Logics. Direct deductive inference: transformation by logical square. Relationships of contradiction and opposition (the most important)

Lecture notes, cheat sheets
Free library / Directory / Lecture notes, cheat sheets

Logics. Direct deductive inference: transformation by logical square. Relationships of contradiction and opposition (the most important)

Lecture notes, cheat sheets

Directory / Lecture notes, cheat sheets

Comments on the article

Table of contents (expand)

40. DIRECT DEDUCTIVE CONCLUSION: LOGICAL SQUARE CONVERSION. RELATIONS OF CONTRADICTIONS AND OPPOSITES

Given the properties of the relationship between the categorical propositions A, E, I, O, which are illustrated by the logical square scheme, one can draw conclusions by establishing the following of the truth or falsity of one judgment from the truth or falsity of another judgment.

The relation of contradiction (contradictority): A-O, E-I.

Since the relationship between contradictory judgments is subject to the law of the excluded middle, from the truth of one judgment follows the falsity of another judgment, from the falsity of one - the truth of the other. For example, from the truth of the universally affirmative proposition (A) "All peoples have the right to self-determination" follows the falsity of the particular-negative proposition (O) "Some peoples do not have the right to self-determination"; from the truth of the particular affirmative judgment (I) "Some court verdicts are acquittal" follows the falsity of the general negative judgment (E) "Not a single court verdict is acquittal."

Conclusions are built according to the schemes:

A → ⌉O; ⌉A → O; E →⌉I;⌉E → I.

Opposite (contrary) relationship: A-E. The truth of one proposition implies the falsity of another, but the falsity of one of them does not imply the truth of the other. For example, from the truth of the generally affirmative proposition (A) “All peoples have the right to self-determination,” the falsity of the generally negative proposition (E) “No people has the right to self-determination” follows. But from the falsity of proposition A, “All court verdicts are acquittal,” the truth of proposition E, “Not a single court verdict is acquittal,” does not follow. This proposition is also false.

Relations between opposite judgments obey the law of non-contradiction.

A → ⌉E, E→ ⌉A, ⌉A → (E ∨ ⌉E), ⌉E → (A ∨ ⌉A).

41. DIRECT DEDUCTIVE CONCLUSION: LOGICAL SQUARE CONVERSION. RELATIONSHIPS OF SUBCONTRARITY AND SUBMISSION

Relation of partial compatibility (subcontrast): I-O. The falsity of one proposition implies the truth of another, but the truth of one of them can entail both the truth and the falsity of another proposition. Both propositions can be true. For example, from the false proposition “Some doctors do not have a medical education” the true proposition “Some doctors have a medical education” follows; from the true proposition “Some witnesses have been interrogated” the proposition “Some witnesses have not been interrogated” follows, which can be either true or false.

Thus, subcontrarian judgments cannot be both false; at least one of them is true:

⌉I → O; ⌉0→I; I → (О ∨ ⌉О); O → (I ∨ ⌉1).

Subordination relationship (A-I, E-O). The truth of the subordinating judgment implies the truth of the subordinate judgment, but not vice versa: the truth of the subordinating judgment does not follow from the truth of the subordinate judgment; it can be true, but it can be false. For example, from the truth of the subordinate proposition A “All doctors have a medical education,” the truth of the subordinate proposition I “Some doctors have a medical education” follows. From a true subordinate proposition "Some witnesses have been examined" one cannot necessarily assert the truth of the subordinate proposition "All witnesses have been examined":

A → I; E → O; I → (A ∨ 1 A); O → (E ∨ 1E).

The falsity of the subordinate judgment follows from the falsity of the subordinate judgment, but not vice versa: from the falsity of the subordinate judgment the falsity of the subordinate does not necessarily follow; it may be true, but it may also be false. For example, from the falsity of the subordinate proposition (O) "Some peoples do not have the right to self-determination" follows the falsity of the subordinate proposition (E) "No people has the right to self-determination." If the subordinating proposition (A) "All witnesses have been examined" is false, then the subordinate proposition (I) "Some witnesses have been examined" may be true, but it may be false (it is possible that no witness has been examined).

In the logical square, the word "some" is used to mean "at least some."

⌉I →⌉ A; ⌉O → ⌉E; ⌉A → (I ∨ ⌉I); ⌉E→ (O ∨ ⌉0).

<< Back: Immediate Deductive Inference: Contrasting with the Predicate

>> Forward: Immediate deductive reasoning: logical square transformation. Relations of subcontrararity and subordination

We recommend interesting articles Section Lecture notes, cheat sheets:

▪ General biology. Crib

▪ Life safety. Crib

▪ History of psychology. Lecture notes

See other articles Section Lecture notes, cheat sheets.

