Logical quality
Encyclopedia
In many philosophies
of logic
statements are categorized into different logical qualities based on how they go about saying what they say. Doctrines of logical quality are an attempt to answer the question: “How many qualitatively different ways are there of saying something?” Aristotle
answers, two: you can affirm something of something or deny something of something. Since Frege, the normal answer in the West, is only one, assertion
, but what is said, the content of the claim, can vary. For Frege asserting the negation
of a claim serves roughly the same role as denying a claim does in Aristotle. Other Western logicians such as Kant
and Hegel answer, ultimately three; you can affirm, deny or make merely limiting affirmations, which transcend both affirmation and denial. In Indian logic
, four logical qualities have been the norm, and Nagarjuna
is sometimes interpreted as arguing for five.
's term logic
there are two logical qualities: affirmation (kataphasis) and denial
(apophasis
). The logical quality of a proposition is whether it is affirmative (the predicate is affirmed of the subject) or negative (the predicate is denied of the subject). Thus "every man is a mortal" is affirmative, since "mortal" is affirmed of "man". "No men are immortals" is negative, since "immortal" is denied of "man"..
. Much of the work previously done by distinguishing affirmation from denial is typically now done through the theory of negation
. Thus, to most contemporary logicians, making a denial is essentially reducible to affirming a negation. Denying that Socrates is ill, is the same thing as affirming that it is not the case that Socrates is ill, which is basically affirming that Socrates is not ill. This trend may go back to Frege although his notation for negation is ambiguous between asserting a negation and denying . Gentzens notation definitely assimilates denial to assertion of negation, but might not quite have a single logical quality, see below.
, in the 2nd or 3rd century CE, argued for the existence of "nonassertive" statements, which indicate suspension of judgment by refusing to affirm or deny anything.. Pseudo-Dionysius the Areopagite
in the 6th century, argued for the existence of "non-privatives" which transcend both affirmation and denial. For example, it is not quite correct to affirm that God is , nor to deny that God moves, but rather one should say that God is beyond-motion, or super-motive, and this is intended not just as a special kind of affirmation or denial, but a third move besides affirmation and denial.
For Kant
every judgment takes one of three possible logical qualities, Affirmative, Negative or Infinite. For Kant, if I say “The soul is mortal” I have made an affirmation about the soul; I have said something contentful about it. If I say “The soul is not mortal,” I have made a negative judgment and thus “warded off error” but I have not said what the soul is instead. If, however, I say “The soul is non-mortal,” I have made an infinite judgment. For the purposes of “General logic” it is sufficient to see infinite judgments as a sub-variety of affirmative judgments, I have said something of the soul, namely that it is not mortal. But from the standpoint of “Transcendental Logic” it is important to distinguish the infinite from the affirmative. Although I have taken something away from the possibilities of what the soul might be like, I have not thereby said what it is or clarified the concept of the soul, there are still an infinite number of possible ways the soul could be. The content of an infinite judgment is purely limitative of our knowledge rather than amplitative of it. Hegel follows Kant in insisting that, at least transcendentally, affirmation and negation are not enough but require a third logical quality sublating them both.
is has long been traditional to claim that there are four kinds of claims. You can affirm that X is so, you can deny that X is so, you can neither-affirm-nor-deny that X is so, or you can both-affirm-and-deny that X is so. Each claim can also take one of four truth-values true, false, neither-true-nor-false, and both-true-and-false. However the tradition is clear that the four kinds of statements are distinct from the four values of statements.. Nagarjuna
is sometimes interpreted as teaching that there is a fifth logical quality besides the four typical of Indian logic, but there are disputing interpretations.
s with empty and non-empty antecedents amounts to a distinction between logical consequences and logical assertion
s. Alternately one might claim that both forms are really just logical assertion
s in the metalanguage, and are not statements at all in the object language, since the turnstile
isn't in the object language. Similarly you might argue that a modern language which includes both an assertion mechanism, and a "retraction" mechanism (such as Diderik Batens' "Adaptive Logics) could be thought of as having two logical qualities "assertion" and "retraction."
