1 Introduction
The topic of definite descriptions (DD) is of wide interest to philosophers, linguists, and logicians. On the other hand, in proof theory and automated deduction the number of formal systems and studies of their properties is relatively modest. In particular, there are several tableau calculi due to Bencivenga, Lambert and van Fraassen [ben:log91], Gumb [gum:des00], Bostock [bos:int00], Fitting and Mendelsohn [fitt:fir98], but all of them introduce DD by means of rather complex rules, and so, are not really in the spirit of tableau methodology. Quite a lot of natural deduction systems for DD have been provided, but only a few of them (namely Tennant’s [ten:des01, ten:des02] and Kürbis’ [kur:bsl19, kur:bsl20] works) deal with DD by means of rules which allow for finer proof analysis and provide normalization proofs. Cutfree sequent calculi for several theories of DD were provided by Indrzejczak [ind:aml18, ind:llp18, ind:aml20, ind:llp20] and recently also by Orlandelli [orlandelli].
The number of theories of DD that have been proposed since Frege’s and Russell’s first accounts (see, e.g., a discussion in [pel:lin00]) is enormous, however what we are concerned with in this paper is an adequate tableau characterization of DD, so due to space restrictions we omit a detailed presentation of different theories of DD and their philosophical or linguistic motivations. In particular, we confine ourselves to only one approach to DD, strongly connected with free logic and commonly called a minimal free description theory (MFD)^{1}^{1}1The reader may find a more finegrained presentation of MFD and its extensions in Lambert’s [lam:des01], Bencivenga’s [benc:free86] or Lehmann’s [leh:fre00] works.. It is based on the socalled Lambert’s axiom (L):
(L) 
In fact, this axiom added to different kinds of free logics leads to significantly different theories of DD. We provide tableau calculi for four kinds of different free logics, called here PFL, NFL, PQFL, and NQFL (where N stands for negative, P for positive, Q for quasi). In negative free logics, in contrast to positive ones, atomic formulas with nondenoting terms are always evaluated as false or, equivalently, all predicates are strict, that is, defined only over denoting terms. Both PFL and NFL characterize absolutely free logics in the sense that variables may also fail to denote. On the other hand, NQFL and PQFL are systems for quasifree logics in the sense that only descriptions can fail to denote; variables are always denoting.
Recently, cutfree sequent calculi for several free logics, yet without DD, have been presented by Pavlović and Gratzl [pavl:21] and by Indrzejczak [ind:des20]. In particular, in the latter work it has been shown that if we restrict instantiation in quantifier rules only to variables, we do not lose completeness, provided that some special rules are added. It makes it possible to characterize NQFL and PQFL by means of classical quantifier rules, which justifies our use of the term ‘quasi free’ (introduced therein).Yet even more importantly, such a restriction on quantifier rules allows us to extend this approach to MFD and preserve cutfreeness (see [ind:llp20]). Since the abovereferenced paper provides a purely prooftheoretic approach, completing the work with the semantic side and suitably defined adequate and analytic tableau systems seems to be a natural next research step. The aim of the present study is to make this step and fill the indicated gap.
We limit our considerations to the logics mentioned above as the most prominent representatives of the family of free logics. PFL is by all means the most popular version of free logic (see, e.g., [benc:free86], [lam:des02], or [leh:fre00]), applied mainly in philosophical studies and as the basis of formalization of modal firstorder logics (see, e.g., Garson [gars:foml06]). The original Lambert’s version of MFD was proposed on the basis of PFL. The basic negative free logic NFL, known also as the logic of existence ([sco:fre00]), was more popular in computer science and foundational studies [ten:des01, ten:des02].
