What is Formal Logic? An Informal Introduction

We all have a vague idea of what logic is. Using ice-cream, I will demonstrate a new hot take (ha-ha) on what logic really is.

Photographer: Rachael Gorjestani | Source: Unsplash

What is Logic?

To say someone ‘isn’t being logical’ is almost the same as saying they are wrong. This isn’t quite right.

Logic is concerned with the relations between statements, and what a correct inference is. If I say that, ‘all men are made of ice-cream. I am a man, therefore I am made of ice-cream’, the statement is baloney, but the logic is correct.

Logic takes statements and abstracts to the general form of their deduction. Restated: if for all x in S, P(x) is true, then if y is in S, P(y) is true. Logic treats propositions such as P(x), or in this case ‘… is made of ice-cream’, like a black box, and merely studies the relationship between these propositions supposing they were true or false.

If I write ‘all men are made of ice-cream. I am made of ice-cream, therefore I am a man’, then this is logically incorrect. It may be that, in your world, there is nothing which is an ice-cream and not a man. Your statement is incidentally correct, but the logic is wrong. This is because from knowing all men are made of ice-cream, you have learnt nothing of whether there are things which are not men which are made of ice-cream.

So logic isn’t concerned with truth in the world, rather it studies the basis of correct inference. Someone who is illogical thus makes incorrect inferences given facts about propositions and objects.

Logic symbols

Logical symbols merely streamline the process, allowing you to make fast deductions and manipulations on equivalent statements. It’s like that age old thing where someone says ‘I don’t not not not think that’, and you are left wondering what the hell is going on. The formal symbols enable rephrasing statements into equivalent ones a lot faster.

Logic in the world: Physics and Economics

It’s worth noting that in the ice-cream themed example above, the mistakes are a bit easier to spot. In the real world logical errors are likely to be hidden. Frankly, most of economics and social sciences research doesn’t progress past the following fallacy (which gives rise to ‘p-hacking’ and an obsession with t values).

‘Every true (and important) relationship between variables has a statistically significant observable effect.

My result is statistically significant.

Therefore I have found a true and important relationship between variables’

The issue is that there are so many bogus results out there you can quite easily get supposedly statistically significant results through messing around with irrelevant variables. I remember a friend telling me how for a long time butter production in one developing country had been an excellent predictor of economic growth in the global economy. I don’t know if this was true, but it serves the point that with enough variables out there some might correlate by chance.

A brief aside: why Science works

If you’re interested, this is why prediction is so important in Science. It reduces the sample space of variables down a lot. If something correlates by chance, then it is highly unlikely to do so repeatedly again in the future. Butter production in Bangladesh may have some genuine causal predictive power… but it might simply be that with millions of industries worldwide, some of them just happen to be correlated. Thus the sample space of observed correlations can get filled with nutty variables, potentially ranging from earwax buds to the market for recycled toenail clippings, none of which have any predictive power.

Unfortunately it’s quite hard to define what the ‘sample space’ is and do formal probabilistic estimates. In physics, prediction is the benchmark. In Economics, hocus pocus is.

The author is Ethan Horsfall, an Economics student at Peterhouse, Cambridge. His interests are primarily Mathematics and Philosophy, and writing flattering bios of himself in the third person

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