Logic is a venerable formal branch of mathematics with well established sets of rules of inference and deduction. There are also formal fallacies within the framework of logic; errors that result if one doesn’t follow the rules of logic. One of the best known and simplest to understand such fallacy is “affirming the consequent” or “converse error”. Simply stated, it takes the form

If A then B; B is true; therefore, A is true.

This inference cannot be made; if A is true, we ‘know’ B is true – that’s contained in the statement “If A then B”. However, “if A then B” says nothing about the state of A given we know B. Therefore, the preceding statement is a formal logical fallacy.

There are many other formal fallacies; but there are more informal logical fallacies as well, those that are not really categorically errors within the rigorous framework of logic. Quite often these might be more readily classified as rhetorical fallacies. One very common such ‘fallacy’ that arises in discourse is the “Slippery Slope” fallacy. In formal logic terms the slippery slope might be written as

If A then B; If B then C; If C then D; …..; If Y then Z

therefore

If A then Z

In concrete terms, it might be something like “If you allow the government to tax the income of the top 1% of wage earners at 3%, then they will want to tax it at 4%; then they will tax the top 2% at 4%; etc; and eventually the top 50% of wage earners will be paying 50% of their income in taxes.” This form of argumentation will often (always?) get labeled as a “slippery slope fallacy” and subsequently dismissed. Here’s the problem – the mathematical statement of the ‘fallacy’ above is NOT a logical fallacy. It is in fact a perfectly valid application of the chain rule. Calling it a logical fallacy in the rhetorical sphere relies on replacing the conditionals in the formal statement with “If A then possibly B” so that each step in the chain is is only a possibility not a necessity.

Will A inevitably lead to Z? Unless one knows that the conditionals are absolutely true, then No. but you can not glibly dismiss (or should not) the possibility as a logical fallacy hence obviating the need for you to think about it and address the downstream implication of each step of the chain. The clean, controlled world of mathematical/logical proofs almost never obtains in any complex system; and human interactions are amongst the most complex systems there are!

Is it fallacious to argue If A then Z if each step of the chain is ‘possibly’ rather than ‘necessarily’? Yes. But is it equally fallacious to dismiss ‘If A then Z’ without examining each step of the chain. Too often, the accusation of ‘slippery slope is a logical fallacy!’ is simply deployed to shut down questions and silence objections via ‘intellectual’ intimidation. Sneeringly saying “Acktually, that’s a logical fallacy, so shut up you unsophisticated rube’ is not an argument. It gets the desired result without having to examine the chain, the accuser thinks it makes them sound/shows them to be smart, and it can short circuit objections from the accused (“it’s a logical fallacy, that sounds very sophisticated, I don’t want to sound dumb and unsophisticated!”)

To get all Bayesian, when someone makes an argument that gets dismissed as a “slippery slope fallacy”, what they are doing, consciously or unconsciously, is applying priors to the conditionals in the chain. Is it “If A then necessarily B”? – no, but given behavior exhibited by similar people in similar situations, it is probably something like “If A then 99% likely B”, so at the end of the day, it is high probability that ‘If A then Z’. Whereas the accuser gets to just dismiss the argument by reflexively repeating ‘logical fallacy’ as if that automagically invalidates every step of the chain. To dismiss the argument properly is to question the proposers selection of priors at each step.

Shorter – Don’t use the ‘slippery slope logical fallacy’ accusation in an argument. It doesn’t make you more sophisticated or smart, it’s just a lazy way of leveraging the appearance of intellectual heft to short-circuit probabilistic (logical) inference that has a decent probability of manifesting in the real world.