Proof and logic

The final part of block D in MS221 of my OU Maths degree is all about mathematical proofs and deduction, which I find absolutely fascinating.  A big part of this block was clarity on some logical fallacies that we encounter all the time and that many people use to trick us into agreeing with their arguments.

XKCD: CorrelationXKCD: Correlation

With one week to go until the General Election in the UK it seems like a good time to revisit logic and proof from both the political and mathematical sides.

Firstly, while it doesn’t appear in mathematics, this is an appropriate point to remind you to be wary of the ad hominem attacks where politicians will deflect by criticising the person rather than their argument.  There are many variants on this, but if someone plays on your dislike of another person to get you to agree then you really should be questioning why they’re doing it!  Try reading a newspaper or watching the news without an example of this cropping up…

So let’s define a few terms:

  • Premise – this is what you assume to be true
  • Propositions – a statement that may either be true or false
  • Fact – something that is actually true (although be wary of items labelled as facts that you cannot verify – doesn’t mean that they are not true but do not assume that something presented as a fact is not actually a premise!)
  • Conclusion – the result asserted from the premises and facts
  • Direct proof – if we work through the premises and can deduce that the desired conclusion is true then we have a direct proof
  • Counter-example – if we have a general statement then a single example is enough to disprove the conclusion
  • Proof by contradiction – we assume that the desired conclusion is false and make deductions until we reach a contradiction.  We can then assume that our assumption that “the desired conclusion is false” is in itself false and hence the desired conclusion is true

It is important to remember that if a conclusion is false then either the argument or the premise(s) are false.

Let’s look at propositions.  Here’s an example:

(A) If I’m drinking a cup of tea then I eat a chocolate biscuit

This is in the form if (a) then (b). So the proposition leads us to the conclusion that every time I drink tea I eat a biscuit.  Although if you caught me drinking tea without a chocolate biscuit to eat then you have a counter-example to disprove the proposition.  However if we look at the statement in a different way:

(B) I’m not drinking tea and I’m not eating a biscuit

On the surface, you might think these are the same but they’re not. For the second statement to be true it is required that I am not drinking tea and I’m not eating a biscuit.  However the first statement can be true whether or not I have any tea.  A subtle but important distinction about phrasing and it’s in this subtlety that many people can be misled.

So let’s change our description into mathematical terms. B can be written as a and b – both must be true or the proposition is false. A can be written as a then b – if a is true then b must also be true, if b is false then a cannot be true but if we know that b is true we cannot deduce whether a is true or false (I may be drinking coffee with my chocolate biscuit, or even not drinking anything at all but raiding the biscuit tin 😉 ).  This is one of the most common fallacies I’ve seen in the political discussions of late.  For example:

If we decrease the number of doctors in hospitals then A&E waiting times will increase

While the increase in waiting times could certainly be due to a decrease in medical professionals (assume that the decrease was on the front line), just knowing that waiting times has increased is not enough to assert that either the number of doctors has decreased or even that this is the only cause.  Which leads me to one of my favourite sayings:

Correlation is not causation

and one of my favourite websites which demonstrates this beautifully.

As part of my revision for the MS221 exam I’d do a future post on proof by exhaustion and induction.

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Dr Janet is a Molecular Biochemistry graduate from Oxford University with a doctorate in Computational Neuroscience from Sussex. I’m currently studying for a third degree in Mathematics with Open University. During the day, and sometimes out of hours, I work as a Chief Science Officer. You can read all about that on my LinkedIn page.