not eventually hold, i.e., is practically unverifiable.

Characteristics (ii): Relation with intuitionism
Some take the following as the conceptual relation between strict finitism and intuitionism: • a statement is verifiable in principle iff it is verifiable in practice with some finite extension of practical resources8 .
We will prove results that seem to formalise this identification scheme.Strict finitistic models with finite frames, arranged in ascending order of practical verification power, can be viewed as the nodes of an intuitionistic model, and vice versa.We will formalise this order using the notion of 'generations'.Let U be a set of strict finitistic finite models.Then the set A W of the variables assertible in W ∈ U determines the assertibility, and hence the practical verifiability of every formula in W .Let us write W W ′ if W is an initial part of another model W ′ and A W ⊆ A W ′ .Then W W ′ means that W ′ represents the same agent as W , but with more verificatory resources, typically from a later generation.We will call G := U, a 'generation structure' ('g-structure').This is an intuitionistic frame with strict finitistic finite models as its nodes, and thereby serves as a framework to express practical verifiability with increasing power.Here two evaluations on G are naturally induced, one intuitionistic, and the other in the strict finitistic style.Since A W summarises practical verifiability in W as a node of G, we define a valuation v by W ∈ v(p) iff p ∈ A W . Then v defines an intuitionistic forcing relation .Also a new evaluation, the 'generation forcing relation' , is defined between the pairs of a W ∈ U and a node of W and the formulas, using the valuation in each W ∈ U for the base case.Implication will be defined so that each W is closed under it: W, k A → B means no matter how power will increase, B comes after A 'in a short while', within the same generation; and hence thus defined is a strict finitistic forcing relation that takes into account the increasement in power.
Validity according to is being forced at all pairs.We will show that an intuitionistic model with the finite verification condition induces a g-structure; and the induced evaluations and always correspond (propositions 2.30, 2.31).Further, we will prove that validity in all g-structures coincides with provability in intuitionistic propositional logic IPC (proposition 2.32).
These results may formalise the aforementioned conceptual relation.Any statement verifiable in principle is verified in practice within finitely many steps with some extension of power; and the totality of the statements verifiable in principle coincides with that of those in practice under extension of power.

The structure of the paper and comparison results
In section 2.1, we define two languages L S and L SF .In 2.2, we define S ∞ with regard to L S and our models of arithmetic via S ∞ ; and look into their semantic properties.The formulas in this context are concrete sentences.To capture validity schematically, we will in section 2.3 use L SF , a standard language of propositional logic with variables, to define and study the strict finitistic semantics.In section 2.4, we define a sequent calculus SF and show that it is sound and complete with respect to the strict finitistic semantics.Our completeness proof is in the usual Henkin-style.
In section 2.5 we conduct a comparison with intermediate logics.Beside the relation mentioned right above, we will prove that our logic is weaker than CPC, but stronger than any proper intermediate logic -in this context, we identify each logic with the set of their theorems, and write SF for strict finitistic logic.An intermediate logic is obtained by adding CPC's theorems to IPC and taking the closure under Modus Ponens and substitution of formulas.A resulting logic is 'proper' if it is not CPC.It is known that the intermediate logics form a lattice under the set-theoretic inclusion, and it is bounded by CPC and IPC as its maximum and minimum, respectively.CPC has a unique direct predecessor called the 'logic of here-and-there' HT, characterised by the class of the intuitionistic models with only 2 nodes ('here' and 'there')9 .We will show HT SF CPC (corollary 2.24, proposition 2.26)10 .Then we move on to how strict finitistic models can be seen as nodes of an intuitionistic model; and end this article with some remarks in section 3.

Strict finitistic propositional logic 2.1 The languages
We use two languages, L SF and L S .L SF is the language of strict finitistic logic.It is a standard language of propositional logic: Var is the countable set of variables, and we define the set Form of all formulas in L SF by We use L S to describe our models of strict finitistic arithmetic.It is a language of quantifier-free arithmetic with no variables.We deal with numbers through their expressions, and they are the terms in L S .We call them after Wright [7,  p.167] 'natural-number denoting expressions' (or 'NDEs') 11 .
Arithmetical propositions are expressed by the formulas in L S .The atoms are ⊥ and the 'equations' between NDEs.We let = be the only predicate symbol, and Eq = {x = y | x, y ∈ NDE}.All formulas in L S are closed, and we define their set ClForm by In both, we let ¬A be an abbreviation of A → ⊥.We note that neither has ⊤.

