Conditional (Probability) Logic

Conditional Logic, an extension of Propositional Logic, sets forth the logic of the indicative conditional, sentences such as:
- If you strike the match, it will light.
- If Anna rode her bike to school yesterday she wore a helmet.
- If it doesn’t rain tomorrow, he’ll play golf in the afternoon.

A Shortcoming of Propositional Logic

The truth-value of a conditional in Propositional Logic is true if and only if its antecedent (the if clause) is false or its consequent (the then clause) is true or both. It makes sense that a conditional is true when antecedent and consequent are both true. And it makes sense that a conditional is false when the antecedent is true and the consequent false. But the idea that a conditional is true when its antecedent is false is not only counterintuitive but leads to the paradoxes of material implication.

Paradoxes of Material Implication

For each of following arguments, valid in Propositional Logic, there’s a scenario in which its premises are true and conclusion false.

The arguments are invalid in Conditional Logic.

Indicative Conditional of Natural Language

John is asked whether Anna wore a helmet riding her bike to school yesterday. John doesn’t know if Anna rode her bike yesterday. But he knows that Anna is very good about wearing a helmet. So, he replies:
- If Anna rode her bike to school yesterday she wore a helmet.

Three scenarios:
1. Anna rode her bike to school yesterday and wore a helmet:
  - John is right and the conditional is true.
2. Anna rode her bike to school yesterday but didn’t wear a helmet:
  - John is wrong and the conditional is false.
3. Anna didn’t ride her bike to school yesterday:
  - John is neither right or wrong and the conditional is neither true or false.
  - The fact that Anna is very good about wearing a helmet is evidence either that if Anna rode her bike to school yesterday she wore a helmet or that if Anna had ridden her bike to school yesterday she would have worn a helmet. That is, it’s evidence either for an indicative conditional or for a counterfactual subjunctive conditional.

Upshot:
- If the antecedent of an indicative conditional is true:
  - the truth-value of the conditional is the same as the truth-value of the consequent.
- If the antecedent of an indicative conditional is false:
  - the conditional is neither true or false;
  - the conditional (or its counterfactual counterpart) can nevertheless be likely (certain, reasonable to believe, doubtful, etc.) based on the evidence.
    - View Epistemic Probability.

Probability Logic Overview

In the 1960s and 70s Ernest W. Adams developed a system of Conditional Logic, which he called Probability Logic, that in a sense replaces the truth-values of Propositional Logic with the real numbers of Probability Theory. Here’s an overview of his system, contrasted with Propositional Logic.

Propositional Logic
- The core notion of Propositional Logic is truth-functionality, the idea that the truth-value of a compound sentence is a function of the truth-values of its component clauses.
- The truth-values of conjunctions, disjunctions, negations, and material conditionals are determined by the truth-values of their sentence letters.
- An argument is valid if there’s no assignment of truth-values to its sentence letters under which the premises are true and the conclusion is false.
- The material conditional is defined as true when its antecedent is false, its consequent true, or both; and false otherwise. That the material conditional is true when its antecedent is false leads to the paradoxes of material implication.

Probability Logic
- The core notion of Probability Logic is epistemic probability, the kind of probability relative to evidence and expressed by terms such as:
  - it is certain that, it is beyond a reasonable doubt that, it’s likely that, it is doubtful that, there’s a remote possibility that, it is impossible that.
  - View Epistemic Probability.
- The probabilities of conjunctions, disjunctions, negations, material conditionals and indicative conditionals are determined by the probabilities of the state-descriptions of their sentence letters.
  - A state-description is a conjunction of sentence letters and negations of sentence letters.
- An argument is p-valid, loosely speaking, if there’s no probability distribution to the state-descriptions of its sentence letters under which the premises are very likely and the conclusion very unlikely.
- The probability of the indicative conditional is defined as the conditional probability of its consequent given its antecedent. That is, the probability of the conditional if A then B = P(A ⇒ B) = P(B|A) = P(A&B) / P(A).

Language of Probability Logic

The language of Adams’ Probability Logic is an extension of the language of Propositional Logic. So it includes sentence letters (A, B, C, ..), parentheses, and the connectives: &, v, ~, and ➞.
Thus the following are sentences of Probability Logic:
- ~(AvB) ➞ C
- ~(A ➞ B)
- (A ➞ B) & (B ➞ A)
It also includes the connective ⇒, symbolizing the indicative conditional, what Adams calls a probability conditional.
Thus the sentences of Probability Logic also include:
- A ⇒ B
- A ⇒ (B &C)
- (A v B) ⇒ C
- A ⇒ ~(A ➞B )
There’s a special restriction on probability conditionals: unlike other connectives, “⇒” has to be the main connective in a sentence; so a probability conditional can’t be within the scope of a another connective. The following are thus not sentences of Probability Logic:
- ~(A ⇒ B)
- (A ⇒ B) & (B ⇒ A)
- (A ⇒ B) v (A ⇒ C)
- A ⇒ (B ⇒ C)

Probability Logic, by Example

To get a feel for Adams’ system we’ll look at a number of arguments and compare why they’re valid or invalid in Propositional Logic versus why they’re p-valid or p-invalid in Probability Logic.

