Tableau metatheory for propositional and syllogistic logics Part - - PowerPoint PPT Presentation

Tableau metatheory for propositional and syllogistic logics

Part IV: Abstract tableau notions: rules, branches, tableaux Tomasz Jarmużek

Nicolaus Copernicus University in Toruń Poland

Logic Summer School, 3th-14th, December 2018, Australian National University

Program of lecture

We describe the main part of tableau metatheory: general tableau notions: ◮ all notions are presented as set–theoretical ones (for example: branches are sequences of sets and tableaux are sets of those sequences) ◮ the rest of tableau notions are defined in a similar, formal way:

• 1. tableau rules
• 2. branches: open, closed, maximal (aka complete)
• 3. tableaux: open, closed, complete
• 4. new notions are also presented — branch consequence relation

(as a very special set of branches) and useless variant of branch.

Tableau language – set of expressions

We need some language of tableau proofs: set of expressions Ex. Firstly, we list symbols:

• 1. indexes/labels — set of natural numbers N
• 2. n–ary functional constants (where n ≥ 1): w1

1 , w1 2 , w1 3 , . . . ,

w2

1 , w2 2 , w2 3 , . . . , w3 1 , w3 2 , w3 3 , . . .

• 3. n–ary predicates (where n ≥ 2): r 2

1 , r 2 2 , r 2 3 , . . . , r 3 1 , r 3 2 , r 3 3 ,

. . . , r 4

1 , r 4 2 , r 4 3 , . . .

• 4. identity symbol: ≡
• 5. semantic negation: ∼.

Tableau language – set of terms

Set of all terms TERM is the least that consists of: wl

k(m1, . . . , ml), where:

◮ k, l, m1, . . . , mk ∈ N ◮ l ≥ 1 ◮ w l

k is a functional constant.

The members of TERM we denote by t1, t2, t3, . . .

Tableau language – set of expressions

Definition (Expressions)

Ex is the least set that consists of the expressions: ◮ r l

k(m1, . . . , ml)

∼ r l

k(m1, . . . , ml)

◮ i ≡ j ∼ i ≡ j ◮ A, t1, . . . , tn ∼ A, t1, . . . , tn for all:

• a. A ∈ For
• b. i, j, k, l, n, m1, . . . , ml ∈ N
• c. t1, . . . , tn ∈ TERM, where n ≥ 1.

When the context is clear, we write: ◮ A, t1, . . . , tn ◮ ∼ A, t1, . . . , tn, removing brackets: .

Fundamental tableau notions: function choosing indexes

Definition (Function choosing indexes)

Function choosing indexes we call a function

• : Ex ∪ TERM ∪ P(Ex ∪ TERM) −

→ P(N) defined by conditions: ◮ ◦(wl

k(m1, . . . , ml)) = {m1, . . . , ml}

◮ ◦(r l

k(m1, . . . , ml)) = ◦(∼ r l k(m1, . . . , ml)) = {m1, . . . , ml}

◮ ◦(i ≡ j) = ◦(∼ i ≡ j) = {i, j} ◮ ◦(A, wl1

k1(xk1 1 , . . . , xk1 l1 ), . . . , wln ho(xho 1 , . . . , xho ln )) =

• (∼ A, wl1

k1(xk1 1 , . . . , xk1 l1 ), . . . , wln ho(xho 1 , . . . , xho ln )) =

{xk1

1 , . . . , xk1 l1 , . . . , xho 1 , . . . , xho ln }

◮ ◦(X) = {◦(y) : y ∈ X}, if X ⊆ Ex ∪ TERM, for all A ∈ For and h, i, j, k, l, o, m1, . . . , ml, xk1

1 , . . . , xk1 l1 , . . . ,

xho

1 , . . . , xho ln ∈ N.

