Logic and Paradox. Lecture 2

1

Pablo Cobreros

Vagueness

Vagueness has to do with the sorites paradox and with borderline cases. Sorites Imagine a long series of patches of color. The first is red and the last is yellow, but adjacent patches in the series cannot be discriminated in color by the naked eye: Table 1: A sorites series a1 ∼ a2 . . .

an−1 ∼ an

. . . ak . . .

red

¬red

red?

Classical logic =⇒ There’s a last red item followed by a non-red item. The negation of this is the “tolerance principle”: ∀x, y(x ∼ y ⊃ (red(x) ⊃ red(y))) Borderline cases. Suppose Tim is a borderline case of the predicate ‘thin’. Then it is unclear whether Tim is thin and this unclarity, it seems, does not depend on a lack of knowledge of the measures of Tim or on a failure of linguistic competence. Table 2: Symmetry in borderline cases assent ‘Tim is thin’ dissent

Thus, there is a sort of symmetry in the dispositions to assent or dissent on whether Tim is thin. Now according to classical semantics the sentences ‘Tim is thin’ and ‘Tim is not thin’ must have an opposite truth-value and this is in tension with the symmetry of our dispositions. Thus, there must be something wrong with classical semantics, or so it seems. Straightforward options. First, one might think classical logic and semantics are ok. This leads to epistemicism. The sorites argument is unsound. Indefiniteness associated to vagueness is a manifestation of ignorance. There’s no symmetry at the level of semantics (Sorensen, Williamson, Fara). Apparent incompatibility with meaning’s being determined by use. Alternatively, we might hold that symmetry of dispositions go in hand with symmetry at the semantic level. Still we have two options: either ’Tim is thin’ is neither true nor false (truth-value gap) or it is both (truth-value glut).

Logic and Paradox. Lecture 2

2

Pablo Cobreros

K3 and LP

2.1

Language, semantics and logic

i) Language: a denumerable set of propositional variables V ar: {p, q, r, . . .}, the logical constants: ¬, ∧, ∨, ⊃, ≡ and two parenthesis: ‘(’ and ‘)’ p | ¬A | (A ∧ B) | (A ∨ B) | (A ⊃ B) | (A ≡ B) ii) Interpretation: A K3/LP interpretation I is a function: V ar 7→ {0, 12 , 1} A K3/LP interpretation I extends to all the formulas of the language according to the following clauses: I(A ∧ B) = min(I(A), I(B)) I(¬A) = 1 − I(A) I(>) = 1 I(⊥) = 0 How should we interpret value 12 ? Does it express a gap (and, thus, a way of failing to be true)? Does it express a glut (and, thus, a way of being true)? If validity is a matter of necessary preservation of truth each reading motivates different consequence relations. Definition 1 (Tolerant and strict truth). A sentence A is tolerantly true in a given interpretation I, just in case I(A) > 0. A is strictly true in I just in case I(A) = 1. Remark: Note that tolerant and strict truth are duals in much the same way as ∃ and ∀ are. Definition 2 (LP-consequence). Γ LP ∆, just in case there is no interpretation I such that I(B) > 0 for all B ∈ Γ and I(A) = 0 for all A ∈ ∆ Definition 3 (Duality). Two consequence relations x and y are duals just in case, Γ x ∆ ⇐⇒ ¬(∆) y ¬(Γ)

Logic and Paradox. Lecture 2

2.2

Pablo Cobreros

Tagged tableaux

In this section we show how to extend tableaux for LP and K3 . This time we need to make use of tagged tableaux : trees where nodes are formed by a pair of a formula and a tag. Motivation. In classical semantics, when we ask about satisfaction, we need to care just about whether sentences are true or false and falsity in a tree can be expressed by inserting the negation of a formula. Now we have, in LP , two ways of being false and, in K3 , two ways of being untrue. We can convey this (relevant!) information in a tableaux making use of tags: ‘t’ for tolerant truth and ‘s’ for strict truth. Γ K3 ∆ means there’s no interpretation where all the Γ’s are strictly true and none of the ∆’s is strictly true. But not being strictly true (< 1) does not mean being strictly false (= 0)! Rather it means: the negation is tolerantly true. Thus, in order to check whether Γ K3 ∆ we will depart from a tagged tree with initial list: Γ, s || ¬∆, t Example 2: Example 1: Proof.

p ⊃ q, p 2LP q p, t p ⊃ q, t ¬q, s ¬p, t

q, t ⊗

Proof.

p ⊃ q K3 ¬q ⊃ ¬p p ⊃ q, s ¬(¬q ⊃ ¬p), t ¬q, t ¬¬p, t p, t ¬p, s ⊗

q, s ⊗

Logic and Paradox. Lecture 2

2.3

Pablo Cobreros

Evaluation

Borderline cases. Both K3 and LP provide a sensitive response to borderline cases and symmetry of dispositions. For K3-ists the right thing to do is refuse to assert ‘Tim is thin’ and refuse to assert ‘Tim is thin’ (same disposition) for LP-ists the right thing to do is to assert both. There is still an annoying thing here: there seems to be just so much symmetry between both responses that one might think that either both are right or neither is. Sorites. The prima facie problematic premise in the sorites argument is the tolerance principle: (Tol)

∀x, y((red(x) ∧ x ∼ y) ⊃ red(y))

That elements in the series are very very similar (∼) looks beyond dispute, so we can concentrate on the conditionals of the form red(an ) ⊃ red(an+1 ) where an and an+1 are adjacent elements of the series. This is what happens: Table 3: Conditionals in a sorites series a1 ∼ a2 . . . red

. . . ak . . . red?

an−1 ∼ an ¬red

1 1 2

0

(The quantifier expression ∀ is defined as a generalization of ∧: it takes the minimum value of all instances) So for K3-ists the argument is valid but unsound and for LP-ists the tolerance principle is valid but the argument is not valid (note the facts in Exercise 7).

Logic and Paradox. Lecture 2

3 3.1

Pablo Cobreros

A three-valued semantics for classical logic The Logic ST

Given the three-valued semantics above, we can take a perspective different to either LP or K3 towards logical consequence. Instead of concentrating on the informal reading of 1 2 we might want to ask about its role in valid arguments. A valid argument is one that does not lead you from very good premises to a very bad conclusion. This motivates the following definition: Definition 4 (ST-consequence). Γ ST ∆, just in case there is no interpretation I such that I(A) = 1 for all A ∈ Γ and I(B) = 0 for all B ∈ ∆ So the third value looks pretty useless in this context. It makes it’s job, however, once we have an interesting language. A L∼ language is a first-order language with a similarity relation ∼P for each vague predicate P . The semantic behavior of a similarity relation ∼P might be given with the following constraint: I(a ∼P b) = 1 just in case |I(P a)−I(P b)| < 1 (that is, when the distance in the truth value of the sentence is less than 1). Although we won’t generally work with first-order languages, the following is an interesting observation ∼ (where ∼ ST is ST over L languages): ∼ ST is a proper extension of CL . ∼ ST ∀x, y((P x ∧ x ∼ y) ⊃ P y) P t, t ∼ u ∼ ST P u ∼ ST is not transitive. The logic ST was proposed by Egr´e, Ripley, van Rooij and Cobreros in Cobreros et al. (2012).

References Cobreros, P., Egr´e, P., Ripley, D., and van Rooij, R. (2012). Tolerant, classical, strict. Journal of Philosophical Logic, 41(2):347–385. Priest, G. (2008). An Introduction to Non-Classical Logic: From If to Is. Cambridge University Press.