Correspondence analysis for strong three-valued logic
A. Tamminga
abstract. I apply Kooi and Tamminga's (2012) idea of correspondence analysis for many-valued logics to strong three-valued logic (k3). First, I characterize each possible single entry in the truth-table of a unary or a binary truth-functional operator that could be added to K3 by a basic inference scheme. Second, I define a class of natural deduction systems on the basis of these characterizing basic inference schemes and a natural deduction system for K3. Third, I show that each of the resulting natural deduction systems is sound and complete with respect to its particular semantics. Among other things, I thus obtain a new proof system for Lukasiewicz's three-valued logic.
Keywords: three-valued logic, correspondence analysis, proof theory, natural deduction systems
1 Introduction
Strong three-valued logic (K3) [1] and Lukasiewicz's three-valued logic L3 [2] have much in common: their truth-tables for negation, disjunction, and conjunction coincide, and they have the same concept of validity. The two logics differ, however, in their treatment of implication: whereas Kleene's implication is definable in terms of negation, disjunction, and conjunction, this does not hold true for Lukasiewicz's implication ( L3 is therefore a truth-functional extension of K3). This fact seriously complicates the construction of proof systems for L3.
In this paper, I present a general method for finding natural deduction systems for truth-functional extensions of K3. To do so, I use the correspondence analysis for many-valued logics that was
presented recently by [3]. In their study of the logic of paradox (LP) [4], they characterize every possible single entry in the truth-table of a unary or a binary truth-functional operator by a basic inference scheme. As a consequence, each unary and each binary truth-functional operator is characterized by a set of basic inference schemes. Kooi and Tamminga show that if we add the inference schemes that characterize an operator to a natural deduction system for LP, we immediately obtain a natural deduction system that is sound and complete with respect to the logic that contains, next to LP's negation, disjunction, and conjunction, the additional operator. In this paper, I show that the same thing can be done for K3.
The structure of my paper is as follows. First, I briefly present K3. Second, I give a list of basic inference schemes that characterize every possible single entry in the truth-table of a unary or a binary truth-functional operator. Third, I define a class of natural deduction systems on the basis of these characterizing inference schemes and a natural deduction system for K3. I show that each of the resulting natural deduction systems is sound and complete with respect to its particular semantics.
2 Strong three-valued logic (K3)
Strong three-valued logic (K3) provides an alternative way to evaluate formulas from a propositional language L built from a set P = {p,p',...} of atomic formulas using negation (-), disjunction (V), and conjunction (A). K3 adds a third truth-value 'none' to the classical pair 'false' and 'true'. In K3, a valuation is a function v from the set P of atomic formulas to the set {0, i, 1} of truth-values 'false', 'none', and 'true'. A valuation v on P is extended recursively to a valuation on L by the following truth-tables for V, and A:
f-
0 1
i i
1 0
fv 0 i 1
0 0 i 1
i i i 1
1 1 1 1
fA 0 i 1
0 0 0 0
i 0 i i
1 0 i 1
An argument from a set n of premises to a conclusion 0 is valid (notation: n |= 0) if and only if for each valuation v it holds that if v(0) = 1 for all ^ in n, then v(0) = 1.
3 Correspondence Analysis for K3
Let be the language built from the set P = {p,p',...} of
atomic formulas using negation (—), disjunction (V), conjunction (A), m unary operators ..., and n binary operators o^ ..., on. It is obvious that L(^)m(0)n is an extension of L. To interpret this extended language, I use K3 's concept of validity, the truth-tables f-, fv, and fA, but also the truth-tables , ..., and the truth-tables /01, ..., f0n. I refer to the resulting logic as K3(~)m(o)n-
To construct a proof system for K3(^)m(o)n, I follow [3]. I first characterize each possible single entry in the truth-table of a unary or a binary operator by a basic inference scheme. To do so, I need the following notion of single entry correspondence [3, p. 722]:
Definition 1 (Single Entry Correspondence). Let n c L(~)m(o)n and let 0 € L^m(o)n. Let x,y,z € {0,i, 1}. Let E be a truth-table entry of the type f^(x) = y or f0(x,y) = z. Then the truth-table entry E is characterized by an inference scheme n/0, if
E if and only if n = 0.
