User:Bripres

Conditional Logic
A conditional is a statement with two clauses in the form of an "If p, Then q" Statement where "p" is the antecedent clause and "q" is the consequent. Many sentences that contain conditionals are context dependent. A sentence of the form "If p then q" is an assertion that the consequent is true, not necessarily in the world as it is, but in the world that it would be if the antecedent were true. The modal operators of necessity $$\Box$$, and possibility, $$\Diamond$$,can illuminate how conditionals behave. To express conditionals of this sort, we need a function which takes a proposition and a possible world into another possible world where it is the case that the antecedent would be true. Which possible worlds that are accessible depends on different accessible relations in possible worlds. Let w1 and w2 be worlds with different constraints, and let w3 be a world where the constraints of w1 hold true but the constraints of w2 world do not hold. So from w1, w3 is an accessible world, but from w2, w3 is not accessible. Accessibility is thus a relative matter in this case. So we can think of accessibility as a relation between worlds, w3 is accessible from w1, but w3 is not accessible from w2. The modal operators of necessity $$\Box$$, and possibility, $$\Diamond$$, correspond to an accessibility relation. For any possible world w1 and sentence $$\phi$$, the sentence $$\Box$$ $$\phi$$ is true at world w1 iff, for every world w2 such that w2 is accessible from w1, $$\phi$$ is true at w2. Similarly $$\Diamond$$ $$\phi$$ is true at w1 iff, for some world w2 such that w2 is accessbile from w1, $$\phi$$ is true at w2. In other words, $$\Box$$ $$\phi$$ is true at w1 iff $$\phi$$ is true at some world accessible from w2 and $$\Diamond$$  $$\phi$$ is true at w1 iff $$\phi$$ is true at some world accessible from w1. There are several different forms of conditionals. Different conditionals deal with modal operators of possbility and necessity to determine an accessibility relation of the form  $$ (M,w) \models p $$> T iff R, w ||p|| ⊆ all q worlds.

INDICATIVE CONDITIONALS
The simplest type of indicative conditionals is the material conditional;“If p then q” has the truth conditionals of “p⊃q” or in other words those of “not both p and not-q”. An indicative conditional (p→q) is not valid if p is true and q is false. The conditional statement “If it is raining then the streets are wet.” will be rejected if it turns out that it is raining and the streets are dry.

More Examples

(a) If you strike the match, the match will light.

(b) If Tweety is a bird, Tweety can fly.

COUNTERFACTUAL CONDITIONALS
Counterfactuals are conditionals in which the one who asserts them typically believes the falsity of the antecedent;

Examples

(a) If John cooked dinner, Mary will eat.

(b) If John cooked dinner and Mary is full, she will not eat.

The counterfactual carries other logical facts. One of these is failure of contraposition: from p →q we cannot in general infer ¬q →¬p. For example, p= “Oswald didn’t kill Kennedy” and q = Oswald was in Texas on 22 November 1963”. If Oswald hadn’t killed Kennedy, then Kennedy would still have been in Texas on that day; but if Kennedy hadn’t been in Texas on that day then Oswald would never have had the chance to kill him. A second is failure of monotonicity: from p →q we cannot in general infer (p^r) →q for arbitrary r. For example p = “I jump out of a high building”, q = “I break my leg”, and r= “I am rescued by an alien spaceship on my way down”. A third is failure of transitivity: from p→q and q→r we cannot infer p→r. For example,p =“Hoover is Russian”, q = “Hoover is a communist”, r = “Hoover is a traitor”.

INDICATIVE CONDITIONALS v. COUNTERFACTUAL CONDITIONALS
The grammatical difference between the two types of conditionals reflects a difference in meaning because it reflects a difference in use. The following example illustrates both differences. Consider the two conditionals:

(a) If Oswald didn't shoot Kennedy, then someone else did.

(b) If Oswald hadn't shot Kennedy, then someone else would have.

It seems acceptable to hold (a) as acceptable: we all know that somebody killed Kennedy, so we know that if it wasn’t Oswald it was someone else. But it also seems that (b) is false, because they think that Oswald acted alone. So on the face of it the two conditionals say different things.

STALNAKER'S LOGIC
Stalnaker's theory applies possible world semantics in modal logic. He uses possible world semantics to modify the belief which the antecedent is premised on in order to make a minimal revision to the actual world to make it the case that the conditional is true. A function f assigns to each world w and sentence Φ a world such that f (Φ,w) that counts as Stalnaker's modification to make Φ true. In possible world semantics, the content of a statement is given by denoting which of the possible worlds would make it true. Where Φ> Ψ is a conditional with antecedentΦ and consequent Ψ, Stalnaker's modifications would specify : Φ> Ψ is true in w iff Ψ is true in f (Φ,w). This makes the semantics of the conditional true relative to the nearest worlds. Stalnaker outlines the following conditions to determine how the nearest possible world is selected where f (Φ,w) = α, where w is the base world and α is the selected world.

