If a company pays dividends on a stock, does that mean that the stock has appreciated in value? why or why not?

Dividend is the amount of share in the earnings paid by the company to its shareholders for their investment in the company. So dividend is paid out of the earnings made by the company and it has no relation with the increase or decrease in the value of the stock.

Hence, If a company pays dividends on a stock, it does not mean that the stock has appreciated in value.

No,  it is not

Dividend only indicated that the company has just earned profit in the last financial period.

A stock would appreciate in value if many investors see it as a great opportunity to earn profits. There are other factors that influenced a stock for it to be appreciated in value, and dividend is just a small portion for it. For example, the company need to had a good cash flow, the company need to maintain a good name in public, the company need to had high asset to debt ratio, etc.