Closely related are intentional objects, which are what thoughts and feelings are about, even if they are not about anything real (such as thoughts about unicorns, or feelings of apprehension about a dental appointment which is subsequently cancelled).[1] However, intentional objects may coincide with real objects (as in thoughts about horses, or a feeling of regret about a missed appointment).
Mathematical objects
Mathematics and geometry describe abstract objects that sometimes correspond to familiar shapes, and sometimes do not. Circles, triangles, rectangles, and so forth describe two-dimensional shapes that are often found in the real world. However, mathematical formulas do not describe individual physical circles, triangles, or rectangles. They describe ideal shapes that are objects of the mind. The incredible precision of mathematical expression permits a vast applicability of mental abstractions to real life situations.
Many more mathematical formulas describe shapes that are unfamiliar, or do not necessarily correspond to objects in the real world. For example, the Klein bottle[2] is a one-sided, sealed surface with no inside or outside (in other words, it is the three-dimensional equivalent of the Möbius strip).[3] Such objects can be represented by twisting and cutting or taping pieces of paper together, as well as by computer simulations. To hold them in the imagination, abstractions such as extra or fewer dimensions are necessary.
In general, a logical antecedent is a sufficient condition, and a logical consequent is a necessary condition (or the contingency) in a logical conditional. But logical conditionals accounting only for necessity and sufficiency do not always reflect every day if-then reasoning, and for this reason they are sometimes known as material conditionals. In contrast, indicative conditionals, sometimes known as non-material conditionals,[5] attempt to describe if-then reasoning involving hypotheticals, fictions, or counterfactuals.
Truth tables for if-then statements identify four unique combinations of premises and conclusions: true premises and true conclusions; false premises and true conclusions; true premises and false conclusions; false premises and false conclusions. Strict conditionals assign a positive truth-value to every case except the case of a true premise and a false conclusion. This is sometimes regarded as counterintuitive, but makes more sense when false conditions are understood as objects of the mind.
False antecedent
A false antecedent is a premise known to be false, fictional, imaginary, or unnecessary. In a conditional sequence, a false antecedent may be the basis for any consequence, true or false.[6]:150–151
The subjects of literature are sometimes false antecedents. Examples include the contents of false documents, the origins of stand-alone phenomena, or the implications of loaded words. Moreover, artificial sources, personalities, events, and histories. False antecedents are sometimes referred to as "nothing", or "nonexistent", whereas nonexistent referents are not referred to.[7]:5–258
Art and acting often portray scenarios without any antecedent other than an artist's imagination. For example, mythical heroes, legendary creatures, gods and goddesses.
False consequent
A false consequent, in contrast, is a conclusion known to be false, fictional, imaginary, or insufficient. In a conditional statement, a fictional conclusion is known as a non sequitur, which literally means out of sequence. A conclusion that is out of sequence is not contingent on any premises that precede it, and it does not follow from them, so such a sequence is not conditional. A conditional sequence is a connected series of statements. A false consequent cannot follow from true premises in a connected sequence. But, on the other hand, a false consequent can follow from a false antecedent.
As an example, the name of a team, a genre, or a nation is a collective term applied ex post facto to a group of distinct individuals. None of the individuals on a sports team is the team itself, nor is any musical chord a genre, nor any person America. The name is an identity for a collection that is connected by consensus or reference, but not by sequence. A different name could equally follow, but it would have different social or politicalsignificance.
In metaphysics and ontology, Austrian philosopher Alexius Meinong advanced nonexistent objects in the 19th and 20th century within a "theory of objects". He was interested in intentional states which are directed at nonexistent objects. Starting with the "principle of intentionality", mental phenomena are intentionally directed towards an object. People may imagine, desire or fear something that does not exist. Other philosophers concluded that intentionality is not a real relation and therefore does not require the existence of an object, while Meinong concluded there is an object for every mental state whatsoever—if not an existent then at least a nonexistent one.[8]
Many objects in fiction follow the example of false antecedents or false consequents. For example, The Lord of the Rings by J.R.R. Tolkien is based on an imaginary book. In the Appendices to The Lord of the Rings, Tolkien's characters name the Red Book of Westmarch as the source material for The Lord of the Rings, which they describe as a translation. But the Red Book of Westmarch is a fictional document that chronicles events in an imaginary world.
Objects of the mind are frequently involved in the roles that people play. For example, acting is a profession which predicates real jobs on fictional premises. Charades is a game people play by guessing imaginary objects from short play-acts.
Imaginary personalities and histories are sometimes invented to enhance the verisimilitude of fictional universes, and/or the immersion of role-playing games. In the sense that they exist independently of extant personalities and histories, they are believed to be fictional characters and fictional time frames.
Science fiction is abundant with future times, alternate times, and past times that are objects of the mind. For example, in the novel Nineteen Eighty-Four by George Orwell, the number 1984 represented a year that had not yet passed.
Calendar dates also represent objects of the mind, specifically, past and future times. In The Transformers: The Movie, which was released in 1986, the narration opens with the statement, "It is the year 2005." In 1986, that statement was futuristic. During the year 2005, that reference to the year 2005 was factual. Now, The Transformers: The Movie is retro-futuristic. The number 2005 did not change, but the object of the mind that it represents did change.
Deliberate invention also may reference an object of the mind. The intentional invention of fiction for the purpose of deception is usually referred to as lying, in contrast to invention for entertainment or art. Invention is also often applied to problem solving. In this sense the physical invention of materials is associated with the mental invention of fictions.
