The argument:
Everyone likes Mandy. Mandy likes nobody but Andy. Therefore, Mandy and Andy are the same person
Some notes on deductive reasoning:
When we say: ‘Everyone likes Mandy,’ then ‘everyone’ is ‘everyone’. I can pick any representative of the ‘Everyone’ set at random, including Mandy, without violating the truth of the sentence, and this is what I’ll do.
Since we do not use reflexive pronouns in logic (i.e. herself, etc), the sentence: ‘Mandy likes herself,’ is simply ‘Mandy likes Mandy.’ We will use both versions below in spoken explanations.
All arguments are set in a universe of discourse, which is the set of things being talked about on a given occasion. In our case, the universe of discourse is simply ‘persons’.
Analysis:
The first premise says: ‘Everyone likes Mandy.’ Let F be our first premise:
F = {… Betty likes Mandy, Sarah likes Mandy, Clive likes Mandy, Mandy likes Mandy, Sandra likes Mandy, Eddie likes Mandy, …}
The second premise says: ‘Mandy likes nobody but Andy.’ This can be paraphrased as ‘Mandy likes only Andy.’ The adverb ‘only’ requires that we show that not only does Mandy like Andy, but that she does not like anybody else. Let S be our second premise:
S = {… Mandy does not like Betty, Mandy does not like Sarah, Mandy does not like Clive, Mandy likes Andy, Mandy does not like Sandra, Mandy does not like Eddie, …}
Another way of saying ‘Mandy likes only Andy’ is, ‘If a person is not Andy, then Mandy does not like him / her.’ Mandy herself, however, is a person. We can legitimately then substitute Mandy for ‘person’ in our conditional sentence.
Our argument boils down to this:
Mandy likes herself.
If Mandy is not Andy, then Mandy does not like herself.
We can use contraposition in our ‘If …, then …’ sentence and simply flip around the sides while remembering to change the signs. Thus, we get:
Mandy likes herself.
If Mandy likes herself, then Mandy is Andy.
Many people find it easier to change the order of the premises. Let’s do that:
If Mandy likes herself, then Mandy is Andy.
Mandy likes herself.
Using Modus Ponens (affirming the antecedent), we infer:
Mandy is Andy.
In real life, there are of course many Mandies, whereas we take Mandy to be a unique individual. However, this is an irrelevance as we can easily assign a number or some other unique designation to each individual in our set. Further, in Natural language (English, Polish, etc) when we say ‘Everyone likes Mandy,’ we probably think of Mandy as standing apart from everyone else. This can be easily expressed in logic as ‘For every person, such that that person is different from Mandy, every person likes Mandy,’ but this is not what the original premise says. Finally, natural language is imprecise. Conversation overheard on a train last week:
She: Shopping is no fun if you don’t have money.
He: You can’t shop at all if you don’t have money, I would have thought.
She: I mean, you know, when you have little money.