This contribution addresses probability relations and hopes to establish a rational component to induction and partial beliefs.
In deductive logic, a conclusion is true just in case the premises are true. An example of this is a categorical syllogism: if all As are Bs, and all Bs are Cs, then it can be deduced that all As are Cs. In deductive logic the movement of reasoning is from general to specific.
With inductive reasoning, empirical observations are made and from these generalizations are induced. For example, if a person throws a brick at a person's head at 10 second intervals, eventually the person may get into the rhythm of ducking at the interval to avoid being hit by the brick. Maybe this goes on for 19 bricks. Maybe at the 20th interval the person ducks in anticipation but is surprised that no brick is thrown. Point: it does not follow that because the brick was thrown 19 times that it will be thrown on the 20th. Similarly, a person can have an intuition that the world is flat, and from this make the claim that the world is flat. Yet we know that this would be a fallacious belief as scientific observations, satellite imaging, and thought have given us much more evidence to think the earth is a sphere. Yet, inductive arguments can be better or worse depending on the evidence in its favor. Indeed, based on a series of precedents one can have a highly educated guess that the brick will be thrown at the 20th interval, that there will not be a tornado in Toronto tomorrow, and that the Sun will rise tomorrow, but these are not and are not supposed to be deductively valid. As a result, the logical thinker can never be certain that the sun will rise tomorrow the way he can be certain that a categorical or disjunctive syllogism is logically valid.
Yet, to say that inductive arguments are non-rational would be misleading. In order to give a rational component to induction we will need to bring in probability. In short, we will need to extend our notion of a logical relation to include probability relations in order to establish how intuitive knowledge can form the basis of rational beliefs that fall short of knowledge. As Keynes has pointed out, a probability statement expresses a logical relationship between proposition ‘p’ and proposition ‘h’ (h is usually a conjunction of propositions). A man who knows ‘h’ and perceives a logical relationship between p and h is justified in believing p with a degree of belief which corresponds to that of the logical relationship. As such, probability is a logical relation holding between propositions which are similar to, although weaker than, logical consequences.
A proposition may be probable to a certain set of data and highly improbable to another set of data which includes the first set as a part. For example, if the only fact you know about a boxer is that his fight ended 30 seconds into the first round, it is quite probable with respect to these data that someone got knocked out or technically knocked out. If you find out after there was a power outage at ringside during his fight, then the probability that someone got knocked out, on combined data, is smaller. We can also think through the following example: given that X is an inhabitant of Europe it follows that he lives either in Britain or in France or in Germany or…Here, one can assign probabilities to person X living in each state based on certain conditions—perhaps relative populations being one of those conditions. Indeed, conditions are critical. To every event there is a finite set of conditions relative to which the event is certain to happen or certain not to happen. If the evidence reveals more of these conditions, we shall say that we have a stronger case, and we can extract probabilities and assign relative values to consequences. Some have argued that one way of quantifying probability relations in a real sense, could be to have the degree of belief in a proposition 'p', which a particular man has at a particular time, by the rate at which he would be prepared to bet upon 'p' being true. In this sense, there is a continuum: falsehood<--l--l--l--l--> truth with various probabilities in between. It is argued, therefore, that the test of relative strength of belief can be quantified and thus measured by a bet a person is willing to stake on their belief.
All in on this point: in matters of induction we cannot hope for anything more than probable conclusions and, therefore, the logical principles of induction must be the laws of probability. In this way, we are able to link intuitions and partial beliefs to a rational base grounded in probabilities.