Negative quasifree logic NQFL is known as the definedness logic (or the logic of partial terms) by Beeson [bee:fre00] and Feferman [fef:fre00]. It has also been extensively studied and applied in computer science. Although it was originally developed in the context of constructive mathematics to deal with partial untyped combinatory and lambda calculi, Feferman rightly noticed that it works without changes in the classical setting (in fact, he was concerned only with classical semantics in [fef:fre00]. PQFL is a positive variant of NQFL, that is, not requiring that all predicates are strict. It is interesting that its intuitionistic restricted version (no identity and DD) was studied prooftheoretically by Baaz and Iemhoff [baa:fre00] and recently by Maffezioli and Orlandelli [maf:fre00].
NQFL^{–} is a variant of NQFL but formulated in the language without the existence predicate. Although the latter can be defined in all the considered logics, it is handy to keep it as primitive. However, in [ind:des20] it was shown that in quantifier rules for all free logics with identity, instantiation terms may be restricted to variables. That opens a possibility of discarding the existence predicate and simplifying the rules, at least for NQFL. Thus, this logic is presented here in two variants: as NQFL with the existence predicate (which allows to compare it with the remaining logics more easily), and then as NQFL^{–} in an existencefree version with simpler rules. In fact NQFL^{–} with the rules for descriptions on classical foundations appears to be equivalent also to the formalization of Russellian theory of descriptions provided by Kalish, Montague and Mar [kali:log64]; (see Indrzejczak [ind:des21] for a detailed explanation).
Lambert’s axiom (L) was used as a basic way of formalizing DD in all the abovementioned logics, except for PQFL. However, on the ground of NFL, (and NQFL) it yields quite a strong theory of DD of essentially Russellian character. This follows from the fact that in NFL (NQFL) (L) is equivalent to the following formula:
(R) 
(R) expresses the Russellian approach to characterizing DD and it was often attacked as being too strong. The lefttoright implication encodes that if we state something about a DD, it implies that this description denotes. According to Strawson’s wellknown criticism, if a DD is used as an argument of a predicate, its existence and uniqueness is presupposed rather than implied. Lambert’s axiom is in general weaker than (R) and in PFL (PQFL) implies only the righttoleft implication of (R) which is commonly acceptable. The equivalence of (L) and (R) in NFL is a consequence of the fact that in NFL all predicates are strict, so the statement of an atomic formula implies that all terms occurring in it are denoting (see [ind:llp20]).
Due to space limitations, we confine ourselves to logics which are founded on the classical core. Interestingly, cutfree sequent calculi in [ind:llp20], after restricting sequents to at most one formula in the succedent and small refinements of some rules for DD, may also characterize their intuitionistic versions. In the case of tableaux adequate with respect to a given semantics, however, such small refinements do not suffice to obtain intuitionistic versions. Hence, we postpone completing this task, as well as the characterization of MFD on the basis of neutral free logics, to future work. In the latter case even the standard sequent calculus is not sufficient for a satisfactory prooftheoretic characterization.
In what follows, after a brief characterization of the syntax and semantics in Section 2, in Section 3 we provide five tableau calculi for the logics PFL, PQFL, NFL, NQFL, and NQFL^{–}. Adequacy of all systems is established in Section 4. In Section 5 we briefly compare our tableau calculi with alternative approaches, in particular with sequent calculi by Indrzejczak [ind:llp20]. Finally we discuss some possible advantages of using DD instead of functional terms and present further lines of research.
2 Preliminaries
2.1 Syntax
For the logics PFL, NFL, PQFL, NQFL we consider sentences, that is, formulas with no free variables, built in the standard firstorder language with identity and the unary existence predicate treated as logical constants and with no function symbols as primitives. The vocabulary of consists of:

a countably infinite set of bound individual variables ,

a countably infinite set of parametric (free) individual variables ,

a countably infinite set of ary predicate symbols , for any nonnegative integer ;

a set of propositional connectives: , ,

the universal quantifier ,

the definite description operator ,

the identity relation ,

the existence predicate ,

left and right parentheses: (, ).
In the case of NQFL^{–}we discard the existence predicate from the language and refer to such a restricted language as .
A set of terms and a set of formulas (in the language of deduction) are defined simultaneously by the following contextfree grammars:
where , , , , and . The existential quantifier and other boolean connectives are introduced as standard abbreviations. Note that the absence of function symbols as primitives in and is due to the fact that they can be simulated by using the operator in the sense that every term of the form can be represented as . On the other hand, not every (proper) description can be expressed using functional terms. For example, descriptions like ‘the winner of the ultimate fight’, ‘the bear we have seen recently’ can only be represented by constants.
2.2 Semantics
By a model we mean a structure , where is a (possibly empty) subset of and for each argument predicate , . An assignment is defined as for PFL, NFL, and as for PQFL, NQFL, and NQFL^{–}. Thus, in proper free logics variables may fail to denote, which is not possible in quasifree logics. An variant of agrees with on all arguments, save, possibly, . We will write to denote the variant of with . The notion of interpretation of a term under an assignment is defined simultaneously with the notion of satisfaction of a formula under , in symbols :
,  
,  
iff 
, and for any variant of , if , then ,  
iff  (and , for NFL, NQFL, and NQFL^{–}),  
iff  (and , for NFL, NQFL, and NQFL^{–}),  
iff  ,  
iff  ,  
iff  and ,  
iff  , for all , 
where , , , and .
A formula is called satisfiable if there exist a model and a valuation such that . A formula is valid if, for all models and valuations , . In the remainder of the paper, instead of writing , we will write .
3 Tableau Calculi
In this section, we present tableau calculi for the considered logics for definite descriptions. For each logic we denote the tableau calculus for by .
A tableau generated by a calculus , for , is a derivation tree whose nodes are assigned formulas in a respective (deduction) language. A branch of is a simple path from the root to a leaf of . For brevity, we identify each branch with the set of formulas assigned to nodes constituting .
Our tableau calculi are composed of rules whose general form is as follows: , where is the set of premises and each , for , is a set of conclusions. If a rule has more than one set of conclusions, it is called a branching rule. Otherwise it is nonbranching. Thus, if a rule is applied to occurring on , splits into branches: . A rule with as the set of its premises is applicable to occurring on a branch if it has not yet been applied to on . A set is called expanded if has already been applied to . A term is called fresh on a branch if it has not yet occurred on . We call a branch closed if the inconsistency symbol occurs on . If is not closed, it is open. A branch is fully expanded if it is closed or no rules are applicable to (sets of) formulas occurring on . A tableau is called closed if all of its branches are closed. Otherwise is called open. Finally, is fully expanded if all its branches are fully expanded. A tableau proof of a formula is a closed tableau with at its root. A formula is tableauvalid (with respect to the calculus ) if all fully expanded tableaux generated by with at the root are tableau proofs of . A tableau calculus is sound if, for each formula , whenever is tableauvalid wrt , then it is valid. is complete if, for each formula , whenever is valid, then it is tableauvalid wrt .
When presenting the rules, we adopt the following notational convention:

metavariables , stand for arbitrary formulas in (or if NQFL^{–}is considered),

metavariables represent arbitrary terms present on a branch,

metavariables , denote fresh parameters,

metavariables , , stand for an arbitrary parameters present on a branch,

an expression represents the result of a correct substitution of all free occurrences of within with a term ,