Accordingly, each of the nine possible single entries in a truth-table f^ for a unary operator ~ and each of the twenty-seven possible entries in a truth-table f0 for binary operator o is characterized by an inference scheme (I do the binary operator case first):
Theorem 1. Let € L(~)m(0)n ■ Then
0 iff —0 A = -(0 o U(0, 0) = { i iff —0 A(0 o V—(0 o ^)\= x
1 iff —0 A = 0 o ^
fo(0,i) =
fo(0, 1) =
fo(i, 0) =
fo(i, i) =
fo(i, 1) =
fo(1, 0) =
fo(1, i) =
fo(1, 1) =
0 iff —0 \= (0 V —0) V —(0 o 0)
1 iff —0, (0 O 0) V -(0 O 0) \= 0 V—0 1 iff —0\=(0 V—0) V (0 O 0)
0 iff —0 A 0\= —(0 o 0)
1 iff —0 A 0, (0 O 0) V —(0 O 0)\= X 1 iff —0 A 0 \= 0 O 0
0 iff —0\=(0 V—0) V—(0 O 0)
1 iff —0, (0 O 0) V —(0 O 0) \= 0 V—0 1 iff —0\=(0 V—0) V (0 O 0)
0 iff = (0 V —0) V (0 V —0) V —(0 O 0)
1 iff (0 O 0) V —(0 O 0) = (0 V —0) V (0 V —0) 1 iff = (0 V —0) V (0 V —0) V (0 O 0)
0 iff 0\=(0 V—0) V—(0 O 0)
1 iff 0, (0 O 0) V —(0 O 0) = 0 V —0 1 iff 0\=(0 V—0) V (0 o 0)
0 iff 0 A —0 \= —(0 o 0)
1 iff 0 A —0, (0 O 0) V —(0 O 0) = X 1 iff 0 A —0 = 0 O 0
0 iff 0\=(0 V—0) V—(0 o 0)
1 iff 0, (0 O 0) V —(0 O 0) = 0 V —0 1 iff 0 = (0 V —0) V (0 O 0)
0 iff 0 A 0 = —(0 O 0)
1 iff 0 A 0, (0 O 0) V —(0 O 0) = X 1 iff 0 A 0 = 0 O 0.
Proof. Case fo(0,0) = 0. Suppose that —0 A—0 = —(0 o 0). Then there is a valuation v such that v(—0 A —0) = 1 and v(—(0 O
0)) = 1. Then v(0) = 0, v(0) = 0, and v(0 o 0) = 0. Therefore, it must be that fo(0, 0) = 0.
Suppose that —0 A—0 \= —(0 o 0). Then —p A—q \= —(p o q), where p and q are atomic formulas. Then for every valuation v it holds that if v(—p A —q) = 1, then v(—(p o q)) = 1. Then for every valuation v it holds that if v(p) = 0 and v(q) = 0, then v(po q) = 0. Therefore, it must be that fo(0,0) = 0.
Case fo(1,i) = i. Suppose that 0, (0o0) V—(0o0) = 0V—0. Then there is a valuation v such that v(0) = 1, v((0O0)V—(0O0)) = 1 and v(0 V —0) = 1. Then v(0) = 1, v(0) = i, and v(0 o 0) = i. Therefore, it must be that fo(1, i) = i.
Suppose that 0, (0o0) V—(0o0) \= 0V—0. Then p, (poq) V —(poq) \= qV—q, where p and q are atomic formulas. Then for every valuation v it holds that if v(p) = 1 and v((p o q) V —(p o q)) = 1, then v(q V —q) = 1. Then for every valuation v it holds that if v(p) = 1 and v(q) = i, then v(p o q) = i. Therefore, it must be that
fo(1, i) = i.
The other cases are proved similarly. □
Theorem 2. Let 0,0 e L(~)m(o)n ■ Then
0 iff — 0 \= — ~ 0 f~ (0) = { i iff —0, (~ 0 V — ~ 0)\= 0 1 iff —0 j=~ 0
f 0 iff
f~(i) = { i iff
1 iff
f 0 iff
f~(1) = < i iff
1 iff
\= (0 V—0) V — ~ 0 0 V — ~ 0) \= 0 V —0
\= (0 V —0)V ~ 0 0 \= — ~ 0
0, 0 V — ~ 0) \= 0 0 0.
Proof. Adapt the proof of the previous theorem. □
As a result, given concept of validity and its truth-tables f-,, fv, and fA, each unary operator (1 < fc < m) is characterized by
the set of three basic inference schemes that characterize the three single entries in its truth-table f^k and each binary operator ol (1 < l < n) is characterized by the set of nine basic inference schemes that characterize the nine single entries in its truth-table f0l. The inference schemes that characterize a truth-table are independent.