The models of arithmetic
In this section we provide our models of strict finitistic arithmetic, which are counterparts of Wright's strict finitistic trees.Our main focus is on what kind of sentence is valid in them.To see it, we will define a structure we call S ∞ .Having it among the models does not change what is valid in all models (proposition 2.10), and the models' properties will be plain from seeing it.S ∞ is one fixed rooted tree-like structure, where NDEs and equations are assigned to the nodes.S ∞ is a tuple T, ≤ T , M, E , where T, ≤ T is a finitelybranching tree each of whose branches is an ω-sequence; M is a mapping from T to P(NDE); E is one from T to P(Eq).Conceptually, M (t) with t ∈ T (ideally) stands for the set of the NDEs the agent has actually constructed by stage t; E(t) the set of the equations, the arithmetical facts, they have actually verified and learnt by t.Letting t 0 be the root, we set M (t 0 ) = {0} and E(t 0 ) = ∅; and for each t ∈ T and x, y ∈ M (t), (i) if S(x) / ∈ M (t), then we let there be a direct successor ] is the natural number that z ∈ NDE denotes, then we let there be a direct successor t ′ such that (We assume [[ ]] in the background in the natural way.)So the agent proceeds one by one.S ∞ is not officially a model of arithmetic, but it serves as a useful tool for our investigation.
Closed formulas in L S are evaluated at each node of S ∞ .We interpret Wright's forcing conditions as follows12 .Definition 2.1 (Actual verification conditions) Let t ∈ T .For any p ∈ Eq, t |= S∞ p iff p ∈ E(t).For any A, B ∈ ClForm, We may omit subscripts in general, if no confusion arises.We note that negation behaves intuitionistically: We note that this system deals with actual construction of terms as well as verification of sentences, unlike the abstract semantic system defined in section 2.3.t |= S∞ A → B means that if A is actually verified, then so will B.But for A → B to hold, the agent does not have to know this.If A ∈ Eq, t |= S∞ A means that the agent has verified A by, and knows A at step t.Otherwise it only reflects our judgement about the agent's verification (cf.section 1.2).
We define validity in S ∞ with two additional notions, assertibility and prevalence.The latter two stand for practical verifiability (cf.section 1.2).Definition 2.2 (Validity, assertibility, prevalence) An A ∈ ClForm is valid in S ∞ if t |= A for all t ∈ T ; and assertible in S ∞ if t |= A for some t ∈ T .An x ∈ NDE is prevalent in S ∞ if, for any t ∈ T , there is a t ′ ≥ T t such that x ∈ M (t ′ ).An A ∈ ClForm is prevalent in S ∞ if for any t ∈ T , there is a t ′ ≥ T t such that t ′ |= A. We write |= V S∞ A, |= A S∞ A and |= P S∞ A for that A is valid, assertible and prevalent in S ∞ , respectively.Accordingly, |= V 0 = 0, since t 0 |= 0 = 0. We are supposing that the ideal agent of S ∞ does not know the equation at the beginning.We will soon introduce models of arithmetic (as elements of S fin ) with various initial knowledge.The notion of semantic consequence is defined via that of validity.We see that assertibility can be calculated just as classical truth-values, and validity is recursively characterisable.
Proof By prevalence property.For (ii, a-c), look at t 0 .
Therefore |= V A → B iff A → B is classically valid; and ¬A ∨ ¬¬A, ¬¬A → A and ((A → B) → A) → A are valid in S ∞ .A general relation with CPC will be given as corollary 2.24 in terms of the semantics defined in section 2.3.
In the introduction (section 1.2), we saw three conditions of practical constructibility.There is a fourth: every statement in strict finitistic reasoning must be 'actually weakly decidable' in the sense that it is either assertible or not assertible [7, p.133]14 .We suggest that |= V ¬A ∨ ¬¬A well conforms to this requirement, as it is equivalent to that |= A A or |= A A.
Also, for any assertible A and non-assertible B, we have t 0 |= 0 = 0 → A and t 0 |= V ¬B, even if they contain huge NDEs.As noted (section 1.2), these do not reflect the agent's knowledge at t 0 , but display our judgements about their verification which one must commit to, given the definition of S ∞ .
S ∞ lacks two principles, the law of excluded middle and Modus Ponens.
(i) is the case even for concrete sentences, unlike in intuitionistic reasoning.Since the nodes represent actual construction steps, there naturally are unverified sentences that will be verified.Similarly, Modus Ponens fails since the conclusion implies A is verified at the beginning.We note that |= V S∞ B → A and |= V S∞ B, however, imply |= A S∞ A for all A and B. These principles in fact hold for the formulas with 'stability'.We say that A is stable in S ∞ if ¬¬A |= V S∞ A; and unstable otherwise.Namely, assertibility implies validity for the stable formulas.One can easily see Although we have not succeeded in recursively characterising the stable formulas, we can present a class with stability.Define Proposition 2.7 Every ClST formula is stable in S ∞ .

