 |
Imperative logic is the field of logic concerned with imperatives. In contrast to declaratives, it is not clear whether imperatives denote propositions or more generally what role truth and falsity play in their semantics. Thus, there is almost no consensus on any aspect of imperative logic.
One of a logic's principal concerns is logical validity. It seems that arguments with imperatives can be valid. Consider:
However, an argument is valid if the conclusion follows from the premises. This means the premises give us reason to believe the conclusion, or, alternatively, the truth of the premises determines truth of the conclusion. Since imperatives are neither true nor false and since they are not proper objects of belief, none of the standard accounts of logical validity apply to arguments containing imperatives.
Here is the dilemma. Either arguments containing imperatives can be valid or not. On the one hand, if such arguments can be valid, we need a new or expanded account of logical validity and the concomitant details. Providing such an account has proved challenging.[ citation needed ] On the other hand, if such arguments cannot be valid (either because such arguments are all invalid or because validity is not a notion that applies to imperatives), then our logical intuitions regarding the above argument (and others similar to it) are mistaken. Since either answer seems problematic, this has come to be known as Jørgensen's dilemma, named after Jørgen Jørgensen (da).
While this problem was first noted in a footnote by Frege, it received a more developed formulation by Jørgensen. [1] [2]
Deontic logic takes the approach of adding a modal operator to an argument with imperatives such that a truth-value can be assigned to the proposition. For example, it may be hard to assign a truth-value to the argument "Take all the books off the table!", but ("take all the books off the table"), which means "It is obligatory to take all the books off the table", can be assigned a truth-value, because it is in the indicative mood.
Alf Ross observed that applying the classical rule of disjunction introduction under the scope of an imperative operator leads to unintuitive (or apparently absurd) results. [3] [4] When applied to simple declaratives, the result appears to be valid deduction.
However, a similar inference does not seem to be valid for imperatives. Consider:
Ross's paradox highlights the challenge faced by anyone who wants to modify or add to the standard account of validity. The challenge is what we mean by a valid imperative inference. For valid declarative inference, the premises give you a reason to believe the conclusion. One might think that for imperative inference, the premises give you a reason to do as the conclusion says; While Ross's paradox seems to suggest otherwise, its severity has been subject of much debate.
The semantics for Deontic logic requires that all obligations in the domain of discourse be fulfilled in an acceptable possible world; The conclusion "It is obligatory to clean your room or burn the house down" does not falsify the premise "It is obligatory to clean your room". In addition, based on the context, it may also be true that "It is obligatory to not burn the house down", in which case any acceptable possible world must have "Your room is cleaned" and "The house is not burnt down" to be both true.
Some strands of this debate connect it to Hans Kamp's paradox of free choice, in which disjunction introduction leads to absurd conclusions when applied under the scope of a possibility modal.
The following is an example of a pure imperative inference:
In this case, all the sentences making up the argument are imperatives. Not all imperative inferences are of this kind. Consider again:
Notice that this argument is composed of both imperatives and declaratives and has an imperative conclusion.
Mixed inferences are of special interest to logicians. For instance, Henri Poincaré held that no imperative conclusion can be validly drawn from a set of premises which does not contain at least one imperative. [5] While R.M. Hare held that no declarative conclusion can be validly drawn from a set of premises which cannot validly be drawn from the declaratives among them alone. [6] There is no consensus among logicians about the truth or falsity of these (or similar) claims and mixed imperative and declarative inference remains vexed.
Aside from intrinsic interest, imperative logic has other applications. The use of imperatives in moral theory should make imperative inference an important subject for ethics and metaethics.
In propositional logic, modus ponens, also known as modus ponendo ponens or implication elimination or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q.P is true. Therefore Q must also be true."
In logic, more precisely in deductive reasoning, an argument is sound if it is both valid in form and its premises are true. Soundness also has a related meaning in mathematical logic, wherein logical systems are sound if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.
The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic.
Deductive reasoning, also deductive logic, is the process of reasoning from one or more statements (premises) to reach a logical conclusion. It consists of drawing particular conclusions from a general premise or hypothesis.
"What the Tortoise Said to Achilles", written by Lewis Carroll in 1895 for the philosophical journal Mind, is a brief allegorical dialogue on the foundations of logic. The title alludes to one of Zeno's paradoxes of motion, in which Achilles could never overtake the tortoise in a race. In Carroll's dialogue, the tortoise challenges Achilles to use the force of logic to make him accept the conclusion of a simple deductive argument. Ultimately, Achilles fails, because the clever tortoise leads him into an infinite regression.
Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.
Understood in a narrow sense, philosophical logic is the area of philosophy that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. Some theorists conceive philosophical logic in a wider sense as the study of the scope and nature of logic in general. In this sense, philosophical logic can be seen as identical to the philosophy of logic, which includes additional topics like how to define logic or a discussion of the fundamental concepts of logic. The current article treats philosophical logic in the narrow sense, in which it forms one field of inquiry within the philosophy of logic.