4 Natural deduction systems
I now use the characterizations of the previous section to construct proof systems for truth-functional extensions of K3. First, I define a natural deduction system NDr3 which I later show to be sound and complete with respect to K3 (this is a corollary of my main theorem). Second, on the basis of ND^3 and Theorems 1 and 2, I define a natural deduction system for the logic K3(~)m(o)n as follows: for each unary operator (1 < k < m) I add its three characterizing basic inference schemes as derivation rules to ND^3 and for each binary operator oi (1 < l < n) I add its nine characterizing inference schemes as derivation rules to ND^3. Third, I show, using a Henkin-style proof, that the resulting natural deduction system is sound and complete with respect to the logic K3(~)m(o)n-
My proof-theoretical study of K3 closely follows Kooi and Tam-minga's (2012) proof-theoretical study of LP. In fact, to construct natural deduction systems for extensions of K3 and to prove their soundness and completeness, I only slightly adapt Kooi and Tam-minga's definitions, lemmas and theorems on extensions of LP. Let me first define the natural deduction system ND^3
Definition 2. Derivations in the system ND^3 are inductively defined as follows:
Basis: The proof tree with a single occurrence of an assumption 0 is a derivation.
Induction Step: Let D, D1, D2, D3 be derivations. Then they can be extended by the following rules (double lines indicate that the rules work both ways):
1For the notational conventions, see [5].
Di D2
fi -fi
fa
EFQ
Di D2 D D
fi Al AEi AE2
fi A fa fi fa
[fi]u
D D Di d2 D3
fi V/i A V/2 -x-X. -yEu
0 V fa 0 V fa x
D D D
0 -(0 V fa) -(0 A fa)
==== DN ^===== DeMv ===== DeMA --0 -0 A-fa -0 V -fa
On the basis of ND^3, I now define a natural deduction system for the logic K3(^)m(o)n. The Theorems 1 and 2 tell me that each truth-table f-k is characterized by three basic inference schemes and that each truth-table f0l is characterized by nine basic inference schemes. I obtain a new natural deduction system for the logic K3(~)m(o)n by adding to ND^3 these characterizing basic inference schemes as derivation rules.
More specifically, for each basic inference scheme fai,...,faj/0 that characterizes an entry f-k (x) = y in the truth-table f-k, I add the derivation rule
Di Dj
—-0-— R-ky)
to the natural deduction system ND^3. Similarly, for each basic inference scheme faifaj/0 that characterizes an entry f0l (x, y) = 2 in the truth-table f0l, I add the derivation rule
Di Dj
fai ••• faj
Rot (x,y,z)
to the natural deduction system ND^3.
For instance, assume that fo(0,0) = 0 is one of the truth-table entries in fo. Then, because Theorem 1 tells me that fo(0, 0) = 0 is characterized by the basic inference scheme —0 A —0/—(0 o 0), I add the derivation rule
D
—0 A
Ro(0,0, 0)
to the natural deduction system ND^3.
In this way, I define the system ND^3 + IJ k=i{R~k(x,V) : f-k (x) = V} + Uri=i{Rol (x,v,z) : foi (x,v) = z}, which I refer to as ND^3(^)m(o)n. I now show that this natural deduction system is sound and complete with respect to the logic K3(~)m(o)n.
4.1 Soundness of ND^3(^)m(o)n
A conclusion 0 is derivable from a set n of premises (notation: n h 0) if and only if there is a derivation in the system ND^3(^)m(o)n of 0 from n.
The system's local soundness is easy to establish:
Lemma 1 (Local Soundness). Let n, n', n" c £(^)m(o)n and let
0,0 e L(~)m(o)n . Then
(i) If 0 e n, then n \= 0
(ii) If n \= 0 and n' \= —0, then n, n' \= 0
(iii) If n \= 0 and n' \= 0, then n, n' \= 0 A 0
(iv) If n \= 0 A 0, then n \= 0
(v) If n \= 0 A 0, then n \= 0
(vi) If n \= 0, then n \= 0 V 0
(vii) If n \= 0, then n \= 0 V 0
(viii) If n \= 0 V 0 and n', 0 \= % and n'', 0 \= x, then n, n', n'' \= x
(ix) n \= 0 if and only if n \= ——0
(x) n \= —(0 V 0) if and only if n \= —0 A —0
(xi) n \= —(0 A 0) if and only if n \= —0 V —0.
Theorem 3 (Soundness). Let n c £(^)m(o)n and let 0 e
L(~)m (o)n ■ Then
If n h 0, then n \= 0.
