Trees without Models - logic.ff.cuni.cz · Trees without Models Truth-Valuational Modal Logic Through Semantic Tableaux Peter Susanszky Department of Philosophy Central European University

Trees without ModelsTruth-Valuational Modal Logic Through Semantic Tableaux

Peter Susanszky

Department of PhilosophyCentral European University

PhDs in Logic X, Prague, May 4th 2018

Peter Susanszky (CEU Philosophy) Trees without Models PhDs in Logic X 1 / 23

Outline

1 Preliminaries

2 The tableau system

3 Semantics

4 Weak completeness

5 Sketch of proof

6 Extension to strong completeness


Outline

1 Preliminaries


3 Semantics

4 Weak completeness

5 Sketch of proof



Preliminaries

• A research programme by Leblanc (cf. Dunn) in 60’s-70’s, but goingback further: from models to valuational functions.

• In essence, a truth-valuational semantics for modal propositional logicreplaces ‘worlds’ with truth-valuations.

• Using an axiomatic system, one can show strong completeness fortruth-valuational semantics through the construction of thevaluational equivalents of ‘canonical models’ (e.g. Ben-Yami)

• In this presentation, I will sketch how to prove the weak completenessof a prefixed tableau deductive system (for the logic K) relative totruth-valuational semantics through modal Hintikka or downwardsaturated sets.

• Extensions to other systems (T, K4, B, S4, S5) are straightforwardbut extension to strong completeness is not as straightforward as onewould hope.


Preliminaries







Preliminaries







Preliminaries







Preliminaries







Kripke and truth-valuational semantics

• In standard Kripke (or model-theoretic) semantics, a model consistsof a non-empty set of ‘possible worlds’, a relation defined on the setthat is usually called the ‘accessibility’ relation between worlds, anda relation between formulas and worlds, which, intuitively, gives usthe ‘truth-at-a-world’ relation.

• In truth-valuational semantics, we exchange this triple for a simplerdouble (called a ‘valuational framework’), where we only have anon-empty set of so-called ‘modal valuations,’ functions fromformulas to truth-values, and a relation defined on the set which wewill call the ‘alternative to’ relation.

• Importantly, since modal valuations are functions, there can be no‘valuational framework’ with two distinct modal valuations that assignthe same values to all formulas of the language (cf. Kripke andDunn).












Outline

1 Preliminaries


3 Semantics

4 Weak completeness

5 Sketch of proof



The tableau system

• Prefixed model tableaux were developed by Fitting (cf. Fitch andSmullyan).

• An L-modal tableau for X is an ordered dyadic tree with anoccurrence of a formula preceded by a sequence of integers (i.e., aprefix) at each point, built with the appropriate L-rules for the systemfrom the origin point at which we have an occurrence of 1 X.

• A closed branch of a modal tableau has, for some formula X andprefix ω, ωX and ω¬X on it. A closed modal tableau has all of itsbranches closed, otherwise it is open.

• A tableau proof of X is a closed modal tableau for 1¬X.

• We will not elaborate on the specific rules. In general, for thecompleteness proof to go through, it is enough to have any set ofrules that, combined with an algorithm, always produces a tableauwhose open branches (if any) constitute a Hintikka set.


The tableau system







The tableau system







The tableau system







The tableau system







An example of a K-modal tableau

`K �((X ∧ (X → Y ))→ Y )

1.2.3.4.5.6.7.

8.

1 ¬�((X ∧ (X → Y ))→ Y )1.1 ¬((X ∧ (X → Y ))→ Y )1.1 (X ∧ (X → Y )) ∧ ¬Y

1.1 X ∧ (X → Y )1.1 X1.1 ¬Y

1.1 X → Y

1.1 ¬X⊗5, 8

1.1 Y⊗6, 8


The intuitive idea behind modal tableaux

• Intuitively, prefixes encode which valuation in the framework we arereasoning about, while the rules of the system show, given an initialformula, what other formulas (usually of smaller complexity) must betrue, if the initial formula is true.