Now we define our models of arithmetic. A tuple
and E ′ are the restrictions of ≤ T , M and E, respectively, to T ′ , (c) T ′ , ≤ T ′ is a tree with its root t ′ 0 and (d) each branch is {t ∈ T | t ′ 0 ≤ T t < T s} for some s ∈ T , and (ii) for any p ∈ Eq, if p ∈ E ′ (t) for some t ∈ T ′ , then for any s ∈ T ′ , there is an s ′ ≥ T ′ s such that p ∈ E ′ (s ′ ) (the atomic prevalence condition).
Let S fin denote the class of all models of arithmetic, and the class S consist in S fin and S ∞ .We carry over the forcing conditions and the semantic notions.For each S ∈ S fin , we define a forcing relation between the nodes of S and ClForm in the same way, and write |= S instead of |= S∞ .Validity, assertibility, prevalence and semantic consequence in each S ∈ S fin are defined likewise, except that we consider only the nodes of S. We will write |= V S A etc., respectively.By |= V S ′ etc. for a class S ′ ⊆ S, we mean the relations |= V S ′ etc. hold, respectively, for all S ′ ∈ S ′ .A ∈ ClForm is stable in S ∈ S if ¬¬A |= V S A. Each S ∈ S fin inherits its properties from S ∞ .Plainly |= S persists.The atomic prevalence is extended to all complex formulas.Proposition 2.8 (Full prevalence property) For all S ∈ S fin and A ∈ ClForm, |= A S A implies |= P S A.
The other properties follow from the full prevalence in their respective forms.We note that proposition 2.5 (ii, a) now is that for all atomic A, |= V S A iff t ′ 0 |= S A. Also, proposition 2.6 takes the following form.Proposition 2.9 (i) |= V S A ∨ ¬A for some S ∈ S fin and A. (ii) For some S ∈ S fin and A and B, |= V S B → A and |= V S B do not imply |= V S A.
Proof Take any S with an A unstable in S. Let B be A → A.
Although S ∞ is not officially a model of arithmetic, adding it to S fin does not change the totality of the valid formulas.Define for each t ∈ T , T ′ := {t | t 0 ≤ T t ′ ≤ T t}.Then T ′ defines a model of arithmetic.We call it the contraction model S ∞ |t of S ∞ with respect to t.Given an A ∈ ClForm, let Eq(A) ⊆ Eq be the set of the equations occurring in A. Since Eq(A) is finite, for each t ∈ T , there is a contraction node t ′ ≥ T t with respect to A such that for all p ∈ Eq(A), if |= A S∞ p, then t ′ |= S∞ p.Let T A be the set of those nodes.

Proposition 2.10 (i) For any t ∈ T A , |=
Proof (i-ii) Both by induction.Use proposition 2.5 and that T B∧C , T B∨C , T B→C ⊆ T B ∩ T C .(ii) will use (i).(iii) Immediate from (ii).

