Arguments are the bricks from which philosophical theories are constructed; logic is the straw that binds those bricks together. Good ideas are worth little unless they are supported by good arguments—they need to be rationally justified, and this cannot be done properly without firm and rigorous logical underpinning. Clearly presented arguments are open to assessment and criticism, and it is this continual process of reaction, revision and rejection that drives philosophical progress.
An argument is a rationally sanctioned move from accepted foundations (premises) to a point that is to be proved or demonstrated (the conclusion). The premises are the basic propositions that must be accepted, provisionally at least, so that an argument can get underway. The premises themselves may be established in several ways, as a matter of logic or on the basis of evidence (i.e. empirically), or they may be the conclusions of previous arguments; but in any case they must be supported independently of the conclusion in order to avoid circularity. The move from premises to conclusion is a matter of inference, the strength of which determines the robustness of the argument. The business of distinguishing good inferences from bad is the central task of logic.
The role of logic Logic is the science of analyzing argument and of establishing principles or foundations on which sound inferences can be made. As such its concern is not with the particular content of arguments but with their general structure and form. So, given an argument such as “All birds are feathered; the robin is a bird; therefore the robin is feathered,” the logician abstracts the form “All Fs are G; a is an F; so a is G,” in which the particular terms are replaced by symbols and the strength of the inference can be determined independently of the subject matter. The study of logic formerly focused primarily on simple inferences of this kind (syllogisms), but since the beginning of the 20th century it has been transformed into a highly subtle and sophisticated analytical tool.
Up until the end of the 19th century the science of logic had proceeded, with only minor embellishments en route, very much along the path on which Aristotle had set it over 2000 years earlier. The model of proper reasoning was taken to be the syllogism, an inference made up of three propositions (two premises and a conclusion), most famously: “All men are mortal; Greeks are men; therefore Greeks are mortal.” Syllogisms were exhaustively classified according to their “form” and “figure” in such a way that valid and invalid types could be distinguished. The limitations of traditional logic were decisively exposed by the work of the German mathematician Gottlob Frege, who introduced notions such as quantifiers and variables that are responsible for the far greater scope and power of modern “mathematical” logic (so called because, unlike traditional logic, it is able to represent all mathematical reasoning).
Deduction The example given above (“All birds are feathered …”) is a deductive argument. In this case the conclusion follows from (is entailed by) the premises, and the argument is said to be “valid.” If the premises of a valid argument are true, the conclusion is guaranteed to be true, and the argument is said to be “sound.” The conclusion of a deductive argument is implicit in its premises; in other words, the conclusion does not “go beyond” its premises or say any more than is already implied by them. Another way of putting this, which reveals the argument’s underlying logical character, is that you cannot accept the premises and deny the conclusion without contradicting yourself.
Induction The other main way of moving from premises to conclusion is induction. In a typical inductive argument, a general law or principle is inferred from particular observations of how things are in the world. For instance, from a number of observations that mammals give birth to live young, it might be inferred, inductively, that all mammals do so. Such an argument can never be valid (in the sense that a deductive argument can) in that its conclusion does not follow necessarily from its premises; in other words, it is possible for the premises to be true but the conclusion false (as it is in the example given, where the conclusion is shown to be false by the existence of egg-laying mammals such as the platypus). This is because inductive reasoning always moves beyond its premises, which never entail a given conclusion but only support it or make it probable to some degree. So inductive arguments are generalizations or extrapolations of various kinds: from the particular to the general; from the observed to the unobserved; from past and present events or states of affairs to future ones.
“‘Contrariwise,’ continued Tweedledee, ‘if it was so, it might be; and if it were so, it would be: but as it isn’t, it ain’t. That’s logic.’”
Lewis Carroll, 1871
Inductive reasoning is ubiquitous and indispensable. It would be impossible to live our everyday lives without using observed patterns and continuities in the past and present to make predictions about how things will be in the future. Indeed, the laws and assumptions of science are often held to be paradigmatic cases of induction (see Science and pseudoscience). But are we justified in drawing such inferences? The Scottish philosopher David Hume thought that we are not—that there is no rational basis for our reliance on induction. Inductive reasoning, he argued, presupposes a belief in the “uniformity of nature,” according to which it is assumed that the future will resemble the past when relevantly similar conditions obtain. But what possible grounds could there be for such an assumption, except inductive ones? And if the supposed uniformity of nature can only be justified in this way, it cannot itself—without circularity—be used in defense of induction. In a similar vein, some have tried to justify induction on the basis of its past successes: basically, it works. But the supposition that it will continue to work in the future can only be inferred inductively from its past successes, so the argument cannot get off the ground. In Hume’s own view, we cannot help but reason inductively (and indeed he does not suggest that we shouldn’t reason in this way), but he insists that our doing so is a matter of custom and habit and is not rationally justified. The so-called “problem of induction” that Hume left behind, especially as it impacts upon the foundations of science, remains an area of active debate to this day.
“The prisoner will be hanged at dawn, by next Saturday at the latest, and will not know in advance the day on which the sentence is to be carried out.” It sounds bad, but the wily prisoner has a comforting train of thought. “The hanging can’t be on Saturday because I would know that in advance if I were still alive on Friday. So the latest day it can be is the Friday. But it can’t be then, because I would know that if I were still alive on the Thursday …” And so he works his way back through all the days to the present and is relieved to discover that the execution cannot take place. So it comes as a bit of a shock to the prisoner when he is in fact hanged on the following Tuesday.
Paradox or fallacy? Well, perhaps both. The story (known as the prediction paradox) is paradoxical because an apparently impeccable line of reasoning gives rise to a conclusion that is manifestly false, as the rueful prisoner discovers. Paradoxes typically involve seemingly sound arguments that lead to apparently contradictory or otherwise unacceptable conclusions. Sometimes there is no way of avoiding the conclusion, and this may call for reexamination of various attendant beliefs and assumptions; or some fallacy (error of reasoning) may be spotted in the argument itself. Either way, paradoxes demand philosophical attention because they invariably point to confusions or inconsistencies in our concepts and reasoning.
Some of the most famous paradoxes (several of which are discussed in the following pages) have been amazingly resistant to solution and continue to perplex philosophers.
the condensed idea
Infallible reasoning?
| timeline | |
|---|---|
| c.350BC | Forms of argument |
| c.300BC | The sorites paradox |
| AD1670 | Faith and reason |
| 1739 | Science and pseudoscience |
| 1905 | The king of France is bald |