Arguments and inferences
Logic is commonly defined in terms of arguments or inferences as the study of their correctness. An argument is a set of premises together with a conclusion. An inference is the process of reasoning from these premises to the conclusion. But these terms are often used interchangeably in logic. Arguments are correct or incorrect depending on whether their premises support their conclusion. Premises and conclusions, on the other hand, are true or false depending on whether they are in accord with reality. In formal logic, a sound argument is an argument that is both correct and has only true premises. Sometimes a distinction is made between simple and complex arguments. A complex argument is made up of a chain of simple arguments. This means that the conclusion of one argument acts as a premise of later arguments. For a complex argument to be successful, each link of the chain has to be successful.
Arguments and inferences are either correct or incorrect. If they are correct then their premises support their conclusion. In the incorrect case, this support is missing. It can take different forms corresponding to the different types of reasoning. The strongest form of support corresponds to deductive reasoning. But even arguments that are not deductively valid may still be good arguments because their premises offer non-deductive support to their conclusions. For such cases, the term ampliative or inductive reasoning is used. Deductive arguments are associated with formal logic in contrast to the relation between ampliative arguments and informal logic.
A deductively valid argument is one whose premises guarantee the truth of its conclusion. For instance, the argument "(1) all frogs are amphibians; (2) no cats are amphibians; (3) therefore no cats are frogs" is deductively valid. For deductive validity, it does not matter whether the premises or the conclusion are actually true. So the argument "(1) all frogs are mammals; (2) no cats are mammals; (3) therefore no cats are frogs" is also valid because the conclusion follows necessarily from the premises.
According to an influential view by Alfred Tarski, deductive arguments have three essential features: (1) they are formal, i.e. they depend only on the form of the premises and the conclusion; (2) they are a priori, i.e. no sense experience is needed to determine whether they obtain; (3) they are modal, i.e. they hold by logical necessity for the given propositions, independent of any other circumstances.
Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference. Rules of inference specify the form of the premises and the conclusion: how they have to be structured for the inference to be valid. Arguments that do not follow any rule of inference are deductively invalid. Modus ponens is a prominent rule of inference. It has the form "p; if p, then q; therefore q". Knowing that it has just rained (
p
{\displaystyle p}
) and that after rain the streets are wet (
p
→
q
{\displaystyle p\to q}
), one can use modus ponens to deduce that the streets are wet (
q
{\displaystyle q}
).
The third feature can be expressed by stating that deductively valid inferences are truth-preserving: it is impossible for the premises to be true and the conclusion to be false. Because of this feature, it is often asserted that deductive inferences are uninformative since the conclusion cannot arrive at new information not already present in the premises. But this point is not always accepted since it would mean, for example, that most of mathematics is uninformative. A different characterization distinguishes between surface and depth information. The surface information of a sentence is the information it presents explicitly. Depth information is the totality of the information contained in the sentence, both explicitly and implicitly. According to this view, deductive inferences are uninformative on the depth level. But they can be highly informative on the surface level by making implicit information explicit. This happens, for example, in mathematical proofs.
Ampliative arguments are arguments whose conclusions contain additional information not found in their premises. In this regard, they are more interesting since they contain information on the depth level and the thinker may learn something genuinely new. But this feature comes with a certain cost: the premises support the conclusion in the sense that they make its truth more likely but they do not ensure its truth. This means that the conclusion of an ampliative argument may be false even though all its premises are true. This characteristic is closely related to non-monotonicity and defeasibility: it may be necessary to retract an earlier conclusion upon receiving new information or in light of new inferences drawn. Ampliative reasoning plays a central role in many arguments found in everyday discourse and the sciences. Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. The support they provide for their conclusion usually comes in degrees. This means that strong ampliative arguments make their conclusion very likely while weak ones are less certain. As a consequence, the line between correct and incorrect arguments is blurry in some cases, such as when the premises offer weak but non-negligible support. This contrasts with deductive arguments, which are either valid or invalid with nothing in-between.
The terminology used to categorize ampliative arguments is inconsistent. Some authors, like James Hawthorne, use the term "induction" to cover all forms of non-deductive arguments. But in a more narrow sense, induction is only one type of ampliative argument alongside abductive arguments. Some philosophers, like Leo Groarke, also allow conductive arguments as another type. In this narrow sense, induction is often defined as a form of statistical generalization. In this case, the premises of an inductive argument are many individual observations that all show a certain pattern. The conclusion then is a general law that this pattern always obtains. In this sense, one may infer that "all elephants are gray" based on one's past observations of the color of elephants. A closely related form of inductive inference has as its conclusion not a general law but one more specific instance, as when it is inferred that an elephant one has not seen yet is also gray. Some theorists, like Igor Douven, stipulate that inductive inferences rest only on statistical considerations. This way, they can be distinguished from abductive inference.
Abductive inference may or may not take statistical observations into consideration. In either case, the premises offer support for the conclusion because the conclusion is the best explanation of why the premises are true. In this sense, abduction is also called the inference to the best explanation. For example, given the premise that there is a plate with breadcrumbs in the kitchen in the early morning, one may infer the conclusion that one's house-mate had a midnight snack and was too tired to clean the table. This conclusion is justified because it is the best explanation of the current state of the kitchen. For abduction, it is not sufficient that the conclusion explains the premises. For example, the conclusion that a burglar broke into the house last night, got hungry on the job, and had a midnight snack, would also explain the state of the kitchen. But this conclusion is not justified because it is not the best or most likely explanation.
