C. I. Lewis' Non-Normal Systems (S1-S3)

2023-2 Modal Logic (Segment 3)

 

Q) Explain the system S1. In what sense is it the case that the “unrestricted” rules of Necessitation does not hold in S1. Does pc tautology necessitation hold in S1 ? What is the Cresswell argument (See my SS1 paper) that S1 has no theorems of the form LL(p⊃ p)? Explain the semantic system for S2 and S3 and S4 where one introduces odd vs normal worlds. Note that both S3 and S4 have transitivity and reflexivity. What happens to world generation in non-normal worlds.

#3) In Principia Mathematica by Russell and Whitehead, there exist two propositions notoriously known as the paradoxes of implication. The first one is p ⊃ (q ⊃ p), and the second one is ~p ⊃ (p ⊃ q). Both can be easily derived from PC tautologies, as follows:

1. ├ pc ~p ∨ p                               pc (taut)
2. ├ pc (~p ∨ p) ∨ ~q                   1, addition
3. ├ pc ~p ∨ (~q ∨ p)                   2, Assoc, comm
4. ├ pc p ⊃ (q ⊃ p)                       4, Def ⊃

Likewise,

1. ├ pc p ∨ ~p                               pc (taut)
2. ├ pc (p ∨ ~p) ∨ q                     1, addition
3. ├ pc p ∨ (~p ∨ q)                     2, Assoc
4. ├ pc ~p ⊃ (p ⊃ q)                     3, Def ⊃

And as a corollary, we obtain the following proposition that seems irrational.

1. ├ pc p ⊃ (q ⊃ p)                  
2. ├ pc ~p ⊃ (p ⊃ q)      
3. ├ pc p ∨ ~p                              pc (taut)
4. ├ pc (q ⊃ p) v (p ⊃ q)             1, 2, 3, Derived rule (CD)

To settle this paradox, C. I. Lewis proposed a novel system which is called S1 by introducing a concept of strict implication, expressed as ‘–Ȝ.’ Whereas material implication, ‘p ⊃ q,’ means nothing but ‘~p ∨ q (not p or q),’ Lewis aimed to capture implication in a strict sense where q follows from p.

According to Cresswell, Lewis’ (α –Ȝ β) can be defined as L(α ⊃ β) or ~M(α ∧ ~β), when written in our modern notation (cf. Hughes and Cresswell 1996: 195). Based on this understanding (referred to as cS1 when necessary), Cresswell compared the system S1 with T and concluded that S1 “contains nearly all of the basis of T but not quite all” (Hughes and Cresswell 1968: 223). In fact, like T, every well-formed formula (wff) of PC is a wff of S1, notwithstanding that S1 is not constructed as an extension of PC. Moreover, S1 has not only the rule of L-M Interchange (LMI) but also Lp ⊃ p as its theorem (Hughes and Cresswell 1968: 223-224). However, Cresswell points out that S1 is weaker than T because the “unrestricted” rule of Necessitation does not hold in S1. In other words, ⊢S1 α ⇒ ⊢S1 Lα does not hold in S1; rather the rule of necessitation should be restricted to the case where a is PC tautology, i.e., ⊢pc α ⇒ ⊢S1 Lα. Take, for instance, L(p ∧ q .⊃. q ∧ p). It is easy to see both S1 and T have this theorem, as follows:

├ T L(p ∧ q .⊃. q ∧ p)
1. ├ T p ∧ q .⊃. q ∧ p         PC
2.├ T L(p ∧ q .⊃. q ∧ p)      1, N

├ cS1 L(p ∧ q .⊃. q ∧ p)
1.├ cS1 p ∧ q .–Ȝ. q ∧ p            CresswellAxS1.1
2.├ cS1 L(p ∧ q .⊃. q ∧ p)        Cresswell Def –Ȝ

(Hughes and Cresswell 1968: 217; Landini 2023: 6).

However, although T also includes LL(p ∧ q .⊃. q ∧ p) as its theorem through the application of the rule of Necessitation, L(p ∧ q .–Ȝ. q ∧ p) cannot be considered a theorem of S1. For the rule of Necessitation in S1 cannot be generalized to apply to all theorems of S1 (Hughes and Cresswell 1968: 227).

By the same token, Cresswell asserts that S1 contains no theorems of the form LLα, for any wff α (Hughes and Cresswell 1968: 227; Landini 2023: 6). Cresswell’s argument is grounded in his proof that accepting any theorems of the form LLα would result in the unrestricted rule of Necessitation. The proof in cS1 is as follows:

1. Suppose ├ cS1 LLα
2. Suppose ├ cS1 Lβ
3. ├ cS1 Lp ⊃ p                                     TS 1.18
4. ├ cS1 LLα ⊃ Lα                                 TS 1.18[Lα/p]
5. ├ cS1 Lα                                             1, 4 MP
6. ├ cS1 Lp .⊃. q –Ȝ p                           TS 1.22
7. ├ cS1 Lα.⊃. β –Ȝ α                            TS 1.22[α/p, β/q]
8. ├ cS1 β –Ȝ α                                       5, 7 MP
9. ├ cS1 Lβ .⊃. α –Ȝ β                           TS 1.22[β/P, α/q]
10. ├ cS1 α –Ȝ β                                     2, 9 MP     
11. ├ cS1 α = β                                       8, 10 conjunction, Def =
12. ├ cS1 LLβ                                         1, 11 Eq

It is worth noting that all the axioms of S1 are equivalent to wffs of the form Lβ (Hughes and Cresswell 1968: 228). Therefore, we will get the derived rule, ├ cS1 Lβ ⇒ ├ cS1 LLβ, which is the same as the unrestricted rule of Necessitation.