is an abbreviation for ,

‘DD’ is an abbreviation for ‘definite description’.
The rules for tableau calculi , , , , and are presented in Figures 1 and 2. Intuitively, if a rule’s name contains ‘’ and the name of an operator, it is an elimination rule which removes the operator from the processed formula. On the other hand, if a rule’s name contains ‘’ and the name of an operator, it is an introduction rule which adds to the branch an expression featuring this operator. Moreover, we have three closure rules which close the branch as inconsistent, and two special analytic cut rules which make it possible to compare denotations of variables and definite descriptions.
PFL  PQFL  NFL  NQFL  NQFL^{–}  
, , , ,  
nonempty domain assumption 
A few words of comment on the rules displayed in Figure 1 are in order. The propositional core of the calculi is known from tableaux for classical propositional logic. The rule closes a branch when a propositional inconsistency occurs thereon, whereas the remaining two closure rules, and rest on reflexivity of identity (possibly in a restricted form). The rules and are standard rules for quantifier elimination in firstorder logic. The remaining two rules for , namely and , reflect the semantic condition saying that a term replacing a variable after quantifier elimination must denote an existing object. While in quasifree logics it is ensured by the definition of valuation, in the remaining (absolutely free) logics it needs to be secured by a separate existence formula. Note that all quantifier elimination rules admit only parameters as instances of bound variables. The rule scheme ensures the substitutability of identical terms within arbitrary formulas, often called Leibniz’ principle. One of its side effects is a guarantee that is symmetric in all calculi. and , occurring only in , which lacks the existence predicate , make sure that each definite description occurring in a true atomic formula has a unique and existing denotation, by equating it with a fresh variable (which is always denoting in NQFL^{–}). and are a restricted form of analytic cut which, for each definite description and denoting variable checks whether their denotations are identical or distinct. works similarly to and with the caveat that it equates with a fresh variable a definite description that is known to be denoting. , which is present only in , enforces reflexivity of identity among denoting terms. Intuitively, it allows us to prove that, for each nondenoting term , a formula holds in NFL. The rules and reflect the semantic condition stating that each term which is an argument of a true atomic NQFLformula, or each definite description occurring in such an NFLformula, is denoting. , on the other hand, refers to the definition of valuation in PQFL and NQFL, where variables are always mapped to existing objects. The rule introduces a fresh variable which is assumed to denote, provided that there are no parameters on the branch. Consequently, it guarantees that the nonempty domain assumption is satisfied, should we make it. The first pair of rules, and , eliminate an occurrence of a definite description provided that it appears as an argument of an identity. In a formula defining the definite description must hold of , hence this formula is present in both conclusions. A definite description is subsequently compared to each parameter occurring on a branch. If we assume that they are equal, it is also equal to (the right conclusion), otherwise does not hold of , so we obtain its negation. In we assume that a denoting parameter and a definite description have distinct denotations. It is either because the formula defining the definite description does not hold of (the left conclusion) or because some other object satisfies this formula. To state the latter a fresh parameter is introduced which satisfies , yet it is not equal to . The second pair of rules, and , being a part of the calculi for proper free logics, work similarly, with the caveat that we need to additionally ensure, using the existence predicate , that respective variables occurring in the premises of the rules are denoting. In PFL and NFL variables are not automatically guaranteed to denote, so such an additional condition is necessary for bringing the rules in line with the semantic condition for proper definite descriptions.
Since the rules in all calculi are closed under subformulas modulo substitution, adding single negations and adding equality to two terms already present on the branch one of which being a definite description and another one being a parameter, one can think of the calculi as analytic in an extended sense of the term.
4 Soundness and Completeness
In order to prove soundness and completeness of the calculi , , , , and we need two wellknown lemmas which we recall without proofs (see, e.g., [EbFlTh:96, Sect. III.4 and III.8]).
Lemma 1 (Coincidence Lemma).
Let , let be a model, and let be assignments. If for each free variable occurring in , then iff .
Lemma 2 (Substitution Lemma).
Let , , and let be a model. Then iff .
4.1 Soundness
Let be a rule from a calculus . We say that is sound if whenever is satisfiable, then is satisfiable, for some .
Lemma 3.
For each all rules of are sound.
Proof.
We confine ourselves to showing soundness of the rules for definite descriptions. The proof of the remaining cases can be found in the Appendix.
To prove soundness of assume that is satisfiable, for , that is, there exists a model and an assignment such that . Let , then and by the satisfaction condition , and for any variant of , if , then . The first conjunct guarantees, by Substitution Lemma, that , which holds for both conclusions. The second conjunct yields, for any , that either or . The former case yields the left conclusion, whereas the latter case yields the right one. To show that is sound assume that is satisfiable for . Then, there exists a model and an assignment such that . It means that . By the satisfaction condition , or for some variant of , but . In the first case, by Substitution Lemma, , so the left conclusion is satisfied. If the second holds, then by Coincidence Lemma and Substitution Lemma we have that but for some fresh .
Proofs for and , respectively, are conducted analogically with the following caveat. In PFL and NFL variables are not automatically guaranteed to denote, so the existence of a referrent object needs to be ensured externally. This is done by placing a variable in the scope of the existence predicate . ∎
Now we are ready to prove the following theorem.
Theorem 4.1 (Soundness).
The tableau calculi , , ,, and are sound.
Proof.
To show that for each formula , where , if is tableauvalid, then it is valid. Let be a proof of , that is, a closed tableau with at the root. Each branch of has at the leaf, which is clearly unsatisfiable. By Lemma 3 we know that all the rules of are satisfiability preserving, and so, going from the bottom to the top of , at each node we have an unsatisfiable set of formulas. Thus, (a singleton set consisting of) is unsatisfiable. By the well known duality between satisfiability and validity we obtain that is valid. ∎
4.2 Completeness
In this section, we prove that, for each , is complete. To that end we show that every open and fully expanded branch of a tableau satisfies some syntactic conditions. Then we show how to construct an structure and a function out of such an open and fully expanded branch, and show that is an valuation, and is an model satisfying, for each formula occurring on , .
We assume that for each , the calculus can be accompanied by a suitable fair procedure in the sense that whenever a rule can be applied, it will eventually be applied. For example, an algorithm from [fitt:fir98], with added steps for additional rules, can be applied to . Thus, a fully expanded, possibly infinite, branch is closed under rule application.
Let be an open and fully expanded branch of a tableau , where . Let , , and be the sets of, respectively, all terms occurring on (that is, parameters and definite descriptions), all bound variables occurring on , and all parameters occurring on . We define a binary relation on in the following way:
Proposition 1.
is an equivalence relation.
Proposition 2.
For any , if , then iff , for all formulas .
So equipped, we are ready to prove the cornerstone result of this section.
Lemma 4 (Satisfaction Lemma).
Let be a tableau, for , and let be an open and fully expanded branch of . Then there exists a structure and a function such that:
() 
Proof.
We first show how to construct and . The latter object is assumed to serve as an assignment, which is normally defined for . The values of bound variables, however, are arbitrary, so for convenience we introduce an extra object that will further play the role of their value. First we define and .