Inductive reasoning is a method of reasoning in which a body of observations is synthesized to come up with a general principle. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from deductive reasoning. If the premises are correct, the conclusion of a deductive argument is certain; in contrast, the truth of the conclusion of an inductive argument is probable, based upon the evidence given.
Deontic logic is the field of philosophical logic that is concerned with obligation, permission, and related concepts. Alternatively, a deontic logic is a formal system that attempts to capture the essential logical features of these concepts. It can be used to formalize imperative logic, or directive modality in natural languages. Typically, a deontic logic uses OA to mean it is obligatory that A, and PA to mean it is permitted that A, which is defined as .
In philosophy, a formal fallacy, deductive fallacy, logical fallacy or non sequitur is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy where deduction goes wrong, and is no longer a logical process. This may not affect the truth of the conclusion, since validity and truth are separate in formal logic.
Alan Ross Anderson (1925–1973) was an American logician and professor of philosophy at Yale University and the University of Pittsburgh.
In metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle. It was first proven, informally, by Gottlob Frege in his 1884 Die Grundlagen der Arithmetik and proven more formally in his 1893 Grundgesetze der Arithmetik I. The theorem was re-discovered by Crispin Wright in the early 1980s and has since been the focus of significant work. It is at the core of the philosophy of mathematics known as neo-logicism.
Logic is the formal science of using reason and is considered a branch of both philosophy and mathematics and to a lesser extent computer science. Logic investigates and classifies the structure of statements and arguments, both through the study of formal systems of inference and the study of arguments in natural language. The scope of logic can therefore be very large, ranging from core topics such as the study of fallacies and paradoxes, to specialized analyses of reasoning such as probability, correct reasoning, and arguments involving causality. One of the aims of logic is to identify the correct and incorrect inferences. Logicians study the criteria for the evaluation of arguments.
A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agree or disagree with the premise in question, and in doing so understand the logical assumptions of the argument. If a premise is logically false, then the conclusion, which follows from all of the premises of the argument, must also be false—unless the conclusion is supported by a logically valid argument which the reader agrees with. Therefore, if the reader disagrees with any one of the argument's premises, they have a logical basis to reject the conclusion of the argument.
The paradoxes of material implication are a group of formulae that are intuitively false but treated as true in systems of logic that interpret the conditional connective as material conditional. On the material implication interpretation, a conditional formula is true unless is true and is false. If natural language conditionals were understood in the same way, that would mean that the sentence "If the Nazis won World War Two, everybody would be happy" is true. Given that such problematic consequences follow from a seemingly correct assumption about logic, they are called paradoxes. They demonstrate a mismatch between classical logic and robust intuitions about meaning and reasoning.
In logic and philosophy, an argument is a series of statements, called the premises, intended to determine the degree of truth of another statement, the conclusion. The logical form of an argument in a natural language can be represented in a symbolic formal language, and independently of natural language formally defined "arguments" can be made in math and computer science.
In logic, specifically in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called well-formed formulas.
Philosophy of logic is the area of philosophy that studies the scope and nature of logic. It investigates the philosophical problems raised by logic, such as the presuppositions often implicitly at work in theories of logic and in their application. This involves questions about how logic is to be defined and how different logical systems relate to each other. It includes the study of the nature of the fundamental concepts used by logic and the relation of logic to other disciplines. According to a common characterization, philosophical logic is the part of the philosophy of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic. But other theorists draw the distinction between the philosophy of logic and philosophical logic differently or not at all. Metalogic is closely related to the philosophy of logic as the discipline investigating the properties of formal logical systems, like consistency and completeness.
Logical consequence is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.
Logic is the study of correct reasoning or good arguments. It is often defined in a more narrow sense as the science of deductively valid inferences or of logical truths. In this sense, it is equivalent to formal logic and constitutes a formal science investigating how conclusions follow from premises in a topic-neutral way or which propositions are true only in virtue of the logical vocabulary they contain. When used as a countable noun, the term "a logic" refers to a logical formal system. Formal logic contrasts with informal logic, which is also part of logic when understood in the widest sense. There is no general agreement on how the two are to be distinguished. One prominent approach associates their difference with the study of arguments expressed in formal or informal languages. Another characterizes informal logic as the study of ampliative inferences, in contrast to the deductive inferences studied by formal logic. But it is also common to link their difference to the distinction between formal and informal fallacies.
Well-known paradoxes | |||||||
|---|---|---|---|---|---|---|---|
| Philosophical | |||||||
| Logical |
| ||||||
| Economic |
| ||||||
| Decision theory | |||||||