Philosophy
Philosophy is the study of general and fundamental problems, such as those connected with existence, knowledge, values, reason, mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational...
of logic
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
statements are categorized into different logical qualities based on how they go about saying what they say. Doctrines of logical quality are an attempt to answer the question: “How many qualitatively different ways are there of saying something?” Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
answers, two: you can affirm something of something or deny something of something. Since Frege, the normal answer in the West, is only one, assertion
Assertion
The term assertion has several meanings:* Assertion , a computer programming technique* Logical assertion, logical assertion of a statement* Proof by assertion, an assertion as opposed to an argument...
, but what is said, the content of the claim, can vary. For Frege asserting the negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
of a claim serves roughly the same role as denying a claim does in Aristotle. Other Western logicians such as Kant
KANT
KANT is a computer algebra system for mathematicians interested in algebraic number theory, performing sophisticated computations in algebraic number fields, in global function fields, and in local fields. KASH is the associated command line interface...
and Hegel answer, ultimately three; you can affirm, deny or make merely limiting affirmations, which transcend both affirmation and denial. In Indian logic
Indian logic
The development of Indian logic dates back to the anviksiki of Medhatithi Gautama the Sanskrit grammar rules of Pāṇini ; the Vaisheshika school's analysis of atomism ; the analysis of inference by Gotama , founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna...
, four logical qualities have been the norm, and Nagarjuna
Nagarjuna
Nāgārjuna was an important Buddhist teacher and philosopher. Along with his disciple Āryadeva, he is credited with founding the Mādhyamaka school of Mahāyāna Buddhism...
is sometimes interpreted as arguing for five.
Aristotle's two logical qualities
In AristotleAristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
's term logic
Term logic
In philosophy, term logic, also known as traditional logic or aristotelian logic, is a loose name for the way of doing logic that began with Aristotle and that was dominant until the advent of modern predicate logic in the late nineteenth century...
there are two logical qualities: affirmation (kataphasis) and denial
Denial
Denial is a defense mechanism postulated by Sigmund Freud, in which a person is faced with a fact that is too uncomfortable to accept and rejects it instead, insisting that it is not true despite what may be overwhelming evidence.The subject may use:* simple denial: deny the reality of the...
(apophasis
Apophasis
Apophasis refers, in general, to "mention by not mentioning". Apophasis covers a wide variety of figures of speech.-Apophasis:...
). The logical quality of a proposition is whether it is affirmative (the predicate is affirmed of the subject) or negative (the predicate is denied of the subject). Thus "every man is a mortal" is affirmative, since "mortal" is affirmed of "man". "No men are immortals" is negative, since "immortal" is denied of "man"..
Making do with a single logical quality
Logical quality has become much less central to logical theory in the twentieth century. It has become common to use only one logical quality, typically called logical assertionLogical assertion
A logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent....
. Much of the work previously done by distinguishing affirmation from denial is typically now done through the theory of negation
Negation
In logic and mathematics, negation, also called logical complement, is an operation on propositions, truth values, or semantic values more generally. Intuitively, the negation of a proposition is true when that proposition is false, and vice versa. In classical logic negation is normally identified...
. Thus, to most contemporary logicians, making a denial is essentially reducible to affirming a negation. Denying that Socrates is ill, is the same thing as affirming that it is not the case that Socrates is ill, which is basically affirming that Socrates is not ill. This trend may go back to Frege although his notation for negation is ambiguous between asserting a negation and denying . Gentzens notation definitely assimilates denial to assertion of negation, but might not quite have a single logical quality, see below.
Third logical qualities
Logicians in the western traditions have often expressed belief in some other logical quality besides affirmation and denial. Sextus EmpiricusSextus Empiricus
Sextus Empiricus , was a physician and philosopher, and has been variously reported to have lived in Alexandria, Rome, or Athens. His philosophical work is the most complete surviving account of ancient Greek and Roman skepticism....