Convenient fictions also occur in science.
Science
The theoretical posits of one era's scientific theories may be demoted to mere objects of the mind by subsequent discoveries: some standard examples include phlogiston and ptolemaic epicycles.
This raises questions, in the debate between scientific realism and instrumentalism about the status of current posits, such as black holes and quarks. Are they still merely intentional, even if the theory is correct?
The situation is further complicated by the existence in scientific practice of entities which are explicitly held not to be real, but which nonetheless serve a purpose—convenient fictions. Examples include field lines, centers of gravity, and electron holes in semiconductor theory.
A reference that names an imaginary source is in some sense also a self-reference. A self-reference automatically makes a comment about itself. Premises that name themselves as premises are premises by self-reference; conclusions that name themselves as conclusions are conclusions by self-reference.
In their respective imaginary worlds the Necronomicon, The Hitchhiker's Guide to the Galaxy, and the Red Book of Westmarch are realities, but only because they are referred to as real. Authors use this technique to invite readers to pretend or to make-believe that their imaginary world is real. In the sense that the stories that quote these books are true, the quoted books exist; in the sense that the stories are fiction, the quoted books do not exist.
In propositional logic, affirming the consequent, sometimes called converse error, fallacy of the converse, or confusion of necessity and sufficiency, is a formal fallacy of taking a true conditional statement, and invalidly inferring its converse, even though that statement may not be true. This arises when the consequent has other possible antecedents.
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent.
In propositional logic, modus ponens, also known as modus ponendo ponens, 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 propositional logic, modus tollens (MT), also known as modus tollendo tollens and denying the consequent, is a deductive argument form and a rule of inference. Modus tollens takes the form of "If P, then Q. Not Q. Therefore, not P." It is an application of the general truth that if a statement is true, then so is its contrapositive. The form shows that inference from P implies Q to the negation of Q implies the negation of P is a valid argument.
In logic or, more precisely, 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.
In mathematics and logic, a vacuous truth is a conditional or universal statement that is true because the antecedent cannot be satisfied. It is sometimes said that a statement is vacuously true because it does not really say anything. For example, the statement "all cell phones in the room are turned off" will be true when no cell phones are in the room. In this case, the statement "all cell phones in the room are turned on" would also be vacuously true, as would the conjunction of the two: "all cell phones in the room are turned on and turned off", which would otherwise be incoherent and false.
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. it is impossible for the premises to be true and the conclusion to be false.
Denying the antecedent, sometimes also called inverse error or fallacy of the inverse, is a formal fallacy of inferring the inverse from the original statement. It is committed by reasoning in the form:
In logic and mathematics, necessity and sufficiency are terms used to describe a conditional or implicational relationship between two statements. For example, in the conditional statement: "If P then Q", Q is necessary for P, because the truth of Q is guaranteed by the truth of P. Similarly, P is sufficient for Q, because P being true always implies that Q is true, but P not being true does not always imply that Q is not true.
In classical logic, a hypothetical syllogism is a valid argument form, a syllogism with a conditional statement for one or both of its premises.
In logic and mathematics, the logical biconditional, sometimes known as the material biconditional, is the logical connective used to conjoin two statements P and Q to form the statement "P if and only if Q", where P is known as the antecedent, and Q the consequent. This is often abbreviated as "P iff Q". Other ways of denoting this operator may be seen occasionally, as a double-headed arrow, a prefixed E "Epq", an equality sign (=), an equivalence sign (≡), or EQV. It is logically equivalent to both and , and the XNOR boolean operator, which means "both or neither".
In mathematical logic, a sequent is a very general kind of conditional assertion.
The barbershop paradox was proposed by Lewis Carroll in a three-page essay titled "A Logical Paradox", which appeared in the July 1894 issue of Mind. The name comes from the "ornamental" short story that Carroll uses in the article to illustrate the paradox. It existed previously in several alternative forms in his writing and correspondence, not always involving a barbershop. Carroll described it as illustrating "a very real difficulty in the Theory of Hypotheticals". From the viewpoint of modern logic it is seen not so much as a paradox than as a simple logical error. It is of interest now mainly as an episode in the development of algebraic logical methods when these were not so widely understood, although the problem continues to be discussed in relation to theories of implication and modal logic.
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.
A premise or premiss is a proposition—a true or false declarative statement—used in an argument to prove the truth of another proposition called the conclusion. Arguments consist of two or more premises that imply some conclusion if the argument is sound.
In logic and mathematics, contraposition refers to the inference of going from a conditional statement into its logically equivalent contrapositive, and an associated proof method known as proof by contraposition. The contrapositive of a statement has its antecedent and consequent inverted and flipped.
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 are connected 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.
↑ Burger, E. B., & Starbird, M., "A One-Sided, Sealed Surface—The Klein Bottle", in E. B. Burger, M. Starbird, T. Stonebarger, & T. Dunton, Joy of Thinking: The Beauty and Power of Classical Mathematical Ideas (Chantilly, VA: The Teaching Company, 2003).
↑ Murray, R., with Walker, J. J., & Wheeler, G. B., Murray's Compendium of Logic—with an Accurate Translation, and a Familiar Commentary (Dublin: M. W. Rooney, 1852; London: Simpkin and Marshall, 1852), pp. 150–151.
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