It is also worth noting that Cresswell’s argument is valid insofar as we accept his definition of –Ȝ, viz., (a –Ȝ b)=ᵈᶠ L(a ⊃ b). However, this definition is illicit as it cannot apply to all wffs (Landini 2023: 2). Instead, we may construct the system called SS1 where we use relevant logic to interpret C. I. Lewis’ strict implication as relevant entailment by defining L in the following ways:

L² =ᵈᶠ ~p –Ȝ p
L³ =ᵈᶠ p⊃p .–Ȝ. P
L⁴ =ᵈᶠ p –Ȝ p .–Ȝ. P

Then we can get theorems like L²L²(s ⊃ s), L²L³(s ⊃ s), L³L²(s ⊃ s), and L³L⁴(s ⊃ s) (Landini 2023: 7). Consequently, Cresswell’s argument is refuted within the framework of SS1.

 As is widely known, Lewis’ S1, S2 and S3 are intertwined with the notion of non-normal (or odd) worlds. S2 is obtained from S1 by adding M(p ∧ q) .–Ȝ. Mp ∧ Mq, and S3 is obtained from S2 by adding p –Ȝ q .–Ȝ. Lp –Ȝ Lq (Hughes and Cresswell 1996: 200). Like S1, the unrestricted rule of Necessitation does not hold in both S2 and S3. Namely, they are non-normal in that both “are compatible with (though they do not contain) the axiom MMα” (Hughes and Cresswell 1996: 201).

Contrastingly, the normal modal systems (such as T) are not compatible with the axiom MMα at all. The reason is simple: all the normal modal systems have the unrestricted rule of Necessitation. So, if α is a theorem of such systems, we can easily derive LLα as well. For LLα ≡ ~MM~α, we can also obtain ~MMα for any wff α by using LMI (L-M Interchange) and US (The Rule of Uniform Substitution). Thus, the axiom MMα cannot hold in any normal modal systems.

In the frames of non-normal modal systems, however, worlds where every proposition is possible can exist, and these are referred to as non-normal (or odd) worlds. Contrary to dead ends, Mα is always true and Lα is always false in such worlds. Therefore, S2 and S3 frames are defined differently from frames in normal modal systems.

Particularly, an S2 frame allows for (but does not necessarily require) the existence of one or more non-normal world and necessarily requires at least one normal word. Additionally, (∃wⁿ)(∀wᵒ)(wⁿRwᵒ) and (∀wⁿ)(wⁿRwⁿ) hold in S2 where wⁿ and wᵒ represent normal and odd worlds, respectively. S3 is the same as S2 except for the additional requirement that the accessibility relation is transitive. Thus, S2 has reflexivity, and S3 has both transitivity and reflexivity.

Now, to define an S2 frame, we represent it as a triple <W, R, N> instead of the conventional <W, R> used in normal modal systems. Here, W is a set of worlds that may include wᵒ. N is a proper subset of W, exclusively containing wⁿ. Then an S2 frame <W, R, N> would be: (1) “for every w ∈ W there is some w ∈ N” (by (∃wⁿ)(∀wᵒ)(wⁿRwᵒ)), and (2) “R is reflexive over N” (by (∀wⁿ)(wⁿRwⁿ)). Likewise, an S3 frame <W, R, N> can be obtained from an S2 frame by adding (3) “R is transitive over N.” When it comes to defining an S2 and S3-model, the distinctive feature of them is that if w is non-normal, then V(Lα,w) = 0, and V(Mα,w) = 1 in every case. Therefore, Cresswell redefines [VL] as follows: “for any wff, α, and for any w ∈ W, V(Lα, w) = 1 if w ∈ N and for every w’ such that wRw’, V(α, w’) = 1. Otherwise V(Lα,w) = 0” (Hughes and Cresswell 1996: 202).

Lewis’ S4 is obtained from S1 by adding the extra axiom, Lp –Ȝ LLp (Hughes and Cresswell 1968: 236). S4 is a proper extension of S1-S3 and achieves full unrestricted necessitation (Hughes and Cresswell 1968: 236; Landini 2023: 2). To show this, we can give a proof as follows:

1. ├ S4 p –Ȝ q .–Ȝ. L(p –Ȝ q)                          TS 4.1
2. ├ S4 p –Ȝ p .–Ȝ. L(p –Ȝ p)                          1[p/q]
3. ├ S4 p –Ȝ q = L(p ⊃ q)                              TS 1.15
4. ├ S4 p –Ȝ p  .–Ȝ. LL(p ⊃ p)                        2, 3 Eq
5. ├ S4 p –Ȝ p                                                 TS 1.2
6. ├ S4  LL(p ⊃ p)                                          4, 5 MP

For S4 has a theorem of the form LLα, it would result in the unrestricted rule of Necessitation, as we previously proved. Therefore, Lewis’ S4 is deductively equivalent to modern modal S4.

 

References

Cresswell, M. & Hughes, G. (1968) An Introduction to Modal Logic. London: Methuen.

Cresswell, M & Hughes, G. (1996) A New Introduction to Modal Logic. London: Routledge.

Landini G. (2023) “Lewis’ Strict Implication as Relevant Entailment,” In Progress, pp. 1-29.