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Hume proceeds to distinguish two "objects of human reason or enquiry ... Relations of Ideas, and Matters of Fact." Any relation of ideas is "either intuitively or demonstrably certain." Such relations are mathematical, and a proposition about a relation of ideas is true by virtue of its logical status—it is necessarily true. So, for example, the proposition "three times five is equal to the half of thirty" expresses a necessary relationship between the number determined by multiplying 3 by 5 and the number 30. The negation of the proposition cannot be consistently maintained.
Matters of fact, on the other hand, are those "objects of human reason" to which necessity does not attach. A proposition about matters of fact is not necessarily true. As Hume asserts, "The contrary of every matter of fact is still possible; because it can never imply a contradiction." Whereas it is contradictory to assert that a square does not have four sides, it is not contradictory to assert that the sun will not rise tomorrow.
Those propositions reflecting relations of ideas are known to be true by "the mere operation of thought, without dependence on what is any where existent in the universe." Those propositions reflecting matters of fact cannot be known in the same way. Hume's plan is to determine "what is the nature of that evidence, which assures us of any real existence and matter of fact, beyond the present testimony of our senses, or the records of our memory."
First, "the nature of that evidence," Hume asserts, is a causal relation. People believe events are causally related, so that knowledge of one event prompts one to believe the second event is real. The two events are not connected by the logic that guides relations of ideas—they are not inferred by employing the principle of noncontradiction. Instead, the two events are connected by the supposition "that there is a connection between the present fact and that which is inferred from it." That connection is causation, and it "arises entirely from experience, when we find, that any particular objects are constantly conjoined with each other."
Hume offers several examples to illustrate his point that experience provides the exclusive occasion for discovering cause-and-effect relations. Without experimentation, one cannot know that "two smooth pieces of marble ... will adhere together," and no amount of analysis will yield the contradiction, either—that is, that they will not stick together.
Hume points out that the first human being would not be able to think their way from looking at water to its potentially suffocating effect or from looking at fire to its potentially scorching effect. In other words, experiential conclusions are not founded on demonstrative reasoning. Instead, experience alone makes joining events together in a causal relation possible. In other words, Hume wants next to investigate how knowledge of cause and effect comes about. These events are entirely distinct from each other. As Hume points out, "every effect is ... distinct ... from its cause." Moreover, assigning the correct cause-and-effect relationship, when other arrangements could be asserted, requires "the assistance of observation and experience."
A single experience of two events cannot legitimately be causally connected. At first, causal judgments, also called "causal inference" and "inductive inference," are "arbitrary." It is only after regular experiences of this type that such a general association is made "by means of reasonings from analogy, experience, and observation." It is through the vivacity, force, and frequency of two events experienced together that one becomes increasingly confident in expecting the second when the first happens.
A matter-of-fact inference is, then, a causal inference. A causal inference, in turn, is based entirely on experience. Nowhere does pure reason play a role. The next question is this: what is the foundation, or justification, for experiential inferences?
Hume summarizes the questions asked and answered thus far in Section 3. Reasoning about matters of fact is "founded on the relation of cause and effect." The foundation of causal reasoning is experience. Hume notes, however, that the foundation of experiential reasoning "implies a new question." It is not any reasoning at all, he asserts, that provides justification for experiential reasoning, since "our conclusions from [the experience of the operations of cause and effect] are not founded on reasoning, or any process of the understanding."
Conclusions drawn from experience are not rationally justified—they cannot be justified through relations of ideas or matters of fact. Hume has already argued that reason alone cannot account for causal inferences. Inductive reasoning itself fares no better for two reasons. First, Hume reiterates the point that people are ignorant of the connection between "the sensible qualities and the secret powers" of an object. The sensible qualities of bread do not reveal its powers of nourishment. Hence, these powers are "secret." Second, Hume reiterates the point that experiences are entirely separated. Any one experience "can be allowed to give direct and certain information of those precise objects only, and that precise period of time, which fell under its cognizance." Hume does not see how one is justified, without an intermediate step, in extending one experience into the future, as is done when one infers from a past event that a present or future event will be sufficiently like it to bring about the same effects.
