# Logical Form of an Appeal to Probability

How can you express an "Appeal to Probability" argument in a logical notation? Feel free to use any forms or renditions of logic, including APL, as I know there are different symbols that can be used.

I came up with this: ♢P > ♢Q ⊢ P ∧ ¬Q. Do you think it makes sense?&

• The use of '>' symbol in this context is unfamiliar to me. But I think you're trying to say that the possibility of P happening is greater than the possibility of Q, etc. That makes sense. The measure has to be defined carefully though. If we go naively about this, every propositional letter will have the probability (2^n / 2), where n is the number of propositional letters occurring in a formula. Something more is needed. – Hunan Rostomyan Aug 1 '14 at 21:55
• Thank you for your comment. I reposted this question here to see if an apl example is possible. codegolf.stackexchange.com/questions/35566/… . I am a novice; would you mind expanding on why every propositional letter will have the probability (2^n / 2), where n is the number of propositional letters occurring in a formula? Many regards; this is very exciting for me. – Akiva Aug 1 '14 at 22:02
• In a two-valued setting such as classical logic, every propositional letter is either true or false. That means that the space of the valuations/interpretations of a formula with n propositional letters will consist of 2^n assignments of truth-values to those propositional letters. This space is usually represented as a truth-table, a Karnaugh map, or some such diagram. If the formula is P, the table has 2^1 rows, one that assigns true to P and one that assigns false. If the formula is (P & Q), there will be 2^2 rows: 00, 01, 10, 11. Continued... – Hunan Rostomyan Aug 1 '14 at 22:11
• ...Continued. Now you just observe that any such table of n proposition letters will have exactly half of its rows s.t. P is true in them and half of its rows s.t. P is false in them. The same is true of the other letters. The same is true of any classical propositional formula. The modal case is not an exception. – Hunan Rostomyan Aug 1 '14 at 22:13
• That was a perfect explanation; I understand clearly now. Many thanks. – Akiva Aug 1 '14 at 23:53

There seem to be two versions of the fallacy. The first says: φ is possible, therefore φ will be true at some point in the future. The second: φ is probable, therefore φ will be true at some point in the future:

Variant 1. An appeal to possibility: Poss(φ) ⊢ F(φ);

Variant 2. An appeal probability: Prob(φ) ⊢ F(φ).

Depending on your account of possibility and probability, these will have different explications. I will sketch a way of explicating (1), leaving you with Yalcin 2010 to work out how to explicate (2).

What we need is a modal language with both alethic (e.g. Poss) and temporal (e.g. F) operators. I propose to use a temporal language and define the needed alethic modality in it instead of taking it as an undefined symbol. As usual, we start with the definition of the language:

Definition 3. (Language) Given a propositional letter 'p', the well-formed formulas of Priorean temporal language are generated by the following grammar:

φ   :=   p   |   φ′   |   ¬φ   |   (φ ∧ φ)   |   G(φ)   |   H(φ).

We read G(φ) as it "henceforth φ" and H(φ) as "hitherto φ"; the rest of the operators are familiar. For everyday research purposes you would then define the semantics of this language on what are called flows of time and prove things about the logic. Here we're concerned with defining things only, so let's define the operators that we need in order to express (1) and (2):

Definition 4. (Future Temporal Possibility) To say that "φ will happen" is to say that "it's not the case that: henceforth ¬φ". This is the motivation behind the definitions:

F(φ)    =df   ¬G¬(φ),

P(φ)    =df   ¬H¬(φ).

Now we define the alethic possibility operator Poss using this language.

Definition 5. (Alethic Possibility) To say that "φ is possible" is to say that "either φ is true or φ was true or φ will be true". This is the motivation behind the definition:

Poss(φ)    =df   P(φ) ∨ φ ∨ F(φ).

This says nothing more than that if φ is possible, then there is at least some point in time (either now, in the past, or in the future) at which φ holds. To explicate Variant (2), this operator has to be strengthened. The explication of Variant (1) is then the following:

Variant 1. (Explication) P(φ) ∨ φ ∨ F(φ) ⊢ F(φ).

The very form of it suggests its counterexample: take any proposition p that has happened in the past and will never happen again (e.g. "Julius Caesar is being stabbed"). That means that P(p) is true. Therefore P(p) ∨ p ∨ F(p) is true. But it's not the case that Caesar will be stabbed again, so F(p) is false.

References

van Benthem, J. (2010) Modal Logic for Open Minds, Stanford, CSLI Lecture Notes #199.
Holliday, W.H. (2012) Modal Reasoning, Lecture Course (Spring), UC Berkeley.
Yalcin, S. (2010) "Probability Operators".