# Predicate Logic - Translations

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.

• It depends on the convention of the use of brackets. – Salomo Apr 8 '15 at 21:38

## 1 Answer

What is the difference between $∀x({\rm lecturer}(x)→{\rm human}(x))$ and $∀x\,{\rm lecturer}(x)→{\rm human}(x)$?

## 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.

• I completely forgot about free and bound variables here! Thank you for your answer , this really helped clearing up my understanding. – Nubcake Apr 9 '15 at 12:49