Proof. By induction on the depth of derivations. The local soundness of the rules of the basic natural deduction system ND^3 follows from the previous lemma. For each unary operator ^k (1 < k < m) the local soundness of the three derivation rules in {R^k(x,v) : f~k(x) = y} follows from Theorem 2. For each binary operator ot (1 < l < n) the local soundness of the nine derivation rules in {Roi(x,v,z) : foi(x,y) = z} follows from Theorem 1. □
4.2 Completeness of NDf3(^)m(o)n
In my completeness proof, consistent prime theories are the syntactical counterparts of valuations:
Definition 3. Let n c . Then n is a consistent prime
theory (CPT), if
(i) n = L(~)m(o)n (consistency)
(ii) If n h 0, then 0 e n (closure)
(iii) If 0 V 0 e n, then 0 e n or 0 e n (primeness).
The syntactical counterpart of the truth-value of a formula under a valuation is a formula's elementhood in a consistent prime theory:
Definition 4. Let n c £(^)m(o)n and let 0 e L(~)m(o)n. Then 0's elementhood in n (notation: e(0, n)) is defined as follows:
i>, n)
if 0 e n and —0 e n if 0 e n and —0 e n if 0e n and —0e n if 0 e n and —0 e n.
To ensure that in the presence of an operator the notion of ele-menthood behaves in comformity with the operator's truth-tables, I need the following lemma:
Lemma 2. Let n be a CPT and let 0,0 e L(~)m(o)n. Then
(i) e(0, n)= 0
(ii) f-(e(0, n)) = e(—0, n)
(iii) fv(e(0, n),e(0, n)) = e(0 V 0, n)
(iv) fA(e(0, n),e(0, n)) = e(0 A 0, n)
(v) Uk (e(0, n)) = e(~k 0, n) for 1 < k < m
(vi) foi(e(0, n),e(0, n)) = e(0 oi 0, n) for 1 < l < n.
e
Proof.
(i) Suppose e(0, n) = 0. Then 0 € n and -0 € n. Then n h 0 and n I—'0. By the rule EFQ, it must be that n h — for all
— € L(~)m(o)n. By closure, — € n for all — € L(-)m(o)n. Then n = L(-)m(0)n. Contradiction.
(ii) Suppose e(0, n) = 0. Then 0 € n and -0 € n. By closure and the rule DN, -0 € n and --0 € n. Hence, e(-0, n) = 1 = f-(0) = f-(e(0, n)).
Suppose e(0, n) = i. Then 0 € n and -0 € n. By closure and the rule DN, -0 € n and --0 € n. Hence, e(-0, n) =
i = f-(i) = f-(e(0, n)).
Suppose e(0, n) = 1. Then 0 € n and -0 € n. By closure and the rule DN, -0 € n and --0 € n. Hence, e(-0, n) = 0 = f-(1) = f-(e(0, n)).
(iii) I prove the cases for (1) e(0, n) = 0 and e(—, n) = 0, (2) e(0, n) = i and e(—, n) = i, and (3) e(0, n) = 1 and e(—, n) = i. The other six cases are proved similarly.
(1) Suppose e(0, n) = 0 and e(—, n) = 0. Then 0 € n,
— € n, -0 € n, and € n. By primeness, 0 V — € n. By closure and the rules AI and DeMv, -(0 V —) € n. Hence, e(0 V n) = 0 = fv(0,0) = fv(e(0, n),e(—, n)).
(2) Suppose e(0, n) = i and e(—, n) = i. Then 0 € n,
— € n, -0 € n, and € n. By closure and the rule Vli, 0 V — € n. By closure and the rules AI and DeMv, -(0 V —) € n. Hence, e(0 V n) = i = fv(i,i) = fv(e(0, n),e(—, n)).
(3) Suppose e(0, n) = 1 and e(—, n) = i. Then 0 € n,
— € n, -0 € n, and € n. By closure and the rule Vli, 0 V — € n. By closure and the rules AEi and DeMv, -(0V—) € n. Hence, e(0V—, n) = 1 = fv(1, i) = fv(e(0, n),e(—, n)).
(iv) Analogous to (iii).
(v) There are three cases for each (1 < k < n). (For readability, the subscript k is dropped in the remainder of this proof.) I prove the case for e(0, n) = 0. The other two cases are proved similarly.
Suppose e(0, n) = 0. Then 0 e n and —0 e n. There are three cases:
(1) Suppose R^(0,0) is one of the three rules for ~ in NDK3(~)m(o)n. Then f^(0) = 0. By closure and the rule R^(0, 0), it must be that — ~ 0 e n. By (i), it must be that ~ 0 e n. Therefore, e(~ 0, n) = 0 = f~(0) = U(e(0, n)).
(2) Suppose R^ (0,i) is one of the three rules for ~ in NDLP)m(o)n. Then f^(0) = i. By closure, the fact that n is a CPT, and the rule R^(0, i), it must be that ~ 0 V — ~ 0 e n. By closure and the rules VI\ and VI2, ~ 0 e n and — ~ 0 g n. Therefore, e(~ 0, n) = i = fo(0) = U(e(0, n)).
