Relevant Logic as Paraconsistent Logic

2023-2 Modal Logic (Segment 3)

 

Q) Explain why that (∀x)Roox yields (∀x)Roxo and collapses Relevent logic into classical logic. If is sufficient (for the present purposes) to just prove something that doesn’t hold in Relevant Logic, e.g., ¬A ∧ (A ∨ B) .–Ȝ. B. Why does it seem not to be viable to take relevant logic to be a universal logic (a logic that can capture all others?)

#2) Generally speaking, Relevant Logic is a non-classical logic aiming to represent the concept of implication that we have before we learn classical logic (Edwin 2022). In Routley–Meyer semantics, (p –Ȝ q)ᵒ is defined as (∀xy)(Roxy .⊃. pˣ ⊃ qʸ). Here, Roxy can be interpreted in terms of a theory of information: ‘o’ is an information-theoretic channel between sites ‘x’ and ‘y’ where a channel is a conduit through which information is transferred, and each site is a receiver of information (Edwin 2022). We can then say Roxy represents a situation where information at ‘x’ flows at ‘y.’

Notably, Relevant Logic is a branch of paraconsistent logic that invalidates the principle of explosion (Graham, Tanaka, and Weber 2022). The principle of explosion is a logical law asserting that any propositions can be inferred from contradiction, which is universally valid in many logical systems, including Classical logic. Suppose that we have any theorems of the form of contradiction, say, p ∧ ~p.

1.  Suppose ├ pc   p ∧ ~p
2. ├ pc    p                                         1, simplification
3. ├ pc    ~p                                       1, simplification
4. ├ pc    p ∨ q                                 2, addition
5. ├ pc   q                                          3, 4 Disjunctive syllogism

In Relevant Logic, however, the principle of explosion does not hold because relevant logicians believe that “the real problem […]  is not to do with the contradictory premises but to do with the lack of connection between the premises and the conclusion” (Graham, Tanaka, and Weber 2022). Therefore, instead of using ‘~’ alone, they invented a star operator to interpret ‘¬’ in a relevant sense (Edwin 2022). The negation of A, ¬A, is true at b if and only if A is false at b* (Edwin 2022). So, the translation schema is as follows:

(¬A)ᵇ = ∼(Aᵇ*).

Also, the Routley–Meyer semantics takes the following two axioms containing a star operator.

Ax1: Rabc ⊃ Rac*b*

Ax2: b** = b

Now, it is easy to show that (¬A ∧ (A ∨ B) .–Ȝ. B)ᵒ is not valid in Relevant Logic. Let us track the way of how the star operator blocks the principle of explosion. To prove (∀xy)(Roxy .⊃.  (¬A ∧ (A ∨ B))ˣ ⊃ Bʸ), we would suppose Roab ∧ (¬A ∧ (A ∨ B))ᵃ and try to show Bᵇ. Then, by Inheritance, (¬A ∧ (A ∨ B))ᵇ is obtained. Accordingly, (¬A)ᵇ and (Aᵇ ∨ Bᵇ) are acquired, in turn. In this case, however, using the disjunctive syllogism to get Bᵇ is illicit because (¬A)ᵇ is not ~Aᵇ, but ∼(Aᵇ*), as we previously noted.

It is worth noting that adopting the axiom (∀x)(Roox), which yields (∀x)Roxo, collapses Relevant Logic into Classical Logic. Specifically, this additional axiom will make (¬A ∧ (A ∨ B) .–Ȝ. B)ᵒ valid in the framework of Relevant logic, as follows:

├R (¬A ∧ (A ∨ B) .–Ȝ. B)ᵒ
├R (∀xy)(Roxy .⊃.  ((¬A .∧. A ∨ B)ˣ ⊃ Bʸ)
1. Suppose ├R Roab ∧ (¬A .∧. A ∨ B)ᵃ                                 Show Bᵇ
2. ├R (¬A .∧. A ∨ B)ᵇ                                                           1, Inheritance
3. ├R (¬A)ᵇ ∧ (A ∨ B)ᵇ                                                         2, (A ∧ B)ᵇ = Aᵇ ∧ Bᵇ
4. ├R (¬A)ᵇ                                                                             3, simplification
5. ├R (A ∨ B)ᵇ                                                                       3, simplification
6. ├R (Aᵇ ∨ Bᵇ)                                                                    5, (A ∨ B)ᵇ = Aᵇ ∨ Bᵇ
7. ├R ∼(Aᵇ*)                                                                           4, (¬A)ᵇ = ∼(Aᵇ*)
8. ├R Roob                                                                            (∀x)(Roox), UI
9. ├R Rob*o*                                                                         8, Rabc ⊃ Rac*b*
10. ├R Rob*o* ∧ ∼(Aᵇ*)                                                        9, 7 Conjunction
11. ├R ∼(Ao*)                                                                        10, Inheritance
12. ├R Rob*o                                                                        (∀x)(Roxo), UI
13. ├R Roo*b**                                                                      12, Rabc ⊃ Rac*b*
14. ├R Roo*b                                                                         13, b** = b
15. ├R Roo*b ∧ ∼(Aᵒ*)                                                         14, 11 Conjunction
16. ├R ∼(Aᵇ)                                                                           15, Inheritance
17. ├R Bᵇ                                                                                6, 16, DS
18. ├R Roab ∧ (¬A .∧. A ∨ B)ᵃ .⊃. Bᵇ                                 1-17, CP
19. ├R Roab .⊃. (¬A .∧. A ∨ B)ᵃ ⊃ Bᵇ                                 18, PC
20. ├R (∀xy)(Roxy .⊃. (¬A .∧. A ∨ B)ˣ ⊃ Bʸ)                      19, UG
21. ├R (¬A ∧ (A ∨ B) .–Ȝ. B)ᵒ                                                20, Def –Ȝ

In a nutshell, adding (∀x)(Roox) makes Relevant Logic lose its original purpose, which is to avoid the lack of connection between the premises and the conclusion. Consequently, considering relevant logic to be a universal logic is not viable.

 

References

Edwin, M. (2022) “Relevance Logic,” The Stanford Encyclopedia of Philosophy (Fall 2022 Edition), Edward N. Zalta & Uri Nodelman (eds.), URL = <https://plato.stanford.edu/archives/fall2022/entries/logic-relevance/>.

Graham, P., Tanaka, K., and Weber, Z. (2022) “Paraconsistent Logic,” The Stanford Encyclopedia of Philosophy (Spring 2022 Edition), Edward N. Zalta (ed.), URL = <https://plato.stanford.edu/archives/spr2022/entries/logic-paraconsistent/>.