An argument is valid in Propositional Logic if and only if there’s no assignment of truth-values to its sentence letters under which the premises are true and the conclusion is false.
An argument is p-valid, loosely speaking, if and only if there’s no probability distribution for the state-descriptions of its sentence letters under which the premises are very likely and the conclusion very unlikely.
- A state-description is a conjunction of sentence letters and negations of sentence letters.
An argument is p-valid, strictly speaking, if and only if there’s no probability distribution to the state-descriptions of its sentence letters under which the probability of the conclusion is less than the sum of the probabilities of the premises (minus one less than the number of premises).
- View Formal P-validity

Reverse Disjunctive Syllogism

The argument
- A v B
- A
- So, ~B
is invalid in both Propositional Logic and Probability Logic:

Invalidity

The argument is invalid since the premises are true and the conclusion false when A and B are both true:

P-Invalidity

The argument is p-invalid because there’s a probability distribution for the state-descriptions of A and B under which the premises are very likely and the conclusion very unlikely. That is, there’s a probability distribution for the state-descriptions
- P(A&B)
- P(A&~B)
- P(~A&B)
- P(~A&~B)
under which AvB and A are very likely and ~B very unlikely.
For example:
- the probability distribution
  - P(A&B) = 0.9625
  - P(A&~B) = 0.0125
  - P(~A&B) = 0
  - P(~A&~B) = 0.025.
- under which
  - P(AvB) = P(A&B) + P(A&~B) + P(~A&B) = 0.975
  - P(A) = P(A&B) + P(A&~B) = 0.975
  - P(~B) = P(A&~B) + P(~A&~B) = 0.0375.

It is helpful to think about probability visually.
State-descriptions correspond to regions of a Venn diagram.
Thus the state-descriptions of A and B correspond to the regions of
where the regions are:
- r1 = ~A&~B
- r2 = A&~B
- r3 = A&B
- r4 = B&~A.
Which makes it clear that the probabilities of the premises and conclusion are:
- P(AvB) = r2 + r3 + r4
- P(A) = r2 + r3
- P(~B) = r1 + r2.

I used Mathematica to find the counterexample above — by telling it to find probabilities for r1, r2, r3, and r4 that make the premises > 0.95 and the conclusion < 0.05.
- FindInstance[
  - (r2 + r3 + r4) > 0.95 &&
  - (r2 + r3) > 0.95 &&
  - (r1 + r2) < 0.05 &&
  - r1 + r2 + r3 + r4 = 1.0 &&
  - r1 ≥ 0 && r2 ≥ 0 && r3 ≥ 0 && r4 ≥ 0,
- {r1, r2, r3, r4}]
Which returns {r1 = 0.025, r2 = 0.0125, r3 = 0.9625, r4 = 0.}. A null return, {}, would have meant that no such probability distribution exists.

Summary:

Disjunctive Syllogism

Disjunctive Syllogism (DS) is valid in Propositional Logic and p-valid:
- AvB
- ~A
- So, B

Validity

There’s no assignment of truth-values to A and B under which the premises are true and the conclusion is false, there being only four:
The argument is thus valid.

P-validity

There are two ways of proving that DS is p-valid:
- Derive the conclusion from the premises using rules of inference that guarantee p-validity.
- Use a decision procedure for p-validity that finds that DS is p-valid.

Derivation

Adams’ rules of inference are sound and complete. That is:
- If the conclusion is derivable from the premises, the argument is p-valid.
- If an argument is p-valid, the conclusion is derivable from the premises.
Since there’s a derivation of B from AvB and ~A, the argument is p-valid.
- See Derivation of Disjunctive Syllogism.

Decision Procedure

The other way of proving p-validity is by using a decision procedure, an algorithm that determines, after a finite number of steps, whether the argument is p-valid or p-invalid.

I’ll use the strict definition of p-validity:
- An argument is p-valid if and only if there’s no probability distribution to the state-descriptions of its sentence letters under which the probability of the conclusion is less than the sum of the probabilities of the premises (minus one less than the number of premises).
Thus, for example, if a p-valid argument has two premises each with probability 0.9, the probability of the conclusion can’t be less than 0.9 + 0.9 − 1 = 0.8. And if the probability of the two premises is 0.8, the probability of the conclusion can’t be less than 0.8 + 0.8 −1 = 0.6.
In other words, if the premises of a p-valid argument are reasonable to believe, the conclusion can’t be reasonable to disbelieve.

The decision procedure is some Mathematica code that determines whether there’s a probability distribution under which the probability of B is less than the probability of AvB + the probability of ~A − 1.

Disjunctive Syllogism
- AvB = r2 + r3 + r4
- ~A = (r1 + r4)
- B = (r3 + r4)
FindInstance[
- r2 + r3 + r4 = p1 &&
- (r1 + r4) = p2 &&
- (r3 + r4) = c &&
- c < (p1 + p2 −1) &&
- 1 + r2 + r3 + r4 = 1.0 &&
- r1 ≥ 0 && r2 ≥ 0 && r3 ≥ 0 && r4 ≥ 0,
{r1, r2, r3, r4, p1, p2, c}]
Output = {}.

The code returns {}, meaning that no such probability distribution exists.
So if AvB and ~A are reasonable to believe, it can’t be reasonable to believe that B is false.

Fallacy of Affirming the Consequent

Fallacy of Affirming the Consequent is the argument form:
- Material implication
  - A ➞ B
  - B
  - So, A
- Indicative conditional
  - A ⇒ B
  - B
  - So, A
The argument is invalid and p-invalid.