The strict finitistic Kripke semantics
In this section, we define the 'strict finitistic semantics' to schematically capture validity in S. It is a Kripke-style semantics which resembles the intuitionistic semantics.An intuitionistic frame K, ≤ is a nonempty set K partially ordered by ≤.Throughout this article, we only consider rooted treelike intuitionistic frames.We denote a root by r.An intuitionistic model K, ≤ , v is an intuitionistic frame equipped with a valuation function v : } is finite (the finite verification condition), and (iv) for any p, if k ∈ v(p) for some k, then for any l ∈ K, there is an l ′ ≥ l such that l ′ ∈ v(p) (the atomic prevalence condition).
Let W denote the class of all strict finitistic models, which contains infinite as well as finite ones.Just as the case of S fin , we carry over the forcing conditions and the semantic notions.We write |= V W A etc. for any W ∈ W.However, (i) the forcing relation is now between the nodes of a model W and the open formulas in L SF .Therefore this semantics does not deal with construction, but only verification schematically, just as the intuitionistic semantics.(ii) The basis of the definition of the forcing conditions is now k |= W p iff k ∈ v(p).(iii) Instead of ClST, we will use the class ST ⊆ Form defined by It is easy to see that the semantic properties so far are carried over in their respective forms.Particularly, for every W ∈ W, |= W persists; and the full prevalence property (proposition 2.8) holds (i.e.|= A W A implies |= P W A). W corresponds to S via substitutions in two senses.We call any mapping σ : → Eq a substitution; and denote the set of the variables occurring in A ∈ Form by Var(A).σ(A) denotes the result of simultaneously substituting all p ∈ Var(A) with σ(p).
an abstraction of a part of S ∞ .One can confirm that for any substitution σ faithful to W , there are a T ′ ⊆ T and a bijection f : where ≤ T ′ is the restriction of ≤ T to T ′ .Note that we do not require that T ′ is closed upwards with respect to ≤ T ; or that, on both sides of (i), there is no node between the two.We call such a bijection f σ .Correspondence (ii) above can be generalised to all formulas.Proof Both by induction.(ii) will use (i).
Secondly, the entire class W captures the validity in S. For each S = T 0 , ≤ T0 , M |T0 , E |T0 ∈ S and σ : Var → Eq, we define the copy model Proposition 2.12 For all t ∈ T 0 and A ∈ Form, t |= WS A iff t |= S σ(A).

Proposition 2.13 For any
Proof Fix Γ and ∆, and prove the contraposition.( =⇒ ) If there are a σ and an S such that σ(Γ) |= V S σ(∆), then consider the copy W S of S under σ.Then by proposition 2.12 Γ |= V WS ∆. ( ⇐= ) If there is a W such that Γ |= V W ∆, then consider a substitution faithful to W , and use proposition 2.11.Proposition 2.10 can be sharpened: the concept of semantic consequence in W boils down to that in the class W 2 ⊆ W of the 2-node models.We will appeal to this fact when comparing our logic with intermediate logics.Let W = K, ≤, v ∈ W. Define for each k ∈ K, the contraction model W |k ∈ W 2 to be {r, k}, ≤ ′ , v ′ where r < ′ k, and v ′ (p) = v(p) ∩ {r, k}.Given A ∈ Form, let K A ⊆ K be the set of the contraction nodes k such that for all p ∈ Var(A), Proof Similar to proposition 2.10.
The rest is easy.

A complete sequent calculus SF
A sequent calculus SF is defined, and proven to be sound and complete with respect to the strict finitistic semantics.For any finite sequences Γ and ∆ in Form, Γ ⊢ ∆ is a sequent in SF.In what follows, ¬¬Γ for any Γ denotes {¬¬A | A ∈ Γ}.The initial sequents are ⊥ ⊢ and p ⊢ p for all p ∈ Var.SF has LK's structural rules.The logical rules are: We write Γ ⊢ SF ∆ to mean that Γ ⊢ ∆ is derivable in SF.
Proving SF's soundness is a routine matter.
We note that A ⊢ SF A, A ⊢ SF ¬¬A and ¬¬¬A ⊢ SF ¬A for all A ∈ Form.Also, A → B ⊣⊢ SF ¬A ∨ ¬¬B, and hence ⊢ SF ¬A ∨ ¬¬A.Admissible are We can reproduce the stability result of the ST formulas in this system.Confirm that derivable are (i) Proposition 2.17 ¬¬S ⊢ SF S for all S ∈ ST.
Proof Induction on S. ⊥'s case is easy.Use (i-iii) above.
Here, ¬¬A ⊢ SF A iff ⊢ SF A ∨ ¬A for all A ∈ Form.Therefore ⊢ SF S ∨ ¬S for all S ∈ ST.A, A → S ⊢ SF S is also plain.Inferences involving an ST formula resemble LK.Proposition 2.18 For any S ∈ ST, the following LK rules are admissible.
Proof (→ L * ) Apply (→ L) to the left premise and S ⊢ S. Then cut ¬¬S using ¬¬S ⊢ SF S. Finally cut S using the right premise.(→ R * ) Cut S from ⊢ S, ¬S and the premise.Use ¬S ⊢ SF S → B and B ⊢ SF S → B. Now let us prove the completeness of SF.Given Γ ⊆ Ψ ⊆ Form, we say that Γ is a prime theory with respect to Ψ if, for all A, B ∈ Ψ, (i) A ∨ B ∈ Γ implies A ∈ Γ or B ∈ Γ, and (ii) Γ ⊢ SF A implies A ∈ Γ.Also, we let Var(Γ) = A∈Γ Var(A); and Form(Γ) be the set of all formulas built only from Var(Γ).For Γ, ∆ ⊆ Form, we say Γ, ∆ is consistent if for any Γ ′ ⊆ fin Γ and ∆ ′ ⊆ fin ∆, Γ ′ ⊢ SF ∆ ′ ; otherwise, it is inconsistent.
The proof progresses as follows.First, let Γ, ∆ ⊆ fin Form such that Γ ⊢ SF ∆ be given.Then we construct a prime theory.