Read and write useful comments on this article.

<< Back

Latest news of science and technology, new electronics:

The existence of an entropy rule for quantum entanglement has been proven 09.05.2024

Quantum mechanics continues to amaze us with its mysterious phenomena and unexpected discoveries. Recently, Bartosz Regula from the RIKEN Center for Quantum Computing and Ludovico Lamy from the University of Amsterdam presented a new discovery that concerns quantum entanglement and its relation to entropy. Quantum entanglement plays an important role in modern quantum information science and technology. However, the complexity of its structure makes understanding and managing it challenging. Regulus and Lamy's discovery shows that quantum entanglement follows an entropy rule similar to that for classical systems. This discovery opens new perspectives in the field of quantum information science and technology, deepening our understanding of quantum entanglement and its connection to thermodynamics. The results of the study indicate the possibility of reversibility of entanglement transformations, which could greatly simplify their use in various quantum technologies. Opening a new rule ... >>

Mini air conditioner Sony Reon Pocket 5 09.05.2024

Summer is a time for relaxation and travel, but often the heat can turn this time into an unbearable torment. Meet a new product from Sony - the Reon Pocket 5 mini-air conditioner, which promises to make summer more comfortable for its users. Sony has introduced a unique device - the Reon Pocket 5 mini-conditioner, which provides body cooling on hot days. With it, users can enjoy coolness anytime, anywhere by simply wearing it around their neck. This mini air conditioner is equipped with automatic adjustment of operating modes, as well as temperature and humidity sensors. Thanks to innovative technologies, Reon Pocket 5 adjusts its operation depending on the user's activity and environmental conditions. Users can easily adjust the temperature using a dedicated mobile app connected via Bluetooth. Additionally, specially designed T-shirts and shorts are available for convenience, to which a mini air conditioner can be attached. The device can oh ... >>

Energy from space for Starship 08.05.2024

Producing solar energy in space is becoming more feasible with the advent of new technologies and the development of space programs. The head of the startup Virtus Solis shared his vision of using SpaceX's Starship to create orbital power plants capable of powering the Earth. Startup Virtus Solis has unveiled an ambitious project to create orbital power plants using SpaceX's Starship. This idea could significantly change the field of solar energy production, making it more accessible and cheaper. The core of the startup's plan is to reduce the cost of launching satellites into space using Starship. This technological breakthrough is expected to make solar energy production in space more competitive with traditional energy sources. Virtual Solis plans to build large photovoltaic panels in orbit, using Starship to deliver the necessary equipment. However, one of the key challenges ... >>

Random news from the Archive

Smart watch Canyon CNS-SW71 for outdoor activities 04.04.2019

Canyon, a manufacturer of accessories for mobile devices and PCs, has announced the launch of the CNS-SW71 smart watch with a durable metal case and a practical magnetic clasp. Black and silver colors are available to choose from. The device can be used both for everyday wear and for various sports. There are two types of strap included: a metal bracelet and a TPU strap.

The device is equipped with advanced functionality, including a pedometer and a sleep monitoring system, which allows you to calculate and plan the optimal load. In addition to fitness tracking, the CNS-SW71 smart watch can be a stylish unisex accessory that provides its owner with uninterrupted communication both during training and throughout the day. Thanks to Bluetooth 4.0, the CNS-SW71 watch displays all alerts, including push notifications from mobile devices.

The watch has a color 1,22-inch touch screen with a resolution of 240x240 pixels. The battery with a capacity of 150 mA allows you to use the gadget for up to 15 days without recharging. The metal case protects the accessory from mechanical damage, and IP68 water resistance allows you to withstand immersion in water to a depth of 1 meter for up to 30 minutes.

Data from the device is synchronized by the Canyon Fit app, which is fully compatible with iOS and Android OS. It contains daily statuses of sports activity and detailed statistics of its physiological indicators.

Other interesting news:

▪ Electric car from BMW

▪ Delay global warming

▪ Compact version of the GeForce GTX 970 graphics card

▪ The initiative of a woman affects the relationship in a couple

▪ Smart glasses Recon Jet HUD Pilot Edition

News feed of science and technology, new electronics

Interesting materials of the Free Technical Library:

▪ section of the site Microcontrollers. Article selection

▪ Yukteswar article. Famous aphorisms

▪ article Why are only 12 letters used on Russian license plates? Detailed answer

▪ article Deputy chief physician of the hospital for surgery. Job description

▪ article Stork instead of Crocodile. Encyclopedia of radio electronics and electrical engineering

▪ article Incredible card guessing. Focus Secret

Leave your comment on this article:

All languages of this page

Home page | Library | Articles | Website map | Site Reviews