(3) Suppose R^(0,1) is one of the three rules for ~ in NDLP)m(o)n. Analogous to (1).
(vi) Analogous to (v).
□
Lemma 3 (Truth). Let n be a CPT. Let vn be the function that assigns to each atomic formula p in P the elementhood of p in n: vn(p) = e(p, n) for all p in P. Then for all 0 in L(^)m(o)n it holds that
vn(0) = e(0, n).
Proof. By an easy structural induction on 0. Use the previous lemma. □
Lemma 4 (Lindenbaum). Let n C C(^)m(0)n and let 0 € (0)n. Suppose that n 1/ 0. Then there is a set n* C C(^)m(0)n such that
(i) n C n*
(ii) n*^ 0
(iii) n* is a CPT.
Proof. Suppose that n | 0. Let —i, ■ ■ ■ be an enumeration of (0)n. I define the sequence no, n^ ■ ■ ■ of sets of formulas as follows:
no = n
n =| n U{—t+i}, ifn, U[—t+i}h 0
i+i \ n,, otherwise.
Take n* = U^nnn. Standard proofs show that (i), (ii), and (iii) hold. □
Theorem 4 (Completeness). Let n c £(^)m(0)n and let 0 € If n |= 0, then n h 0.
c(~)m{◦)-. Then
Proof. By contraposition. Suppose n | 0. By the Lindenbaum lemma, there is a CPT n* such that n c n* and n* | 0. Let vn* be the valuation introduced in the truth lemma. By the truth lemma, it holds that vn** (—) = 1 for all — in n and vn** (0) = 1. Therefore, n = 0. □
Corollary 1. The system ND^3 is sound and complete with respect to K3.
Proof. Consider the logic K3- that is obtained from K3 by adding K3's truth-table f- for negation to it. Evidently, K3- is K3. By the soundness and completeness theorems, ND^3- is sound and complete with respect to K3-. It is easy to see that the rules R-(0,1), R-(i,i), and R-(1,0) are derived rules in ND^3. □
5 Lukasiewicz's three-valued logic (L3)
Let me illustrate this general method for finding natural deduction systems for truth-functional extensions of K3 with Lukasiewicz's three-valued logic ( L3). L3 evaluates arguments consisting of formulas from a propositional language LD built from a set P = {p,p',...} of atomic formulas using negation (—), disjunction (V), conjunction (A), and implication (d). L3 has the same valuations as K3: in L3, a valuation is a function v from the set P of atomic formulas to the set {0, i, 1} of truth-values. A valuation v on P is extended recursively to a valuation on LD by the truth-tables for —, V, and A, and the truth-table for D:
fD 0 i 1
0 1 1 1
i i 1 1
1 0 i 1
L3 has the same concept of validity as K3: an argument from a set n of premises to a conclusion 0 is valid (notation: n = 0) if and only if for each valuation v it holds that if v(0) = 1 for all 0 in n, then v(0) = 1.
Theorem 1 tells me that the truth-table fD is characterized by the following nine basic inference schemes:
fD(0, 0) =1 iff —0 A = 0 D 0
fD (0, i) =1 iff -0 = (0 V —0) V (0 D 0)
f D (0,1) =1 iff —0 A 0 = 0 D 0
fD(i, 0) =i iff —0, (0 D 0) V —(0 D 0) = 0 V —0
fD(i,i) - =1 iff = (0 V —0) V (0 V —0) V (0 D 0)
fD (i, 1) =1 iff 0 = (0 V —0) V (0 D 0)
fD(1,0) =0 iff 0 A — 0 = —(0 D 0)
fD(1, i) =i iff 0, (0 D 0) V —(0 D 0) = 0 V —0
fD(1, 1) =1 iff 0 A 0 = 0 D 0.
From Theorems 3 and 4 it follows that the natural deduction
system ND^3D, obtained from adding these nine basic inference schemes as derivation rules to the natural deduction system ND^3, is sound and complete with respect to L3. The general method I
presented in this paper, therefore, makes it easy to find natural deduction systems for truth-functional extensions of K3.
6 Conclusion
Next to Kooi and Tamminga's (2012) proof-theoretical study of LP, the present investigation of K3 is only a second step in the study of many-valued logics using correspondence analysis. At the current stage of research, the following questions seem pressing. Which many-valued logics can be studied using correspondence analysis? Which many-valued logics cannot? Are there some characteristics a many-valued logic must have to be amenable to correspondence analysis?
References
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