(1) For all antecendents Φ and base worlds w, Φ must be true in f (Φ,w).

(2) For all antecedents Φ and base worlds w, f (Φ,w) = λ only if there is no world possible with respect to w in which Φ is true. Following this Stalnaker takes as an axiom, namely the Stalnaker Axiom (SA), for his conditional logic :

(SA) ◊Φ ⊃ [¬(Φ > Ψ) ≡ (Φ > ¬Ψ)].

This axiom in the presence of Stalnaker's logic, validates the conditional excluded middle.

(CEM) (Φ > Ψ) v (Φ > ¬Ψ).

which corresponds to the assumpiton that a single world is selected as the closest antecedent world. If f(Φ, w) is the world selected then either Ψ or ¬Ψ is true.

LEWIS' COUNTERFACTUALS
Conversely, Lewis argued that it is not possible to assume comparative similarity of possible worlds is such that there will always be a unique nearest antecedent world. For example, in an antecedent such as "Bizet and Verdi are compatriots" a world where they are both French and a world where they are both Italian might be equally close to the actual world and they can conjuctively be closer than all other possible worlds. So, Lewis specifies that in comparing possible worlds:

Φ$$\Box$$ -> Ψ is true at w iff, for some (Φ ^ Ψ)-world is closer to w that any (Φ ^ ¬Ψ)-world.

If there is a unique closest Φ, then Stalnaker's logic coincides with Lewis'. But, Lewis notes that it can be the case when there is more than one closest world and thus,

Φ $$\Box$$-> Ψ just in case every nearest Φ world is a Ψ world. So in the example conditional previously stated Lewis would further Stalnaker's analysis to two cases that could be true in possible worlds:

(1) It is not the case that if Bizet and Verdi were compatriots that Bizet would be Italian.

and

(2) It is not the case that if Bizet and Verdi were compatriots that Bizet would not be Italian.

Although this is traditionally thought to be a counter-example to CEM, the sentences in conjuction sound like a contradiction.

LEWIS' INTRODUCTION OF THE MIGHT PRINCIPLE
The two conditionals above in concert sound like a contradiction because a way to negate a conditional is to assert the same claim with a negated consequent as we see in the two claims. However, Lewis introduced the "might" principle to assert that it can be the case that:

¬(Φ$$\Box$$-> Ψ) iff (Φ$$\Box$$-> ¬Ψ) so that the conditionals take the form:

(1') If Bizet and Verdi were compatriots Bizet might be Italian.

and

(2') If Bizet and Verdi were compatriots Bizet might not be Italian.

Lewis provides (1') and (2') to demonstrate that the might principle can hold and in fact not have contradictory claims.

STALNAKER'S DEFENSE OF THE EXCLUDED MIDDLE
Stalnaker responds to Lewis by questioning whether (1) (above) is just as likely to be true as (2)(above) by considering the relative similarity of sharp distinctions in possible worlds to the actual world. He notes that there are various possible valuations that arbitrarily settle indeterminacies. For example, a sentence is true/false in such a vague context just in case it would come out true/false in every one of sharp distinctions of valuations. Stalnaker applies this response to Bizet and Verdi:

(1r) If Bizet and Verdi were compatriots, then Bizet would be Italian.

and

(2r) If Bizet and Verdi were compatriots then Bizet would not be Italian.

and determines that both in conjunction are indeterminate and further their disjuctions are true because they arbitrarily select the nearest possible world.

LEWIS' SPHERES
If ranked possible worlds according to degree of closeness, we can obtain a series of spheres (Lewis' spheres) of the form, Sw1, S w2 ...Swn. This can validate the following contraints on the selection function f.

fA(w)⊆[A]

w ∈ [A] -> w ∈fA(w)

[A]≠ Ø⇒ fA(w) ≠ Ø

fA(w) ⊆ [B] ^ fB(w) ⊆ [A] ⇒fA(w) = fB(w)

fA(w)∩[B] ≠Ø fA^B(w)⊆fA(w)

Further Lewis maintains that, if w is an A world, it is the unique closest A-world to itself. Lewis' spheres imply that w is among the closest A worlds. This constraints asserts that,

w ∈ [A]^w'∈fA (w)⇒ w=w' which corresponds to a conditional of the form φ^ψ $$ \models $$ φ> ψ.

STALNAKER'S SELECTION
Stalnaker assumes that the selection of a closest world is always a singleton unique closest world. His contraint is,

w ∈ fA(w) ^w' ∈ fA(w)⇒ w=w'

This contraints allows the Conditional Excluded Middle to hold or the conditional of the form (Φ > Ψ) v (Φ > ¬Ψ).