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I'm trying to create a first order logic statement, but I have constants in addition to predicates.

Predicates:

$Time(a)$, which states that a is a time;

$After(a,b)$, which states that $a$ is after $b$;

$Together(x,y,a)$ which states that $x$ and $y$ are together at time $a$

Constants:

$now$, $you$, and $I$.

Need to create statement: Before now, you and I used to be together, and after now, you and I will never be together.

Would this be something like: $¬Time(a) ∧ ¬Now → Together(x,y,a) ∧ Now → ¬Together(x,y,a)$

Victor
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    Hi, you recently posted a very similar question at http://math.stackexchange.com/questions/452227/need-help-with-first-order-logic - this site is intended to help you understand how to approach a whole class of problems by yourself more than to answer an array of similar questions for you. You should also include some indication of what you've tried. – not all wrong Jul 25 '13 at 20:18
  • Hi @Sharkos. Sorry about that, Let me add what I've tried then. – Victor Jul 25 '13 at 20:20

1 Answers1

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Take it a piece at a time. We clearly need to be able to be able to say you and I are/were together at time $t$; that’s $\text{Together}(\text{you},\text{I},t)$. Now take the first half of the sentence:

Before now, you and I used to be together.

The most straightforward interpretation of the English is that at every time before now, you and I were together. In other words, if $t$ is a time, and $t$ is before now, you and I were together at time $t$. This isn’t too hard to translate into symbols:

$$\forall t\Big(\text{Time}(t)\land\text{After}(\text{now},t)\to\Big)\text{Together}(\text{you},\text{I},t)\;.$$

Very literally, that’s,

For all $t$, if $t$ is a time, and $\text{now}$ is after $t$, then you and I were together at time $t$,

which in English is a clumsy but understandable way to say that you and I were always together before now.

The second half of the sentence is

After now, you and I will never be together.

See if you can translate that part on your own; the basic structure is really quite similar to that of the first half.

Brian M. Scott
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