• Then, prefixed modal tableaux are ‘counter-framework’ buildingframeworks, and a given algorithm that produces such tableau is acounter-framework ‘builder.’

• Without an algorithm, the ‘builder’ is the person carrying out theproof.












Outline

1 Preliminaries


3 Semantics

4 Weak completeness

5 Sketch of proof



Truth-valuational semantics

Giving a more precise formulation of our semantics, valuational frameworksare defined as follows:

Valuational framework

A valuational framework F is a pair < V,R >, where V is a non-emptyset of modal valuation functions and R is a binary relation defined on V.If v1 and v2 are both in V and v1Rv2, we say that v2 is an alternativevaluation to v1.



Giving a more precise formulation of our semantics, valuational frameworksare defined as follows:

Valuational framework

A valuational framework F is a pair < V,R >, where V is a non-emptyset of modal valuation functions and R is a binary relation defined on V.If v1 and v2 are both in V and v1Rv2, we say that v2 is an alternativevaluation to v1.



Modal valuations are an extension of the Boolean valuations of classicalpropositional logic, defined on the set S of all formulas of the language.

Modal valuations

A modal valuation v relative to a valuational framework F is a Booleanvaluation v1 ∈ V for which the following additional conditions hold for anyX and v2 ∈ V.

1 v1(�X) = t iff for all modal valuations v2 ∈ V, if v1Rvv2, thenv2(X) = t.

2 v1(♦X) = t iff there is a modal valuation v2 ∈ V such that v1Rvv2and v2(X) = t.




Modal valuations







Modal valuations







Modal valuations






L-modal tautologies and L-satisfiability are defined as follows:

L-modal tautology and L-satisfiability

• If the class L of valuational frameworks has as members allframeworks with a specified R, we call any F in L an L-valuationalframework.

• Then, a formula X is an L-modal tautology iff it is true under allvaluations v ∈ V in all L-valuational frameworks.

• A formula X is L-satisfiable iff there is an L-valuational framework Fand a v ∈ V such that v(X) = t.

If L = K, the class has as members every valuational framework. Fromnow on, we focus on K.


































Outline

1 Preliminaries


3 Semantics

4 Weak completeness

5 Sketch of proof



Weak completeness

We want to show the following fact between the system K and the logicK:

Weak completeness

If a proposition is a K-modal tautology, it is a K-theorem. In symbols, forany formula X, if �K X, then `K X.

We show it through its contrapositive, i.e., the following:

Weak completeness [contrapositive]

If X is not a K-theorem, then ¬X is K-satisfiable, or in symbols, if0K X, then F , v �K ¬X.


Weak completeness


Weak completeness






Weak completeness


Weak completeness






Outline

1 Preliminaries


3 Semantics

4 Weak completeness

5 Sketch of proof



Algorithms and Hintikka sets

We first introduce an algorithm which, for any X, builds a modal tableausuch that each of the tableau’s open branches (if any) constitute a modalHintikka set S. Note that in any S we have 1 X. This can be found inFitting.

K-modal Hintikka set S

For any prefix ω, variable P and formulas X,Y :

• not both ωP and ω¬P are in S;

• if ω¬¬X ∈ S, then X ∈ S; if ωX ∧ Y ∈ S, then ωX,ωY ∈ S; ifω¬(X ∧ Y ) ∈ S, then ω¬X or ω¬Y (or both) is in S (and similarlyfor all formulas of form ωX ∨ Y , ωX → Y and their negations);

• if ω♦X ∈ S, so is ω.nX for some n, and if ω¬♦X ∈ S, then so isω.n¬X for every ω.n that occurs in an element of S (and similarlyfor ω�X and its negation).