.
For :

[hence ].
For :

.
For :

.
Next, we define as a function mapping elements from to for PFL and NFL, and as a function from to for PQFL, NQFL, and NQFL^{–}. We let

, for each ;

iff and for any , if , then , for each and ;

.
We need to show that is a properly defined assignment.
Assignment
First, we show that is a properly defined assignment, for being any of the considered logics. First we prove that is a function on . Totality of straightforwardly follows from its definition. Uniqueness of the value assigned by to each element of is a consequence of two facts. First, is an equivalence relation, so equivalence classes of are pairwise disjoint. Secondly, is nonempty. Indeed, without loss of generality we can assume that we check for validity of universally quantified formulas, that is, the input formula is of the form . By expandedness of we get that the rules , , , , and , for , were applied on to the point where an atomic formula or a negated atomic formula with a free term , that is, a parameter or definite description, occurrs on . Such a formula must finally occur on as does not contain the constants and and an atomic formula of is of one of the forms: , , or , where are terms and is an ary predicate symbol. Thus, an equivalence class of such a freely occurring term is an element of .
For we additionally need to show that the image of is included in . But for the first two logics this is a straightforward consequence of presence of the rule in and , which, for each parameter on , introduces to , and the definition of for both logics. In the last case the required inclusion rests solely on the definition of .
Let us now show that ( ‣ 4) holds. The notion of satisfaction in is defined as in Section 2.2. We proceed by induction on the complexity of which is defined as the number of connectives and quantifiers occuring in but not in the scope of the operator. We restrict attention to the cases where and . The proof of the remaining cases can be found in the Appendix.
Let and . Let . By the definition of , , and so, by the definition of , . Thus, by the satisfaction condition for formulas in both logics, . Now let . By expandedness of we know that the rule (NFL) or together with (NQFL) was applied to , thus yielding . By the proof of the case we know that and . Moreover, by the definition of and , . Hence, by the satisfaction condition for formulas, . Finally, let . By expandedness of the rule was applied to , thus yielding , for and being a definite description. Without loss of generality assume that and is a definite description, so we have and . By the definition of and for we get that , and . By the definition of , and . Hence, by the satisfaction condition for formulas, .
Let and . Let . By openness of , and are distinct terms, for otherwise the rule would close . Again, by openness of , , so by the definition of , . Hence, by the definition of , . Thus, by the satisfaction condition for formulas in both logics, , and so, by the satisfaction condition for formulas, . Let . Clearly, either and are distinct, or identical. Assume, first, that and are distinct terms. Then we proceed with the proof similarly to the case for . Now, assume that is of one of the forms . We know that , for otherwise we could apply
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