Fundamental tableau notions: similar sets of expressions

Definition (Similar sets of expressions)

Let X, Y ⊆ Ex be sets of expressions. Let Z ⊆ N. Set X is similar to Y in respect of Z iff there is a bijection ‡ : ◦(X) − → ◦(Y ) (where ◦(X), ◦(Y ) are sets of indexes occurring in expressions of X and Y ) such that: (a) for all x ∈ Z, if x ∈ ◦(X), then ‡(x) = x (b) for all kinds of expressions in Ex:

Fundamental tableau notions: similar sets of expressions

(a) r l

k(m1, . . . , ml) ∈ X iff r l k(‡(m1), . . . , ‡(ml)) ∈ Y

(b) i ≡ j ∈ X iff ‡(i) ≡ ‡(j) ∈ Y (c) A, wl1

k1(xk1 1 , . . . , xk1 l1 ), . . . , wln ho(xho 1 , . . . , xho ln ) ∈ X iff

A, wl1

k1(‡(xk1 1 ), . . . , ‡(xk1 l1 )), . . . , wln ho(‡(xho 1 ), . . . , ‡(xho ln )) ∈ Y

(d) ∼ r l

k(m1, . . . , ml) ∈ X iff ∼ r l k(‡(m1), . . . , ‡(ml)) ∈ Y

(e) ∼ i ≡ j ∈ X iff ∼ ‡(i) ≡ ‡(j) ∈ Y (f) ∼ A, wl1

k1(xk1 1 , . . . , xk1 l1 ), . . . , wln ho(xho 1 , . . . , xho ln ) ∈ X iff

∼ A, wl1

k1(‡(xk1 1 ), . . . , ‡(xk1 l1 )), . . . , wln ho(‡(xho 1 ), . . . , ‡(xho ln ))

∈ Y , for all A ∈ For and h, i, j, k, l, o, m1, . . . , ml, xk1

1 , . . . , xk1 l1 , . . . ,

xho

1 , . . . , xho ln ∈ N.

Fundamental tableau notions: tableau inconsistency

Definition (Tableau inconsistent sets of expressions)

Let X ⊆ Ex. We say that X is tableau inconsistent iff it consists

• ne of pairs of the expressions:

(a) r l

k(m1, . . . , ml), ∼ r l k(m1, . . . , ml)

(b) i ≡ j, ∼ i ≡ j (c) A, t1, . . . , tn, ∼ A, t1, . . . , tn) for all: ◮ A ∈ For and i, j, k, l, m1, . . . , ml, n ∈ N ◮ t1, . . . , tn ∈ TERM. Otherwise, we call set X tableau consistent. We shortly say that X is t-consistent or respectively t-inconsistent.

Model suitable to a set of expressions

Definition (Model suitable to a set of expressions)

Let X ∈ Ex. Let M = {Wi}i∈M, {Rj}j∈N, V ∈ Mi and X ⊆ Ex. Model M is suitable to X iff there are functions: (a) f ′ : N − → M ∪ N (b) f ′′ : N − →

i∈M Wi

such that following conditions are fulfilled:

Model suitable to a set of expressions

(a) if A, wl1

k1(xk1 1 , . . . , xk1 l1 ), . . . , wln ho(xho 1 , . . . , xho ln ) ∈ X, then:

◮ (f ′′(xk1

1 ), . . . , f ′′(xk1 l1 ) ∈ W l1 f ′(k)1, . . . , f ′′(xho 1 ), . . . , f ′′(xho ln ) ∈

W ln

f ′(h)o

◮ W l1

f ′(k)1 × · · · × W ln f ′(h)o ∈ {Wi}i∈M

◮ M, f ′(xk1

1 ), . . . , f ′(xk1 l1 ), . . . , f ′(xho 1 ), . . . , f ′(xho ln ) |

= A

(b) if ∼ A, wl1

k1(xk1 1 , . . . , xk1 l1 ), . . . , wln ho(xho 1 , . . . , xho ln ) ∈ X, then:

◮ (f ′′(xk1

1 ), . . . , f ′′(xk1 l1 ) ∈ W l1 f ′(k)1, . . . , f ′′(xho 1 ), . . . , f ′′(xho ln ) ∈

W ln

f ′(h)o

◮ W l1

f ′(k)1 × · · · × W ln f ′(h)o ∈ {Wi}i∈M

◮ M, f ′(xk1

1 ), . . . , f ′(xk1 l1 ), . . . , f ′(xho 1 ), . . . , f ′(xho ln ) |

= A

Model suitable to a set of expressions

(c) if r l

k(m1, . . . , ml) ∈ X, then f ′′(m1), . . . , f ′′(ml) ∈ Rl f ′(k)

(d) if ∼ r l

k(m1, . . . , ml) ∈ X, then f ′′(m1), . . . , f ′′(ml) ∈ Rl f ′(k)

(e) if i ≡ j ∈ X, then f ′′(i) is equal to f ′′(j) (f) if ∼ i ≡ j ∈ X, then f ′′(i) is not equal to f ′′(j) for all A ∈ For and h, i, j, k, l, o, m1, . . . , ml, xk1

1 , . . . , xk1 l1 , . . . ,

xho

1 , . . . , xho ln ∈ N.

