ICCL Summer School 2008 The logic of generalized truth values. A - PowerPoint PPT Presentation

ICCL Summer School 2008 The logic of generalized truth values. A tour into Philosophical Logic Heinrich Wansing Heinrich Wansing The logic of generalized truth values

1. Generalized truth values 2. The trilattice SIXTEEN 3 3. Hyper-contradictions and Belnap-trilattices 4. Some 7-valued logics 5. Summary Heinrich Wansing The logic of generalized truth values

The set of classical truth values 2 = { F , T } . A classical valuation v 2 (a 2-valuation) is a function from the set of atoms into 2 . Nuel Belnap (1977) suggested that the elements of P ( 2 ) may be viewed as four generalized truth values: N = ∅ – none (“told neither falsity nor truth”); F = { F } – plain falsehood (“told only falsity”); T = { T } – plain truth (“told only truth”); B = 2 = { F , T } – both falsehood and truth (“told both falsity and truth”). 4 = P ( 2 ) = { N , F , T , B } . A function v 4 from the set of propositional variables into 4 (a 4-valuation) is then a generalized truth value function. Heinrich Wansing The logic of generalized truth values

Matthew Ginsberg (1988) introduced the notion of a bilattice and pointed out that Belnap’s four truth values form the smallest non-trivial bilattice. B r i ✻ F T r r r N ✲ t Figure: Bilattice FOUR 2 ≤ t – a truth order ( x ≤ t y – y is as least as true as x ) ≤ i – an information order ( x ≤ i y – y is as least as informative as x ) Heinrich Wansing The logic of generalized truth values

The information order is just set-inclusion, and ≤ t , it is claimed, represents increase in truth. Lattice meet (greatest lower bound) and lattice join (least upper bound) with respect to ≤ t give rise to a conjunction ∧ and a disjunction ∨ , and one can define a unary operation − that satisfies − − x = x and ≤ t -inversion and gives rise to a negation connective ∼ . Definition A � 4 B iff ∀ v 4 � v 4 ( A ) ≤ t v 4 ( B ) � . Heinrich Wansing The logic of generalized truth values

This entailment relation can be axiomatized by the consequence system know as First Degree Entailment, the logic codifying, according to Belnap, how a computer should think. The system is a pair ( L , ⊢ ), where ⊢ is a binary relation (consequence) on the language L satisfying the following postulates (axiom schemes and rules of inference): A 1. A ∧ B ⊢ A A 2. A ∧ B ⊢ B A 3. A ⊢ A ∨ B A 4. B ⊢ A ∨ B A 5. A ∧ ( B ∨ C ) ⊢ ( A ∧ B ) ∨ C A 6. A ⊢ ∼∼ A A 7. ∼∼ A ⊢ A R 1. A ⊢ B , B ⊢ C / A ⊢ C R 2. A ⊢ B , A ⊢ C / A ⊢ B ∧ C R 3. A ⊢ C , B ⊢ C / A ∨ B ⊢ C R 4. A ⊢ B / ∼ B ⊢ ∼ A . Heinrich Wansing The logic of generalized truth values

But, after moving from 2 to P ( 2 ), should one stop at 4 ? An Argument: Consider, e.g., the combination TB (= {{ T } , { F , T }} ) of T and B . This new truth value would then mean “true and true-and-false”. But a repetition of truths gives us no new information and hence is superfluous. Thus, the meaning of TB collapses just into “true-and-false”, and in this way we simply obtain B . This argument is, of course, a non sequitur: TB = {{ T } , { F , T }} is distinct from B = { T } ∪ { F , T } . Heinrich Wansing The logic of generalized truth values

A further generalization of the classical truth values results from taking the power-set of { N , F , T , B } giving the set 16 : 1. N = ∅ 9. FT = {{ F } , { T }} 2. N = { ∅ } 10. FB = {{ F } , { F , T }} 3. F = {{ F }} 11. TB = {{ T }} , { F , T }} 4. T = {{ T }} 12. NFT = { ∅ , { F } , { T }} 5. B = {{ F , T }} 13. NFB = { ∅ , { F } , { F , T }} 6. NF = { ∅ , { F }} 14. NTB = { ∅ , { T } , { F , T }} 7. NT = { ∅ , { T }} 15. FTB = {{ F } , { T } , { F , T }} 8. NB = { ∅ , { F , T }} 16. A = { ∅ , { T } , { F } , { F , T }} . Heinrich Wansing The logic of generalized truth values

Assume that one source of information tells a computer that a sentence is true-only, while another informant supplies inconsistent data, namely that the sentence is both true and false: this is a clear case for TB . If the computer is only told that a sentence is true-only this gives the hitherto unavailable value {{ T }} = T . Support from the literature: Dunn and Hardegree (2001, p. 277): “[T]here can be states of information that are inconsistent, incomplete, or both ”. Heinrich Wansing The logic of generalized truth values

We must, for example, distinguish between the following situations: An informant gives only the information “The sentence is true”: { T } An informant gives only the information “The sentence is true-only”: {{ T }} Heinrich Wansing The logic of generalized truth values

♣♣♣♣ ♣ ❘ ✠ ♣ ♣ ♣ ′′ C C 3 ♣♣♣♣♣♣♣♣ ✒ ■ ♣ ♣ ♣ ❄ ♣ ✲ ✛ C 2 C 4 ′ C ✻ C 1 Figure: A computer network Heinrich Wansing The logic of generalized truth values

