I'm having a hard time translating logic statements into english because most of the time I don't know how to translate a pattern I have not seen before:
There are two relations where
- $lecturer(x)$ x is a lecturer
- $human(x)$ x is a human
I find it extremely difficult to work out what a statement is saying if I change a quantifer or remove the brackets, for example is the following correct?
$$\forall x lecturer(x) \to \forall x human(x)$$ If everything is a lecturer then everything is human?
What is the difference between
$$\forall x (lecturer(x) \to human(x))$$
And
$$\forall x lecturer(x) \to human(x)$$ ?
I don't think the second one is correct in the sense that you cannot write that but how can I tell? Are there any tips for translating statements? I understand the common patterns such as
$$\forall x (A \to B)$$ $$\exists x (A \wedge B)$$
It's just uncommon ones that confuse me or when I start to wonder what will this mean if I remove this bracket? Logic is really my weakpoint.
Operator precedence.
Similarly to the order of operations in arithmetic, in propositional logic the operator precedence is that: $\neg$ has precedence over $\vee$ and $\wedge$, which in turn have precedence over quantifier binding, and $\to$ is evaluated last. (The inclusion of brackets overrides this default order of precedence.)
So $∀x\,{\rm lecturer}(x)→{\rm human}(x)$ is evaluated as if it had implicit braketting like thus: $$\color{blue}{\Big(}∀x_\color{blue}{1}\,{\rm lecturer}(x_\color{blue}{1})\color{blue}{\Big)} \;\to\; {\rm human}(x_\color{blue}{2})$$
Which means the second $x$ is not bound to the quantifier. The statement translates to "If all things are lecturers, then a free variable (named $x_\color{blue}{2}$) is a human".
Where as the explicit bracketing in the first statement $∀x\,\big({\rm lecturer}(x)\to {\rm human}(x)\big)$, means all $x$ symbols in the brackets are bound to the same quantifier. Thus it reads: "If anything is a lecturer, then that thing is a human". Which is to say: "All lecturers are human".
When in doubt, include brackets.