Complex tableau notions

Having a set of expressions Ex we can put very general conditions defining rules. Our rules extend properly a set of expressions and they have also an internal mechanism that blocks extending of t-inconsistent sets. Let us distinguish a set of indexes that plays a role of signs of logical values in our language: LV ⊆ N (for some domain in models Wj). Let Z ⊆ Ex. Z is co–infinite iff N \ ◦(Z) is infinite.

Complex tableau notions

Definition (Rule)

Assume that P(Ex) is the set of all subsets of the set Ex. Let P(Ex)n be n–ary Cartesian product P(Ex) × · · · × P(Ex)

• n

, for some n ∈ N, and let

n∈N P(Ex)n be the union of all such n–ary

Cartesian products that n ≥ 2.

Complex tableau notions

Definition (Rule)

Assume that P(Ex) is the set of all subsets of the set Ex. Let P(Ex)n be n–ary Cartesian product P(Ex) × · · · × P(Ex)

• n

, for some n ∈ N, and let

n∈N P(Ex)n be the union of all such n–ary

Cartesian products that n ≥ 2. ◮ Rule is such a subset R ⊆

n∈N P(Ex)n that if

X1, . . . , Xn ∈ R, then the following conditions are satisfied:

◮ X1 ⊂ Xi, for all 1 < i ≤ n ◮ X1 is t-consistent ◮ if k = l, then Xk = Xl, for all 1 < k, l ≤ n

Rule

Definition (Rule cont.)

◮ (Closure under similarity) for any such subset of expression Y1 that Y1 is similar to X1 in respect of LV, there exist such sets

• f expressions Y2, . . . , Yn, that Y1, . . . , Yn ∈ R and for all

1 < i ≤ n, Yi is similar to Xi in respect of LV

Rule

Definition (Rule cont.)

◮ (Closure under similarity) for any such subset of expression Y1 that Y1 is similar to X1 in respect of LV, there exist such sets

• f expressions Y2, . . . , Yn, that Y1, . . . , Yn ∈ R and for all

1 < i ≤ n, Yi is similar to Xi in respect of LV ◮ (Existence of a core of rule) for some finite set Y ⊆ X1 there exists exactly one such n–tuple Z1, . . . , Zn ∈ R that:

• 1. Z1 = Y
• 2. for any 1 < i ≤ n, Zi = Z1 ∪ (Xi \ X1)
• 3. there does not exist a proper subset U1 ⊂ Y and such n–tuple

U1, . . . , Un ∈ R that for 1 < i ≤ n, Ui = U1 ∪ (Zi \ Z1)

Any such n–tuple Z1, . . . , Zn is called a core of rule R in X1, . . . , Xn

Rule

Definition (Rule cont.)

◮ (Closure under Expansion) for any t-consistent set of expressions Z1 such that:

• 1. X1 ⊂ Z1
• 2. Z1 is co–infinite
• 3. for all 1 < i ≤ n, Xi is not similar in respect of LV to any

subset of Z1: ◮ if n–tuple W1, . . . , Wn is a core of rule R in X1, then:

• 1. there are n − 1 such sets of expressions Z2, . . . , Zn that

Z1, . . . , Zn ∈ R

• 2. and for all 1 < i ≤ n, Wi is similar in respect of LV to

W1 ∪ (Zi \ Z1)

◮ (Closure under Finite Sets) if X1 is a finite set, then for all 1 < i ≤ n, Xi is a finite set

Rule

Definition (Rule cont.)

◮ By saying that a rule R was applied to X1, we mean that for 1 < i ≤ n, exactly one Xi of some X1, . . . , Xn ∈ R was chosen, where 1 < i ≤ n. ◮ By R we denote a set of rules.

Examples of rules

Since we give examples of rules for two–valued modal logics below, we omit indexes for logical values. If we have classical conjunction ∧ in our language, we can take the following rule: R∧ :

X ∪ {(A∧B),i} X ∪ {(A∧B),i, A,i, B,i}.

Examples of rules

Since we give examples of rules for two–valued modal logics below, we omit indexes for logical values. If we have classical conjunction ∧ in our language, we can take the following rule: R∧ :

X ∪ {(A∧B),i} X ∪ {(A∧B),i, A,i, B,i}.