How insane are they? Do they really suggest that the logic of C ′′ is an at least 65536-valued logic? Yes, they do, but . . . before we shall return to this question, we first take a slightly closer look at a certain lattice structure defined on 16 . Definition An n-dimensional multilattice (or simply n-lattice ) is a structure M n = ( S , ≤ 1 , . . . , ≤ n ) such that S is a non-empty set and ≤ 1 , . . . , ≤ n are partial orders defined on S such that ( S , ≤ 1 ) , . . . , ( S , ≤ n ) are all distinct lattices. Heinrich Wansing The logic of generalized truth values

Consider any two distinct partial orders defined on some non-empty set. We say that these relations are mutually independent with respect to these definitions (or are defined mutually independently) iff they are not inversions of each other and the only common terms that are used in both definitions, except of metalogical connectives and quantifiers, are the usual set theoretical terms. Definition A multilattice is called proper iff all its (pairs of) partial orders can be defined mutually independently. Heinrich Wansing The logic of generalized truth values

Formal definition of the ordering relations ≤ i and ≤ t in FOUR 2 . ≤ i : for any x , y ∈ 4 , x ≤ i y iff x ⊆ y . ≤ t : x ≤ t y iff x t ⊆ y t and y f ⊆ x f , where for each element of 4 its ‘truth part’ and its ‘falsity part’ is defined as follows as follows: x t x f := { z ∈ x | z = T } ; := { z ∈ x | z = F } . In FOUR 2 , ≤ t is not just a truth order but rather a ‘ truth-and-falsity order’ in the sense that in order to define ≤ t we must refer to both T and F . Heinrich Wansing The logic of generalized truth values

For every x ∈ 16 we denote by x t the subset of x that contains exactly those elements of x which in turn contain T as an element and by x − t the ‘truthless’ subset of x : x t x − t := { y ∈ x | T ∈ y } ; := { y ∈ x | T / ∈ y } ; and analogously for falsity: x f x − f := { y ∈ x | F ∈ y } ; := { y ∈ x | F / ∈ y } . Definition For every x , y in 16 : x ≤ i y iff x ⊆ y ; x ≤ t y iff x t ⊆ y t and y − t ⊆ x − t ; x ≤ f y iff x f ⊆ y f and y − f ⊆ x − f . Heinrich Wansing The logic of generalized truth values

We obtain a structure that combines the three (complete) lattices ( 16 , ≤ i ), ( 16 , ≤ t ), and ( 16 , ≤ f ) into the trilattice SIXTEEN 3 = ( 16 , ≤ i , ≤ t , ≤ f ). A s NFT FTB s s FT s NFB NTB s s i ✻ NF NT FB TB s s s s s s F T s NB s s N B s N ✲ f t ✲ Heinrich Wansing The logic of generalized truth values

SIXTEEN 3 may also be represented as the structure ( 16 , ⊓ i , ⊔ i , ⊓ t , ⊔ t , ⊓ f , ⊔ f ). On SIXTEEN 3 we might want to consider unary ‘inversion’ operations with the following properties: 1 . t − inversion ( − t ) : 2 . f − inversion ( − f ) : ( a ) a ≤ t b ⇒ − t b ≤ t − t a ; ( a ) a ≤ t b ⇒ − f a ≤ t − f b ; ( b ) a ≤ f b ⇒ − t a ≤ f − t b ; ( b ) a ≤ f b ⇒ − f b ≤ f − f a ; ( c ) a ≤ i b ⇒ − t a ≤ i − t b ; ( c ) a ≤ i b ⇒ − f a ≤ i − f b ; ( d ) − t − t a = a . ( d ) − f − f a = a . 3 . i − inversion ( − i ) : 4 . tf − inversion ( − tf ) : ( a ) a ≤ t b ⇒ − i a ≤ t − i b ; ( a ) a ≤ t b ⇒ − tf b ≤ t − tf a ; ( b ) a ≤ f b ⇒ − i a ≤ f − i b ; ( b ) a ≤ f b ⇒ − tf b ≤ f − tf a ; ( c ) a ≤ i b ⇒ − i b ≤ i − i a ; ( c ) a ≤ i b ⇒ − tf a ≤ i − tf b ; ( d ) − i − i a = a . ( d ) − tf − tf a = a . Heinrich Wansing The logic of generalized truth values

5 . ti − inversion ( − ti ) : 6 . fi − inversion ( − fi ) : ( a ) a ≤ t b ⇒ − ti b ≤ t − ti a ; ( a ) a ≤ t b ⇒ − if a ≤ t − if b ; ( b ) a ≤ f b ⇒ − ti a ≤ f − ti b ; ( b ) a ≤ f b ⇒ − if b ≤ f − if a ; ( c ) a ≤ i b ⇒ − ti b ≤ i − ti a ; ( c ) a ≤ i b ⇒ − if b ≤ i − if a ; ( d ) − ti − ti a = a . ( d ) − if − if a = a . 7 . tfi − inversion ( − tfi ) : ( a ) a ≤ t b ⇒ − tif b ≤ t − tif a ; ( b ) a ≤ f b ⇒ − tif b ≤ f − tif a ; ( c ) a ≤ i b ⇒ − tif b ≤ i − tif a ; ( d ) − tif − tif a = a . Heinrich Wansing The logic of generalized truth values