That intermediate step, or "medium," is what would "enable the mind to draw such an inference, if indeed it be drawn by reasoning and argument." It is not provided by demonstrative reasoning, for reasons already provided in Part 1 but also because it is possible that nature could change. What is taken to be a snowflake, for example, could taste salty or feel hot—there is no way to demonstrate that the possibility is false. In addition, one's beliefs may be incorrect. That leaves inductive reasoning, which Hume here calls "moral reasoning."
Hume recaps what is involved in an inductive inference:
P1: Judgments about matters of fact depend on the idea of cause and effect.
P2: Knowledge of cause and effect, in turn, is derived solely from experience.
P3: Experiential reasoning, in turn, assumes "that the future will be conformable to the past."
Conclusion: Judgments about matters of fact assume that nature is uniform. That is "that the future will be conformable to the past."
Such reasoning, Hume points out, is circular. P3 assumes what the argument intends to prove, namely that past experience justifies judgments about similar future events.
Relations of ideas and matters of fact are the two names Hume gives to modes of reasoning. Human understanding is confined to propositions that are either certain or probable. To assert a relation of ideas is to make a logically necessary judgment. To assert a matter of fact is to make an experiential judgment. A conclusion resulting from relations of ideas is, by its very nature, certain. A conclusion resulting from matters of fact is, by its very nature, probable. The former kind is mathematical, and the latter kind is experiential.
The reasoning confined to relations of ideas involves extracting a proposition from the necessary connection between ideas. More specifically, the reasoning proceeds by way of noncontradiction: Since 3 x 5 = 15 is true and 15 is half of 30 is true, it cannot be false that "three times five is equal to the half of thirty." Reasoning involving matters of fact, on the other hand, involves connecting two ideas that are not knowable by employing the principle of noncontradiction. The evidence for a judgment based on a relation of ideas is not the same kind, then, as one based on a matter of fact.
Hume stresses this distinction in terms of the evidence required for each kind. One who reasons to a conclusion from relations of ideas requires the principle of noncontradiction. One who reasons to a conclusion from matters of fact enlists causation. For example, if a man on a desert island finds a watch, Hume asserts, he would conclude that other people had been there. At the time he finds the watch, he does not see any other humans, but he draws the inference by connecting the present to the past. The present is the effect, a human-made artifact, and the past is the cause, humans having been on the island.
Reason alone, that is, relations of ideas or demonstrative reasoning, cannot yield knowledge of causation. Hume says, "The mind can never possibly find the effect in the supposed cause, by the most accurate scrutiny and examination. For the effect is totally different from the cause, and consequently can never be discovered in it." Consider Hume's original point about impressions. By themselves, that is, absent any unifying principle, impressions are entirely separate from each other. There is no logically formal way to bring them together, as there is in mathematics, where conclusions are effectively extracted from premises—results are already contained within the numbers or figures under investigation. The same cannot be said about matters of fact. Here, the unifying principle, cause and effect, occurs through experience itself.
Moreover, this association does not present itself upon the first occurrence of an entirely new event, as there is no reference for which sort of cause is appropriate. As Hume points out, the event of one billiard ball moving in a straight line toward another does not suggest the subsequent motion of that second ball. Indeed, "even suppose motion in the second ball should by accident be suggested to me, as the result of their contact or impulse; may I not conceive, that a hundred different events might as well follow from that cause?" This, Hume thinks, is the reason why the ultimate causes of things cannot be known. The best that can be achieved is a set of general causes whose scope is considerably narrower.
Although Hume is a skeptical empiricist, he is, nonetheless, an empiricist. In arguing against the claim that reason alone can discern that causes contain their effects, Hume argues against a rationalist tradition going back at least as far as Descartes, who produced a causal argument for the existence of God based on the premise that causes must be at least as real as their effects. According to Hume, logical entailment is not the same as cause-and-effect relations. In other words, the premises in a demonstration entail their conclusion: The figure is a square or a triangle. The figure is not a square. So, the figure is a triangle. Causal reasoning, on the other hand, does not involve the entailment of the effect in the cause: The sun rose yesterday. The sun rose today. So, the sun will rise tomorrow.