If we have diamond ♦ in our language, we can take the following rule. R♦ :

X ∪ {♦A,i} X ∪ {♦A,i, irj,A,j}, where:

• 1. j ∈ ◦(X ∪ {♦A, i})
• 2. for all k ∈ N, {irk, A, k} ⊆ X.

Core of a rule

Definition (Core of a rule)

Let R be a rule and n ∈ N. Let X1, . . . , Xn ∈ R and Z1, . . . , Zn ∈ R. n–tuple Z1, . . . , Zn is a core of rule R in X1, . . . , Xn iff

• 1. Z1 ⊆ X1
• 2. for all 1 < i ≤ n, Zi = Z1 ∪ (Xi \ X1)
• 3. there does not exist a proper subset U1 ⊂ Z1 and such n–tuple

U1, . . . , Un ∈ R that Ui = U1 ∪ (Zi \ Z1), for all 1 < i ≤ n.

Core of a rule

In the case of structurally defined rules: R∧ :

X ∪ {(A∧B),i} X ∪ {(A∧B),i, A,i, B,i}

we can easily distinguish a core of the rule. Here it is (A ∧ B), i}, for all A, B, i.

Non-introducing/introducing rules

The section is because in tableau systems there can appear some special tableau rules that we can call introducing rules — as

• pposed to them, the rest of rules we can call non–introducing

rules.

Non-introducing/introducing rules

The section is because in tableau systems there can appear some special tableau rules that we can call introducing rules — as

• pposed to them, the rest of rules we can call non–introducing

rules. These are not definitions, but the difference between those two kinds of rules is that if we have a set of tableau expressions X, then:

Non-introducing/introducing rules

The section is because in tableau systems there can appear some special tableau rules that we can call introducing rules — as

• pposed to them, the rest of rules we can call non–introducing

rules. These are not definitions, but the difference between those two kinds of rules is that if we have a set of tableau expressions X, then: ◮ in case of non–introducing rule R, when we obtain from X an extended by R set Y , then in no expression in set Y there appears any new symbol of alphabet that is not a part of an expression in set X

Non-introducing/introducing rules

The section is because in tableau systems there can appear some special tableau rules that we can call introducing rules — as

• pposed to them, the rest of rules we can call non–introducing

rules. These are not definitions, but the difference between those two kinds of rules is that if we have a set of tableau expressions X, then: ◮ in case of non–introducing rule R, when we obtain from X an extended by R set Y , then in no expression in set Y there appears any new symbol of alphabet that is not a part of an expression in set X ◮ in case of introducing rule R, when we obtain from X an extended by R set Y , then at least in one expression in set Y there appears a new symbol of alphabet than does not occur in any expression in set X.

RX

Definition

Let X ⊆ Ex. Let R be a rule. By RX we denote a maximal subset of all n-tuples in R, such that their first member is X and if other members of two n-tuples in RX differ, then the rule can be applied more times to the extended set. Formally, RX is such a maximal set of n–tuples that for all n ∈ N, Y1, . . . , Yn ∈ RX iff: ◮ Y1, . . . , Yn ∈ R and Y1 = X ◮ for all Z1, . . . , Zn ⊆ Ex, if Z1, . . . , Zn ∈ RX, then:

◮ for some core Z ′

1, . . . , Z ′ n of R in Z1, . . . , Zn, for some

1 < i ≤ n, Z ′

1 ∪ Yi, . . . Z ′ n ∪ Yi ∈ R.

RX

Having a rule R and a set of expressions X we define a set of n-tuples belonging to R such that after application of R the other n-tuples in R cannot extend the extended set. For example, we have a set

RX

Having a rule R and a set of expressions X we define a set of n-tuples belonging to R such that after application of R the other n-tuples in R cannot extend the extended set. For example, we have a set X = {♦A, 1}

RX

Having a rule R and a set of expressions X we define a set of n-tuples belonging to R such that after application of R the other n-tuples in R cannot extend the extended set. For example, we have a set X = {♦A, 1} We apply the rule for ♦, using the pair in the rule: {♦A, 1}, {♦A, 1, 1r3, ♦A, 3}, and we get: (∗) {♦A, 1, 1r3, ♦A, 3},

RX

Having a rule R and a set of expressions X we define a set of n-tuples belonging to R such that after application of R the other n-tuples in R cannot extend the extended set. For example, we have a set X = {♦A, 1} We apply the rule for ♦, using the pair in the rule: {♦A, 1}, {♦A, 1, 1r3, ♦A, 3}, and we get: (∗) {♦A, 1, 1r3, ♦A, 3}, but from (∗) and the rule for ♦ we cannot get: {♦A, 1, 1r3, ♦A, 3, 1rk, ♦A, k, }, for any k ∈ N, where k = 3.