Now define
The completeness is now established by appealing to the 2-node model.
Proof Contraposition.By the preceding lemmas, we can assure there is a W ∈ W 2 with k |= W A for all A ∈ Γ, and k |= W B for all B ∈ ∆.

Relations with intermediate logics
In this section we will compare SF with intermediate logics.We will establish that HT SF CPC; and then show how strict finitistic models can be seen as nodes of an intuitionistic model.HT is known to be characterised by Proof Use that ⊢ HT A implies |= V V2 A, and apply completeness of SF.
Finally, we will show how strict finitistic finite models can be viewed as nodes of an intuitionistic model that satisfies the finite verification condition.Let W fin ⊆ W be the class of the finite models, U ⊆ W fin be at most countable, ⊆ U 2 and for each W ∈ U, . Plainly, a generation order is a partial order.U, is a generation structure (g-structure) if it is an intuitionistic frame and is a generation order.We only consider rooted tree-like intuitionistic frames.Let G be the class of all g-structures.For any A g-structure U, specifies how an agent's cognitive abilities increase, and represents how their actual verification of basic facts, such as concrete equations, proceeds under the increasement.Each W ∈ U typically represents the same agent from different generations, and a later generation is associated with a larger amount power and more verification steps.
Given a G = U, ∈ G, we induce a mapping v G : Var → P(K G ) from the valuations of U's elements: Proof By persistence of v W in each W and definition of v G .

We define the generation forcing relation
G thus defined formalises the agent's actual verification under the increasing power.W, k G A → B means that if A is verified with some extension in power, then so is B with the same power; W, k G ¬A means A is practically unverifiable with any extension.If A is atomic, W, k G A means that the agent has verified A by, and knows that A at step k of generation W . Otherwise, it only reflects our judgement.
One can verify by induction that G persists both in G and each W just as v G .Therefore Since each W is a strict finitistic model, we have the prevalence property of G inside of them.
Proposition 2.28 (Prevalence in generation) If W, k G A for some k ∈ K W , then for any l ∈ K W , there is an l Proof ( ⇐= ) Follows from prevalence in generation.
We note that |= P W where W ∈ U does not persist in G.One may be too powerless to verify (|= P W ¬p), but may verify later (|= P W ′ p with W W ′ ).Let J denote the class of the intuitionistic models that satisfy the finite verification condition.We see that a G = U, ∈ G induces a corresponding I ∈ J , and vice versa.Given G, we define a mapping v : Var → P(U) by W ∈ v(p) iff there is a k ∈ K W such that W, k G p; and let I G := U, , v .Then v(p) is closed upwards in G, and since each W ∈ U is finite, I G ∈ J follows.So W IG A makes sense for each A ∈ Form and W ∈ U. We mean by V IG A that W IG A for all W ∈ U. IG persists in G, and therefore V IG A iff R G IG A. The correspondence between IG and G can be generalised to all complex formulas.