Hume addresses what is commonly known as the problem of induction: How is past experience a justification for present and future experiential judgments? The short answer is that it's not—at least not a rational or logical justification. When thinking of an empirical justification, one thinks of a reason or set of reasons considered sufficient evidence in support of a claim. An empirical justification is a different kind than a nonempirical, or deductive, justification, which relies on the form or structure of the reasoning rather than the content. Consider the following two examples of formally correct reasoning (also known as logically correct, or valid, reasoning):
Argument 1: Either Lady Gaga or Abraham Lincoln was the 16th president of the United States. Since Lady Gaga was not the 16th president of the United States, it follows that Abraham Lincoln was the 16th president of the United States.
Argument 2: The moon is made of blue or green cheese. Since the moon is not made of blue cheese, it follows that the moon is made of green cheese.
Both are instance of the same form:
p or q
The form of the reasoning is correct: if the premises are true, the conclusion must be true. Another way to put this is as follows: the conclusion can't be false if the premises are true. Determining logical correctness is a matter of determining whether or not the conclusion can be true when the premises are taken as true. Both Argument 1 and Argument 2 are logically correct, despite the fact that the first premise and the conclusion of Argument 2 are false. The justification for the conclusion in both arguments is the arrangement of the premises, not their content.
Argument 1 is logically correct, and the premises are actually true. That makes it a superior argument, overall, to Argument 2. The topic at present, however, does not involve the content of the argument, only its form. Contrast evaluation of the reasoning's form with content, which is what is done with empirical reasoning, or what Hume calls matters of fact.
The justification of the conclusion of any matter-of-fact reasoning depends on the likelihood that the conclusion is true. Another way to put things is that the justification is the experience itself. So, when Hume mentions examples such as predicting the sun will rise tomorrow or that the bread will provide nourishment, the justification for each is prior experience. More specifically, that justification is a cause-and-effect relationship one has previously identified. The problem that Hume points out, however, is that the justification of a cause-and-effect relationship—the prediction that the sun will rise tomorrow or that this slice of bread will provide nourishment—relies on the assumption that cause-and-effect relations are correct. But why should cause and effect be a standard for justifying conclusions for any matter of fact, which already proceeds on the assumption that causal relations are the source of such reasoning? Isn't doing so simply assuming what one wants to prove? In other words, the conclusion drawn from inductive reasoning relies on inductive reasoning, something philosophers call "begging the question."
This section of Hume's Enquiry is particularly tricky, as significantly related concepts have to be separated out while maintaining their associations. First, matters of fact, cause and effect, and experience are all associated. To reiterate a suggestion previously made, one can think of a matter of fact as an empirical judgment, that is, a statement that expresses something about sensations or feelings and their objects. Someone might assert, "I saw a frog leap into a pond." The judgment is the result, at least in part, of an inference about the object, namely that it is a frog. Previous encounters with creatures that look similar to the one in question and do similar things as the one in question precipitate the judgment that this particular creature is a frog and not, say, a toad.
This is an element of Hume's point about eating bread. The bread one eats at any time is not an object previously encountered. It's an entirely new object. Moreover, one does not know the ultimate cause of the nourishment one gets from the bread one eats—be it the slice in question, previous slices, or future slices. Judging that the bread is nutritious displays confidence in the belief that the bread provides nutrition. That belief, however, is not the result of a rational inferential process. After all, the nature of the bread is unknown. So, when one infers that one gains nutrition from eating the bread, one infers that the present experience resembles previous such experiences. Observations in individual instances, such as "This bread was nourishing," are extended to other instances, or even all of them, as happens when universal claims are made, such as "Bread provides nourishment."
On the assumption that inductive inferences are rationally justified, there should be an element that links one experience to another or multiple experiences to a universal judgment. As Hume points out, however, experiential reasoning (what Hume also calls "experimental" and "inductive," "matter of fact" and "cause and effect" reasoning) can't be justified using the very principle that's meant to be justified: "causal reasoning is justified by causal reasoning" is not legitimate.