, in the 2nd or 3rd century CE, argued for the existence of "nonassertive" statements, which indicate suspension of judgment by refusing to affirm or deny anything.. Pseudo-Dionysius the Areopagite
Pseudo-Dionysius the Areopagite
Pseudo-Dionysius the Areopagite, also known as Pseudo-Denys, was a Christian theologian and philosopher of the late 5th to early 6th century, the author of the Corpus Areopagiticum . The author is identified as "Dionysos" in the corpus, which later incorrectly came to be attributed to Dionysius...
in the 6th century, argued for the existence of "non-privatives" which transcend both affirmation and denial. For example, it is not quite correct to affirm that God is , nor to deny that God moves, but rather one should say that God is beyond-motion, or super-motive, and this is intended not just as a special kind of affirmation or denial, but a third move besides affirmation and denial.
For Kant
KANT
KANT is a computer algebra system for mathematicians interested in algebraic number theory, performing sophisticated computations in algebraic number fields, in global function fields, and in local fields. KASH is the associated command line interface...
every judgment takes one of three possible logical qualities, Affirmative, Negative or Infinite. For Kant, if I say “The soul is mortal” I have made an affirmation about the soul; I have said something contentful about it. If I say “The soul is not mortal,” I have made a negative judgment and thus “warded off error” but I have not said what the soul is instead. If, however, I say “The soul is non-mortal,” I have made an infinite judgment. For the purposes of “General logic” it is sufficient to see infinite judgments as a sub-variety of affirmative judgments, I have said something of the soul, namely that it is not mortal. But from the standpoint of “Transcendental Logic” it is important to distinguish the infinite from the affirmative. Although I have taken something away from the possibilities of what the soul might be like, I have not thereby said what it is or clarified the concept of the soul, there are still an infinite number of possible ways the soul could be. The content of an infinite judgment is purely limitative of our knowledge rather than amplitative of it. Hegel follows Kant in insisting that, at least transcendentally, affirmation and negation are not enough but require a third logical quality sublating them both.
The Indian Tradition
In Indian logicIndian logic
The development of Indian logic dates back to the anviksiki of Medhatithi Gautama the Sanskrit grammar rules of Pāṇini ; the Vaisheshika school's analysis of atomism ; the analysis of inference by Gotama , founder of the Nyaya school of Hindu philosophy; and the tetralemma of Nagarjuna...
is has long been traditional to claim that there are four kinds of claims. You can affirm that X is so, you can deny that X is so, you can neither-affirm-nor-deny that X is so, or you can both-affirm-and-deny that X is so. Each claim can also take one of four truth-values true, false, neither-true-nor-false, and both-true-and-false. However the tradition is clear that the four kinds of statements are distinct from the four values of statements.. Nagarjuna
Nagarjuna
Nāgārjuna was an important Buddhist teacher and philosopher. Along with his disciple Āryadeva, he is credited with founding the Mādhyamaka school of Mahāyāna Buddhism...
is sometimes interpreted as teaching that there is a fifth logical quality besides the four typical of Indian logic, but there are disputing interpretations.
More than One Quality Today
Although the distinction between affirmation and denial is rarely supported today, you might try to argue that some other distinctions in the structure of assertion could be thought of as differences of logical quality. One might argue, for instance, that the distinction between sequentSequent
In proof theory, a sequent is a formalized statement of provability that is frequently used when specifying calculi for deduction. In the sequent calculus, the name sequent is used for the construct which can be regarded as a specific kind of judgment, characteristic to this deduction system.-...
s with empty and non-empty antecedents amounts to a distinction between logical consequences and logical assertion
Logical assertion
A logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent....
s. Alternately one might claim that both forms are really just logical assertion
Logical assertion
A logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent....
s in the metalanguage, and are not statements at all in the object language, since the turnstile
Turnstile (symbol)
In mathematical logic and computer science the symbol \vdash has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails"...
isn't in the object language. Similarly you might argue that a modern language which includes both an assertion mechanism, and a "retraction" mechanism (such as Diderik Batens' "Adaptive Logics) could be thought of as having two logical qualities "assertion" and "retraction."