Proposition 2.30 W
Use the induction hypothesis.
Conversely, let I = U * , * , v * ∈ J be given.Then, for each U ∈ U * , we can obtain  Then W U ′ , U ′ GI B. Since U ′ is the maximum of K U ′ , ≤ U ′ , W U ′ , U ′ GI C by the supposition.
These correspondences between G and J yield that the totality of the formulas valid in all G ∈ G coincides with IPC.We note that as easily seen, ⊢ IPC A iff V I A for all I ∈ J .
Proposition 2.32 For all A ∈ Form, (i) V G A iff (ii) V IG A for all G ∈ G iff (iii) ⊢ IPC A. These results may be of philosophical interest.HT SF CPC (cf.corollary 2.24, proposition 2.26) means that SF is too strong to be properly intermediate.Yet it has an intimate relation with IPC: propositions 2.31 and 2.32 seem to endorse the conceptual relation between strict finitism and intuitionism mentioned in section 1.4.The prevalence conditions (cf.propositions 2.4 and 2.8) may naturally be suspected to be responsible, but we do not yet exactly know how they are contributing.

Ending remarks: Further topics
This article only presents our first attempt to classically formalise Wright's strict finitistic logic, and we must leave at least two topics for further investigations.(i) Wright pointed out that the atomic prevalence cannot be assumed in general (cf.section 1.2), and his sketch already included the predicate part.The semantics needs be extended in these two directions.Without the prevalence conditions, the connection with CPC will be lost, and ¬A will separate from A → ⊥, although Peirce's law and ¬A ∨ ¬¬A may remain valid.Investigating what the logic looks like might provide an answer to the philosophical question mentioned above.Meanwhile, the predicate part of our semantics should involve quantification ranging over all objects in the domain, not only those after one node, due to the global nature of negation.We are preparing an article for these purposes, with a sound and complete proof system with respect to the extended semantics.
(ii) We maintained Wright's forcing conditions, and as part of our classical idealisation, accepted any finite lengths of time-gap in implication (cf.section 1.3).Indeed, from the literature it appears hardly possible to philosophically motivate a specific number as a maximum length.However, developing a theory of restricted implication may be of interest.Intuitionistic implication is strict finitistic implication with time-gap 0. If we associate ω to SF because the consequent can come within any n < ω steps, then it might be reasonable to associate 1 to IPC, since 0 < 1.With a theory of implication with various lengths, then we might see a gradation of logics between SF and IPC.
and v U (p) = v * (p) ∩ K U .Define U := {W U | U ∈ U * }; ⊆ U 2 by W U1 W U2 iff U 1 * U 2 ;and G I := U, .Then G I ∈ G.I and G I correspond in the following sense.

Proposition 2 .
31 For all A ∈ Form andU ∈ U * , U I A iff W U , U GI A. Proof Induction.(→) ( =⇒ ) Let U I B → C, and W U ′ , U ′′ GI B with U U ′′ U ′ .Then, since W U ′ , U ′ GI B by persistence, U ′ I B. Therefore U ′ I C.So W U ′ , U ′ GI C, ( ⇐= ) Suppose W U , U GI B → C, U * U ′ and U ′ I B.
r G G A, and hence R G IG A.(ii =⇒ iii) Let H = U * , * , v * ∈ J , and R be H's root.Then we have V H A iff R H A iff W R , R GH A iff W R IG H A. (ii) implies the last.(iii=⇒ i) Prove by induction that for all A, ⊢ IPC A implies V G A. The basis and (∧) are trivial.(∨) By disjunction property of IPC.(→) does not depend on the induction hypothesis.AssumeV I B → C for all I ∈ J .Let G = U, ∈ G.By corollary 2.29, R G , r G G B → C iff R G , k G B → C for some k ∈ K RG .By the assumption we have R G IG B → C. Apply proposition 2.30.
for any t ∈ T , if t |= A holds for all A ∈ Γ, then t |= B for some B ∈ ∆.We write Γ |= V S∞ ∆ for this.We do not define this notion for infinite sets, since if Γ e.g.contains infinitely many equations, then Γ |= V ∆ would hold vacuously.Plainly |= S∞ persists 13 .S ∞ has what we call the prevalence property: assertibility implies prevalence.Proposition 2.4 (Prevalence property) (i) Every NDE is prevalent in S ∞ .(ii) |= A S∞ A implies |= P S∞ A. It follows that t |= ¬A iff s |= A for all s ∈ T .The right side is Wright's original condition for negation, and this justifies abbreviating A → ⊥ as ¬A (sections 1.2, 1.3).We note that |= V ¬¬A iff |= A A: ¬¬A stands for practical verifiability, in contrast to ¬A for practical unverifiability (section 1.2).We also have t |= A → B iff t |= ¬A ∨ ¬¬B.