Another short (and simple) note on the so-called **tacking paradox** from the philosophy of science literature. Continuing on from here and related to a recent blog comments exchange here. See those links for the proper background.

**[ Disclaimer**: written quickly and using wordpress latex haphazardly with little regard for aesthetics…]

Consider a scientific theory with two ‘free’ or ‘unknown’ parameters, a and b say. This theory is a function which outputs predictions . I will assume this is a deterministic function for simplicity.

Suppose further that each of the parameters is discrete-valued and can take values in . Assuming that there is no other known constraint (i.e. they are ‘variation independent’ parameters) then the set of possible values is the set of all pairs of the form

That is, . Just to be simple-minded let’s arrange these possibilities in a matrix giving

This leads to a set of predictions for each possibility, again arranged in a matrix

Now our goal is to determine which of these cases are consistent with, supported by and/or confirmed by some given data (measured output) .

Suppose we define another function of these two parameters to represent this and call it for ‘consistency of’ or, if you are more ambitious, ‘confirmation of’ any particular pair of values with respect to the observed data .

For simplicity we will suppose that outputs a definite value which can be definitively compared to the given . We will then require iff , and otherwise. That is, it outputs 1 if the predictions given and values match, 0 if the predictions do not. Since will be fixed here I will drop , i.e. I will use without reference to .

Now suppose that we find the following results for our particular case

How could we interpret this? We could say e.g. and are ‘confirmed/consistent’ (i.e. ), or we could shorten this to say is confirmed for any replacement of the second argument.

Now the ‘tacking paradox’ argument is essentially:

so

is confirmed, i.e. ‘a=0 & b=0’ is confirmed. But ‘a=0 & b=0’ logically implies ‘b=0’ so we should want to say ‘b=0’ is confirmed. But we saw

and so

is also confirmed, which under the same reasoning gives that ‘b=1’ is confirmed!

**Contradiction!**

There are a number of problems with this argument, that I would argue are particularly *obscured by the slip into simplistic propositional logic reasoning*.

In particular, we started with a clearly defined function of two variables . Now, we found that in our particular case we could reduce some statements involving to an ‘essentially’ one argument expression of the form ‘‘ or ‘ is confirmed’, i.e. we have confirmation for a=0 and b ‘arbitrary’. This is of course just ‘quantifying’ over the second argument – we of course can’t leave any free (c.f. bound) variables. But then we are led to ask

*What does it mean to say ‘b is confirmed’ in terms of our original givens? *

Is this supposed to refer to ? But this is undefined – is of course a function of two variables. Also, b is a free (unbound) variable in this expression. Our previous expression had one fixed and one quantified variable, which is *different to having a function of one variable.*

OK – what about trying something similar to the previous case then? That is, what about saying ? But this is a short for a claim that *both * *and * hold (or that their conjunction is confirmed, if you must). This is clearly not true. Similarly for .

So we can clearly see that *when our theory and hence our ‘confirmation’ function is a function of two variables we can only ‘localise’ when we spot a pattern in the overall configuration*, such as our observation that holds.

So, *while ** the values of the C function (i.e. the outputs of 0 and 1) are ordered (or can be assumed to be), this does not guarantee a total order when it is ‘pulled back’ to the parameter space. That is, ** does not guarantee an ordering on the parameter space that doesn’t already admit an ordering! It also doesn’t allow us to magically reduce a function of two variables to a function of one without explicit further assumptions. Without these we are left with ‘free’ (unbound) variables.
*

(Note: Bayesian statistics does of course allow us to reduce a function of two variables to one *via marginalisation, and given assumptions on correlations*, but this process again illustrates that there is no paradox; see previous posts).

One objection is to say – “well this clearly shows a ‘logic’ of confirmation is impossible”. Staying agnostic with respect to this response, I would instead argue that what it shows is that:

*The ‘logic’ of scientific theories cannot be a logic only of ‘one-dimensional’ simple propositions. A scientific theory is described at the very, very minimum by a ‘vector’ of such propositions (i.e. by a vector of parameters), which in turn lead to ‘testable’ predictions (outputs from the theory). To reduce functions of such collections of propositions, e.g. a function f(a,b) of a pair (a,b) of propositions, to functions of less propositions, e.g. ‘f(a)’, requires the use – again at very, very minimum, of quantifiers over the ‘removed’ variables, e.g. ‘f(a,b) = f(a,-) for all choices of b’. *

Normal probability theory (e.g the use of Bayesian statistics) is still a potential candidate in the sense that it extends to the multivariable case and allows function reduction via marginalisation. Similarly, pure likelihood theory involves concepts like profile likelihood to reduce dimension (localise inferences). While standard topics of discussion in the *statistical* literature (e.g. ‘nuisance parameter elimination’), this all appears to be somewhat overlooked in the *philosophical* discussions I’ve seen.

*So this particular argument* is not, to me, a good one against Bayes/Likelihood approaches.

(I am, however, generally sympathetic to the idea that functions like that above are better considered as *consistency* functions rather than as *confirmation *functions – in this case the, *still fundamentally ill-posed*, paradox ‘argument’ is blocked right from the start since it is ‘reasonable’ for both *‘b=0’* and *‘b=1*‘ to be *consistent *with observations*. *On the other hand it is* still not clear how you are supposed to get from a function of two variables to a function of one*.)

To conclude: ** the slip into the language of simple propositional logic, after starting from a mathematically well-posed problem**, allows one to ‘sneak in’ a ‘reduction’ of the parameter space, but

**like .**

*leaves us trying to evaluate a mathematically undefined function*The tacking ‘paradox’ is thus a ‘non-problem’ * caused by unclear language/notation*.