Back in the 1890's, Arrhenius predicted about 5-6 degrees of warming if carbon dioxide levels were doubled:
The History of Climate Science
Of course, Arrhenius had the disadvantage of working with late 19th century technology. This changed when Dr. Gilbert Plass released the paper in the 1950's: "The Carbon Dioxide Theory of Climatic Change"Through further work Arrhenius determined that if you halved the amount of atmospheric carbon dioxide, the temperature of Europe could drop by as much as 4-5°C. But could such a change, big enough to cause an ice age, occur? He turned to colleague Arvid Hogbom (1857-1940), who had been investigating natural carbon dioxide cycles, to see if it could. Hogbom had, at the time, started to consider carbon dioxide emissions from factories (simple enough if you know, for example, how many tons of coal each factory burns a year). He had been surprised to find that man-made emission rates were very similar to those occurring in nature. Back in the 1890s, that of course represented a tiny fraction of the fossil fuels that we burn today; but what, they asked themselves, might happen if mankind burnt ever-increasing amounts over many centuries? Side-tracking from the ice-age research, Arrhenius ran calculations to see what a doubling of carbon dioxide levels might do to temperatures. He came up with an answer of 5-6°C of warming as a globally-averaged figure.
Back then, at 1890s burning-rates, they didn't see this as a problem: firstly at those rates it would take thousands of years for the doubling to take place and secondly the oceans were thought to be able to absorb five-sixths of the emissions. By the time the hypothesis appeared in a popular book that was published in 1908, the burning-rate had already gone up significantly, so in accordance with that change they revised the doubling-time down to a few centuries, but it was still something of a scientific curiosity, the stuff of after-dinner conversations.
The Carbon Dioxide Theory of Climatic Change - PLASS - 2010 - Tellus - Wiley Online Library
Ok, so now I'm sure your question is that "Well that's what the calculations were in the 1890's, 1950's, but what about today?". Good question. Here is a 2008 paper.By the mid-1950s, scientists had the huge advantage of the calculating power of computers. This made it possible to dissect each layer of Earth's atmosphere and work out how it might absorb infra-red radiation. Physicist Gilbert Plass undertook the task: firstly his work (published as a paper entitled The Carbon Dioxide Theory of Climatic Change, in the journal Tellus in 1956) confirmed that more carbon dioxide would have a warming effect and secondly that doubling levels of that gas would result in a warming of 3-4°C. That, at mid-1950s emissions rates, would be a rise of around 1.1°C per century. Plass wrote that if at the end of the 20th Century the average temperature had continued to rise, it would be "firmly established" that carbon dioxide could cause climate change. But again, the response was luke-warm. The lack of attention to water-vapour and cloudiness led to criticisms of crudeness, and again the matter of the ocean absorbing the extra gas was raised in objection to Plass' suggestion that the extra carbon dioxide would remain in the atmosphere for a thousand years
This is complex stuff, and I don't dare claim to be an expert. However, this SKS article gives a good explanation of our knowledge about climate sensitivity:
How sensitive is our climate?
Climate sensitivity is the estimate of how much the earth's climate will warm in response to the increased greenhouse effect if we double the amount of carbon dioxide in the atmosphere. This includes feedbacks which can either amplify or dampen that warming. This is very important because if it is low, as some climate 'skeptics' argue, then the planet will warm slowly and we will have more time to react and adapt. If sensitivity is high, then we could be in for a very bad time indeed. There are two ways of working out what climate sensitivity is. The first method is by modelling:
Climate models have predicted the least temperature rise would be on average 1.65°C (2.97°F) , but upper estimates vary a lot, averaging 5.2°C (9.36°F). Current best estimates are for a rise of around 3°C (5.4°F), with a likely maximum of 4.5°C (8.1°F). The second method calculates climate sensitivity directly from physical evidence, by looking at climate changes in the distant past:
These calculations use data from sources like ice cores to work out how much additional heat the doubling of greenhouse gases will produce. These estimates are very consistent, finding between 2 and 4.5°C global surface warming in response to doubled carbon dioxide.
All the models and evidence confirm a minimum warming close to 2°C for a doubling of atmospheric CO2 with a most likely value of 3°C and the potential to warm 4.5°C or even more. Even such a small rise would signal many damaging and highly disruptive changes to the environment. In this light, the arguments against reducing greenhouse gas emissions because of climate sensitivity are a form of gambling. A minority claim the climate is less sensitive than we think, the implication being we don’t need to do anything much about it. Others suggest that because we can't tell for sure, we should wait and see.
In truth, nobody knows for sure quite how much the temperature will rise, but rise it will. Inaction or complacency heightens risk, gambling with the entire ecology of the planet, and the welfare of everyone on it.