RX

Having a rule R and a set of expressions X we define a set of n-tuples belonging to R such that after application of R the other n-tuples in R cannot extend the extended set. For example, we have a set X = {♦A, 1} We apply the rule for ♦, using the pair in the rule: {♦A, 1}, {♦A, 1, 1r3, ♦A, 3}, and we get: (∗) {♦A, 1, 1r3, ♦A, 3}, but from (∗) and the rule for ♦ we cannot get: {♦A, 1, 1r3, ♦A, 3, 1rk, ♦A, k, }, for any k ∈ N, where k = 3. So in the RX, where R is the rule for ♦, we have only one pair {♦A, 1}, {♦A, 1, 1rj, ♦A, j}, for some j ∈ N.

Definition (Tableau rules)

Let R be a set of rules. R is a set of tableau rules iff

• 1. R is finite
• 2. for any X ⊆ Ex, if X is finite, then for any rule R ∈ R, each

set RX is finite.

Definition (Tableau rules)

Let R be a set of rules. R is a set of tableau rules iff

• 1. R is finite
• 2. for any X ⊆ Ex, if X is finite, then for any rule R ∈ R, each

set RX is finite. By TR we denote some fixed set of tableau rules.

Branch

A branch is a sequence of sets of expressions: X1 ⊆ X2 ⊆ · · · ⊆ Xn (possibly infinite), where for any 1 ≤ i ≤ n, Xi+1 is a result of application of some rule to set Xi.

Branch

A branch is a sequence of sets of expressions: X1 ⊆ X2 ⊆ · · · ⊆ Xn (possibly infinite), where for any 1 ≤ i ≤ n, Xi+1 is a result of application of some rule to set Xi.

Definition (Branch)

Let K = N or K = {1, 2, . . . , n}, where n ∈ N. Let X be a set of

• expressions. Branch (or branch starting from X) is any string

φ : K − → P(Ex) satisfying the conditions:

• 1. φ(1) = X
• 2. for all i ∈ K: if i + 1 ∈ K, then there exists a rule R ∈ TR

and n-tuple Y1, . . . Yn ∈ R, that φ(i) = Y1 and φ(i + 1) = Yk, for some 1 < k ≤ n.

On–member–branch

From the definition of branch we have a corollary:

Collorary

• Let X ⊆ Ex. Then X1 is a branch.

On–member–branch

From the definition of branch we have a corollary:

Collorary

• Let X ⊆ Ex. Then X1 is a branch.
• Let φ : K −

→ P(Ex) be a branch. Then ({φ(i) : i ∈ K})1 is a branch.

On–member–branch

From the definition of branch we have a corollary:

Collorary

• Let X ⊆ Ex. Then X1 is a branch.
• Let φ : K −

→ P(Ex) be a branch. Then ({φ(i) : i ∈ K})1 is a branch. Instead of ({φ(i) : i ∈ K})1 we will write φK, as

• ne–member–branch made by union of a branch φ : K −

→ P(Ex).

Closed/open branch

Definition (Closed/open branch)

A branch φ : K − → P(Ex) is closed iff φ(i) is a t–inconsistent set, for some i ∈ K. A branch is open iff is not closed.

Maximal branch (aka complete)

Definition (Maximal branch 1 — for finite cases)

Let φ : K − → P(Ex) be a branch. We say that φ is maximal iff

• 1. K = {1, 2, 3, . . . , n}, for some n ∈ N
• 2. there is no branch ψ such that φ ⊂ ψ.

Maximal branch (aka complete)

Definition (Maximal branch 1 — for finite cases)

Let φ : K − → P(Ex) be a branch. We say that φ is maximal iff

• 1. K = {1, 2, 3, . . . , n}, for some n ∈ N
• 2. there is no branch ψ such that φ ⊂ ψ.

Definition (Maximal branch 2)

Let φ : K − → P(Ex) be a branch. We say that φ is maximal iff there is no branch ψ such that φK ⊂ ψ.

