A machine that passes a conversation test by consulting a table keyed on the whole conversation needs one entry for each possible conversation. Count them, see why that makes a table the opposite of a program, and state what Block and Searle each use the table for.
Core ideas
Behaviour that passes a test can be produced in two quite different ways, by a rule that generates it or by a list that stores it. The second is countable.
A lookup table holds a reply for every case it must meet and computes nothing. A program holds a rule from which the replies follow. The same replies can come from either, and the two have very different sizes.
If each turn offers one of n things to say and the conversation lasts r turns, there are nʳ complete conversations, because each turn multiplies the count by n. This is the count of ordered choices with repetition from Combinatorics and Counting on the Discrete Mathematics path.
Block uses the table to argue that passing a behavioural test is not enough for intelligence. Searle uses a room with a rulebook to argue that running a program is not enough for understanding. The device differs, and so does the count that bears on it.
The material
The count first, then the two arguments, then what the count settles and does not.
A lookup table answers by finding its input in a list and returning the reply stored beside it. It does not compute a reply; every reply it will ever give is already written down.
Suppose a machine is to pass the conversation test of the previous lessons by this method. Its reply at any moment depends on everything said so far, so the table has to be keyed on the whole conversation, not on the last remark. If at each turn the judge may say one of n things, and the conversation has r turns, the table needs a reply ready for every sequence of r remarks.
That is a count of ordered choices with repetition: n choices at the first turn, n at the second, and so on, which gives nʳ. Shorter conversations add a little more, because the table must also answer the ones that end early, but the largest term is the whole-conversation count. In the lab, set n to 24 and r to 12.
That is over thirty-six quadrillion entries for a twelve-turn conversation in which the judge has only twenty-four things to say. Slide r down and the table shrinks fast: at three turns it is 13 824, and each further turn multiplies it by 24. A real conversation has a vastly larger stock of remarks and many more turns, so the table grows at a rate no budget can meet.
Compare a program. A program that holds a rule for forming replies can be a few thousand lines and meet every one of those conversations, because it does not store them; it works them out. A table is the opposite of compression. Its size is the number of cases it covers, and a program's size is the length of the idea behind them.
Block takes the table as a thought experiment. Build one anyway, in principle, with every sensible conversation of some length written in. It passes the test with a Bayes factor of 1 and nothing in it is intelligent: it looks up. So behaviour that passes the test is not sufficient for intelligence, and behaviourism is false. The reply is that a table which cannot be built tells us little about what a thing that could be built must be like. Block's rejoinder is that behaviourism was a claim about what is conceptually enough, and an in-principle case refutes a claim of that strength.
Searle's room is a different case. A person who reads no Chinese sits in a room with a rulebook for manipulating Chinese symbols, and passes out replies that a Chinese speaker would accept. The room runs on rules, so its size is the size of the rulebook and not the number of conversations, and the count does not touch it. Searle argues that the person understands nothing, so following the rules cannot be understanding. The standard reply is that the whole system, room and rulebook and person together, understands. Searle's rejoinder is to let the person memorise the rulebook and work in the open, so that the system is the person, who still understands nothing. The lesson does not choose.
What the count settles is narrow. A passing system of any realistic size is not a table, so it generalises, and whether generalising amounts to understanding is the open question. What it does not settle is whether a table, if built, would think.
Interactive
Enumerated while enumeration is possible
Method
Four steps. The second is the one that makes the number large.
Say how many things the judge can say at one turn. Call it n. The lab's n is that number.
The reply may depend on everything said before, so the table needs an entry for each sequence of r remarks, not for each remark.
Ordered, with repetition: n at each turn, so nʳ. Read it off the first row of the lab.
A program that forms replies is the length of its rule, whatever the number of conversations. A table is the length of the number of conversations.
A judge with twenty-four possible remarks per turn, and a conversation of twelve turns.
The table grows with the number of conversations, and the number of conversations grows by a factor of 24 for every turn. So the machine that passes a long test is not a table of this kind in practice. It is something that computes, and Block's table is then an argument about what would follow if one existed.
Given the number of remarks and turns, you should be able to compute the entries a table needs, say why a program of the same behaviour is smaller, and state in a sentence what Block and what Searle infer from their devices.
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