Algorithms and Hintikka sets

We first introduce an algorithm which, for any X, builds a modal tableausuch that each of the tableau’s open branches (if any) constitute a modalHintikka set S. Note that in any S we have 1 X. This can be found inFitting.

K-modal Hintikka set S

For any prefix ω, variable P and formulas X,Y :

• not both ωP and ω¬P are in S;

• if ω¬¬X ∈ S, then X ∈ S; if ωX ∧ Y ∈ S, then ωX,ωY ∈ S; ifω¬(X ∧ Y ) ∈ S, then ω¬X or ω¬Y (or both) is in S (and similarlyfor all formulas of form ωX ∨ Y , ωX → Y and their negations);

• if ω♦X ∈ S, so is ω.nX for some n, and if ω¬♦X ∈ S, then so isω.n¬X for every ω.n that occurs in an element of S (and similarlyfor ω�X and its negation).


Constructing a framework for S

• So far, the proof is exactly as can be found in publications on prefixedmodal tableaux and model-theoretic semantics.

• The following construction, however, differs in two important aspects.

• First, instead of taking the set of prefixes occurring in S as the set ofpossible worlds of the counter-model, we construct a set of modalvaluations with the same cardinality as that of the set of all prefixesoccurring in S.

• Additionally, to ensure an admissible counter-framework isconstructed for each extension to logics other than K, we introduce away of defining the constructed modal valuations in a way that eachis pairwise distinguishable from every other by way of distinct valuesbeing assigned to at least one propositional variable for each pair.





















• First, take the (denumerably infinite) set Q of the variables that donot occur in S and suppose they are ordered p1, p2, .... Suppose the(at most denumerably infinite) set O of all ω occurring in S isordered ω1, ω2, ....

• For every prefix ω that occurs in any formula in S, we introduce afunction vω : S → {t, f}.

• For any prefix ω and propositional variable P occurring in S, ifωP ∈ S, we set vω(P ) = t, and if ωP /∈ S, we set vω(P ) = f .

• For any ωn ∈ O and pk ∈ Q, we set vωn(pk) = t if k ≤ n, and setvωn(pk) = f if n < k.

• If ω1 and ω2 occur in some prefixed formulas in S, we set vω1Rvvω2

iff ω1 is of form ω and ω2 is of form ω.n


































Proving satisfiability and completeness

• Finally, we show that for each formula X and prefix ω, if ωX ∈ S,then vω(X) = t, i.e., X is true under vω.

• The proof is a trivial Hintikka-type induction on the complexity of X.

• We thus show that the set S is satisfiable in a valuational frameworkF . Since 1 X is in any S, we have that there is a F and a v ∈ Vsuch that F , v �K X.

• Completeness follows by a reductio: if X is not a K-theorem, thenthe tableau for 1¬X does not close, which means that ¬X isK-satisfiable.




















Outline

1 Preliminaries


3 Semantics

4 Weak completeness

5 Sketch of proof



Extension to strong completeness

• In the above proof sketch, we have ensured that the constructedstructure for S is a valuational framework by setting the outputvalues for each vω in such a way that in the end they come outatomically distinct from each other.

• We can do this when proving weak completeness, since each formulais of finite length and thus omits denumerably infinite many variables,while any set of prefixes occuring in S is itself at most denumerablyinfinite.

• This breaks down with strong completeness: we may have premisesets which do not omit infinitely many variables, thus we cannot makesure that the constructed structure for S will only positdistinguishable modal valuations as distinct.

• Two ‘tricks’ taken from first-order truth-valuational accounts: freeingup denumerably infinite many variables through reassigning variablesOR extending the language with denumerably infinite many variables.




















The end

(Thanks to Edi Pavlovic for the CEU LATEX theme!)


The end(Thanks to Edi Pavlovic for the CEU LATEX theme!)


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Trees without Models - logic.ff.cuni.cz · Trees without Models Truth-Valuational Modal Logic Through Semantic Tableaux Peter Susanszky Department of Philosophy Central European University