Maximal branch (aka complete)

Definition (Maximal branch 1 — for finite cases)

Let φ : K − → P(Ex) be a branch. We say that φ is maximal iff

• 1. K = {1, 2, 3, . . . , n}, for some n ∈ N
• 2. there is no branch ψ such that φ ⊂ ψ.

Definition (Maximal branch 2)

Let φ : K − → P(Ex) be a branch. We say that φ is maximal iff there is no branch ψ such that φK ⊂ ψ.

Fact

Let φ : K − → P(Ex) be a branch. If φ is maximal in respect to definition of maximal branch 1, then it is maximal in respect to definition of maximal branch 2.

Branch consequence relation

Let Z ⊆ For and i ∈ N. By Z i we mean set {A, w1(i) : A ∈ Z}.

Definition (Branch consequence relation)

Let X ⊆ For and A ∈ For. Formula A is a branch consequence of X (in short: X ⊲TR A) iff there exists such a finite set Y ⊆ X and some index i ∈ N, that any maximal branch starting from the set Y i ∪ {∼ A, w1(i)} is closed.

Tableau

Definition (Branch maximal in a set of branches)

Let Φ be a set of branches and let ψ ∈ Φ. Branch ψ is maximal in set Φ (in short: Φ–maximal) iff there does not exist such a branch φ ∈ Φ, that ψ ⊂ φ.

Tableau

Definition (Tableau)

Let X ⊆ For, A ∈ For and Φ be a set of branches. An ordered triple X, A, Φ is a tableau for X, A (or just tableau) iff there are satisfied conditions: ◮ Φ is a non-empty subset of the set of branches starting from the set X i ∪ {∼ A, w1(i)}, for some index i ∈ N (i.e. if ψ ∈ Φ, then ψ(1) = X i ∪ {∼ A, w1(i)}) ◮ each branch in Φ is Φ-maximal

Tableau

Definition (Tableau cont.)

◮ for any n, i ∈ N and any branches ψ1, . . . , ψn ∈ Φ, if:

◮ i and i + 1 are in domains of functions ψ1, . . . , ψn ◮ for any 1 < k ≤ n and any o ≤ i, ψ1(o) = ψk(o)

then there exists such a rule R ∈ TR and an ordered m-tuple Y1, . . . , Ym ∈ R, where 1 < m, that for all 1 ≤ k ≤ n:

◮ ψk(i) = Y1 ◮ there exists such 1 < l ≤ m, that ψk(i + 1) = Yl.

Useless variant of branch

Look at these examples of branches:

Useless variant of branch

Look at these examples of branches: (⋆) {p ∨ q, 1, ¬¬p, 1} ⊂ {p ∨ q, 1, ¬¬p, 1, p, 1}

Useless variant of branch

Look at these examples of branches: (⋆) {p ∨ q, 1, ¬¬p, 1} ⊂ {p ∨ q, 1, ¬¬p, 1, p, 1} The one–star–branch was produced by application of rule for double negation R¬ ¬: R¬ ¬ :

X ∪ {¬¬A,i} X ∪ {¬¬A,i, A,i}

Useless variant of branch

Look at these examples of branches: (⋆) {p ∨ q, 1, ¬¬p, 1} ⊂ {p ∨ q, 1, ¬¬p, 1, p, 1} The one–star–branch was produced by application of rule for double negation R¬ ¬: R¬ ¬ :

X ∪ {¬¬A,i} X ∪ {¬¬A,i, A,i}

(⋆⋆) {p ∨ q, 1, ¬¬p, 1} ⊂ {p ∨ q, 1, ¬¬p, 1, p, 1} (⋆ ⋆ ⋆) {p ∨ q, 1, ¬¬p, 1} ⊂ {p ∨ q, 1, ¬¬p, 1, q, 1} ⊂ {p ∨ q, 1, ¬¬p, 1, q, 1, p, 1}

Useless variant of branch

Look at these examples of branches: (⋆) {p ∨ q, 1, ¬¬p, 1} ⊂ {p ∨ q, 1, ¬¬p, 1, p, 1} The one–star–branch was produced by application of rule for double negation R¬ ¬: R¬ ¬ :

