Logics. Equivalent judgments. Logical relations between incompatible propositions (most important)

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

Logics. Equivalent judgments. Logical relations between incompatible propositions (most important)

Lecture notes, cheat sheets

Table of contents (expand)

28. EQUIVALENT JUDGMENTS. LOGICAL RELATIONS BETWEEN INCOMPATIBLE JUDGMENTS

Equivalent is a judgment that includes as components two judgments connected by a double (direct and inverse) conditional dependence, expressed by the logical connective “if and only if...

That...". For example: “If and only if a person has been awarded orders and medals (R), then he has the right to wear the appropriate order bars (q)".

The logical characteristic of this judgment is that the truth of the statement about the award (R) is considered as a necessary and sufficient condition for the truth of the statement about the existence of the right to wear order bars (Q). In the same way, the truth of the statement about the existence of the right to wear order bars (Q) is a necessary and sufficient condition for the truth of the statement that the person has been awarded the appropriate order or medal (R). Such mutual dependence can be symbolically expressed by the double implication p ↔ q, which reads: "If and only if рthen q". Equivalence is also expressed by another sign: р ≡

In natural language, including in legal texts, conjunctions are used to express equivalent judgments: “only provided that... then...”, “if and only if... then.. .", "only then... then...", etc.

Judgment р = q true in those cases when both propositions take on the same meaning, being simultaneously either true or false. This means that truth р enough to be true q, and vice versa. The relationship between them is also characterized as necessary, falsity р serves as an indicator of falsehood q, and falsity q points to falsehood р.

Logical relations between incompatible propositions.

Incompatible are propositions A and E, A and 0. E and I, which cannot be true at the same time. There are two types of incompatibility: opposition and contradiction.

1. Opposite (contrary) are propositions A and E, which cannot be true at the same time, but can be false at the same time.

The truth of one of the opposite judgments determines the falsity of the other: A → ⌉E; E → ⌉A. For example, the truth of the proposition “All officers are military personnel” determines the falsity of the proposition “Not a single officer is a military man.” If one of the opposing judgments is false, the other remains undefined - it can be either true or false: ⌉A → (E ∨ ⌉E); ⌉E → (A ∨ ⌉A).

2. Contradictory (contradictory) propositions are propositions A and O, E and I, which at the same time cannot be either true or false.

Contradiction is characterized by strict, or alternative, incompatibility: if one of the judgments is true, the other will always be false; if the first is false, the second will be true. The relationship between such judgments is governed by the law of excluded middle.

If A is recognized as true, then O will be false (A → ⌉O); if E is true, I will be false: (E → ⌉I). And vice versa: if A is false, O will be true (⌉A → O); and if E is false, I will be true (⌉E → I).

<< Back: Types of disjunction

>> Forward: Logical relationships between simple propositions

We recommend interesting articles Section Lecture notes, cheat sheets:

▪ Metrology, standardization and certification. Lecture notes

▪ Quality control. Crib

▪ Foreign literature of the XX century in brief. Part 1. Cheat sheet

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

Samsung SmartThings Wifi Mesh Network System 18.08.2018

Samsung introduced the SmartThings Wifi mesh network system. It is based on Wi-Fi technology, supplemented by AI capabilities. The SmartThings Hub included with the system is said to provide "an intelligently optimized home network, complete with smart home control in every room."

The advantages of SmartThings Wifi include stable coverage and rational distribution of bandwidth between devices in accordance with their capabilities and needs.

The open system Samsung SmartThings makes it easy to automate the operation of various devices and manage or use the application of the same name. SmartThings is compatible with hundreds of third-party devices and services such as lights, door locks, cameras, thermostats, voice assistants, and more.

A set of three SmartThings Wifi routers supporting 802.11a/b/g/n/ac - Wave2 2 x 2 MU-MIMO costs $280, one unit costs $120.

Other interesting news:

▪ The admixture of carbon improves the electrical conductivity of copper

▪ Wacom professional graphics tablets

▪ OLED display with a density of 1000 pixels per inch

▪ NCP1603 Power Factor Correction Controller

▪ Remains of ancient giants found

News feed of science and technology, new electronics

Interesting materials of the Free Technical Library:

▪ site section Parameters of radio components. Article selection

▪ article Money is the nerve of war. Popular expression

▪ What were the specifics of Europe in the early Middle Ages (mid-XI - late XV centuries)? Detailed answer

▪ article Work on metalworking equipment. Standard instruction on labor protection

▪ article Amplifier on the chip TDA7285, 2x0,05 watts. Encyclopedia of radio electronics and electrical engineering

▪ article error after error. Focus Secret

Leave your comment on this article:

All languages of this page

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