X ∪ {¬¬A,i} X ∪ {¬¬A,i, A,i}

(⋆⋆) {p ∨ q, 1, ¬¬p, 1} ⊂ {p ∨ q, 1, ¬¬p, 1, p, 1} (⋆ ⋆ ⋆) {p ∨ q, 1, ¬¬p, 1} ⊂ {p ∨ q, 1, ¬¬p, 1, q, 1} ⊂ {p ∨ q, 1, ¬¬p, 1, q, 1, p, 1} The two– and three–star branches were produced by application of rule for disjunction R∨, and the three–star–branch was additionally produced by rule R¬ ¬. R¬ ∨ :

X ∪ {¬(A∨B),i} X ∪ {¬(A∨B),i,¬A,i,¬B,i}

Useless variant of branch

But the three–star–branch is useless in this sense that if the

• ne–star–branch is closed, then it is too.

Useless variant of branch

But the three–star–branch is useless in this sense that if the

• ne–star–branch is closed, then it is too.

If it is open, then the one–star–branch is open, too.

Useless variant of branch

But the three–star–branch is useless in this sense that if the

• ne–star–branch is closed, then it is too.

If it is open, then the one–star–branch is open, too. Hence all we should know is on the one–star–branch. The presence

• f the three–star–branch in a given tableau is not necessary.

Useless variant of branch

Definition (Useless variant of branch)

Let φ and ψ be such branches that for some numbers i and i + 1 that belong to their domains and for all j ≤ i, φ(j) = ψ(j), but φ(i + 1) = ψ(i + 1). Branch ψ is useless variant of branch φ iff: ◮ there are such a rule R ∈ TR and n–tuple X1, . . . , Xn ∈ R, that φ(i) = X1 and φ(i + 1) = Xj, for some 1 < j ≤ n ◮ there are such a rule R ∈ TR and m–tuple Y1, . . . , Ym ∈ R, where n < m, that ψ(i) = Y1 and:

• 1. ψ(i + 1) = Yk, for some 1 < k ≤ m
• 2. for all 1 < l ≤ n there is such 1 < o ≤ m that o = k

and Xl = Yo.

Useless variant of branch

Definition (Useless variant of branch)

Let φ and ψ be such branches that for some numbers i and i + 1 that belong to their domains and for all j ≤ i, φ(j) = ψ(j), but φ(i + 1) = ψ(i + 1). Branch ψ is useless variant of branch φ iff: ◮ there are such a rule R ∈ TR and n–tuple X1, . . . , Xn ∈ R, that φ(i) = X1 and φ(i + 1) = Xj, for some 1 < j ≤ n ◮ there are such a rule R ∈ TR and m–tuple Y1, . . . , Ym ∈ R, where n < m, that ψ(i) = Y1 and:

• 1. ψ(i + 1) = Yk, for some 1 < k ≤ m
• 2. for all 1 < l ≤ n there is such 1 < o ≤ m that o = k

and Xl = Yo.

Let Φ, Ψ be sets of branches and Φ ⊂ Ψ. Ψ is useless superset of Φ iff for any branch ψ ∈ Ψ \ Φ there is such a branch φ ∈ Φ that ψ is a useless variant of φ.

Complete tableau

Definition (Complete tableau)

Let X, A, Φ be a tableau. X, A, Φ is complete iff:

• 1. all branches in Φ are maximal
• 2. each such set of branches Ψ that:

2.1 Φ ⊂ Ψ 2.2 X, A, Ψ is a tableau

is a useless superset of Φ. Tableau is incomplete iff it is not complete.

Closed/open tableau

Definition (Closed/open tableau)

Let X, A, Φ be a tableau. X, A, Φ is closed iff it satisfies the conditions:

• 1. X, A, Φ is a complete tableau
• 2. all branches in Φ are closed.

Tableau is open iff it is not closed.

Acknowledgments

Most of the presented materials contain results of research supported by National Science Centre, Poland, under grant UMO-2015/19/B/HS1/02478. Some parts, particularly prepared for the visit at ANU, were financially supported by prof. dr. hab. Radosław Sojak, Dean of Departament of Humanities, at Nicolaus Copernicus University in Toruń. I would also like to thank dr. hab. Krzysztof Pietrowicz for inspirations and motivations. Special words of gratitude and thanks for the invitation to ANU and warm hospitality I must direct to prof. dr. Rajeev Goré. Last, but not least I would like to thank my Wife Joasia and our children: Helenka and Kazimierz for their persisting patience, love and spiritual as well as practical support.