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Oparaugo Michael — Wed, 09 Sep 2026 03:43:26 +0000

Property law is the area of law that governs legal rights, including ownership and tenure, in property[1]. Legal systems generally recognise two major kinds of property: property that relates to land, often called real property; and property that does not, which may include personal property. Depending on jurisdiction, personal property either is synonymous with tangible property, which may include money, or can be further divided into tangible and intangible property, which includes intellectual property[2].

Property law relates to the acquisition, divestment, and enforcement of proprietary rights. As these rights usually involve relationships between private individuals, property law is largely an area of private law, although public law aspects of property law include such issues as compulsory land acquisition, wealth redistribution, environmental effects, antitrust or economic competition, indigenous rights, and the human rights to property and housing.

The property law of common law jurisdictions originate in medieval English law, which developed under two separate systems of court, equity and common law, each with its set of proprietary rules. Civil law jurisdictions, on the other hand, trace their proprietary origins to the Roman law, although the two legal traditions interact with and influence one another. The most significant doctrinal influence across the two systems is the reception of the trust into civilian jurisdictions

Doctrine of Estates: A Comprehensive Guide to Property Rights
Definition & meaning
The doctrine of estates is a legal principle originating from English law, which states that individuals do not own land outright. Instead, they hold an "estate" in the land, which gives them certain rights to use and possess it for a specified period. This concept emerged during the Norman conquest of England in 1066, when the feudal system established a hierarchy of land ownership. Under this system, land was granted by a lord to a vassal for a defined time in exchange for services or obligations. Today, while the doctrine remains significant, the number of recognized estates has been limited by the Law of Property Act of 1925.

Types of Estate
In property law, an "estate" refers to the degree, nature, and extent of a person's interest in land or property. The main types of estates are:

Freehold Estates Fee Simple Absolute: The most complete ownership interest possible. The owner has unlimited duration and can transfer the property freely. Life Estate: Ownership lasts for the duration of a person's life. After their death, the property passes to another party (remainderman). Fee Tail: Ownership is inherited by a specific line of heirs (rare in modern law).
Leasehold Estates Estate for Years: Lasts for a fixed, specified period (e.g., a 5-year lease). Periodic Tenancy: Continues for successive periods (e.g., month-to-month) until terminated by notice. Tenancy at Will: Can be ended at any time by either party. Tenancy at Sufferance: Occurs when a tenant remains after the lease has expired without the landlord's consent.
Other Types
Contingent Estate: Ownership depends on the occurrence of a certain event.
Vested Estate: Ownership is secured for a person, but possession may be delayed.
Summary Table
Type of Estate Duration/Condition Example
Fee Simple Absolute Unlimited Outright ownership
Life Estate For a person's lifetime "To A for life"
Estate for Years Fixed period 5-year lease
Periodic Tenancy Renews automatically Month-to-month rental
Tenancy at Will Indefinite, can end anytime Open-ended arrangement
Tenancy at Sufferance After lease expiry, no consent Holdover tenant
These are the main types of estates recognized in property law.
Prepare an inventory of your assets and liabilities
A family inventory is a comprehensive list of information pertaining to your family’s current financial status, such as:

Financial accounts
Investment-related debt
Digital assets
Insurance and annuity
Non-registered investments
Completing an inventory is a key step in developing your estate plan. An inventory will help ensure that all of your assets are accounted for and considered, and that beneficiaries are taken care of.

“It helps to have an overview right at the beginning, to better plan ahead,” says Michelle Lau, a wealth planner at RBC Wealth Management in Asia based in Singapore.

Regarding what you need to include on this list, Octavia Liu, a wealth planner at RBC Wealth Management in Asia based in Hong Kong, says, “Asset types can be very diversified. While bankable assets such as life insurance and property tend to be focused on, there might be items that are valuable to you, like antiques, jewelry or perhaps a book collection.”

The inventory should also identify the ownership structure (for example, sole or joint ownership) of these assets, as well as any beneficiary designations applicable to the assets.

“It is also important to be cognizant of where the assets are located,” adds Lau.

She explains that estate and inheritance taxes may vary across jurisdictions. “That might potentially complicate the probate process, because there will be legal processes that need to be completed in each jurisdiction where the assets are located,” she says.

Draft a list of your estate-planning objectives Estate objectives can act as a guide and help to ensure your estate plan accurately addresses both your personal and financial goals.

The following questions can help you establish your objectives:

Who are the beneficiaries of the estate?
When do you want your beneficiaries to receive their inheritance – immediately or at some future date?
How do you want to pass your assets to your beneficiaries? There are a variety of options to consider: an inter vivos trust, also known as a living trust, is created by the grantor during his or her lifetime; a testamentary trust, which is formed after the death of the grantor; and simpler approaches, including outright gifts and inheritances.
Do you want to leave a portion for your own retirement plan and/or to charities?

Determine the actions needed to achieve your objectives After identifying and evaluating your estate objectives, the next step is to determine the components of your estate plan that will address these goals.

Having a will prepared is one of the most important steps to take in estate planning. It outlines who will oversee your estate, and how your assets will be distributed after you’re gone.

Lau points out that a will is important because asset-holding structures may not cover every type of asset that you want to pass on.

While a will is a good start in crafting an estate plan, Liu notes that it does not cover every objective.

Other potential elements of your action plan may include changes in the legal ownership of assets (for example, placing the assets into an inter vivos trust, which is created while a person is still alive in order to name the beneficiaries of property and assets upon death), a full review of beneficiary designations for your registered plans and life insurance policies and possibly the gifting of assets prior to death.

Consult with advisors to support and help implement components of your plan To ensure your estate plan is properly executed, you may require the assistance of professionals such as an accountant, a lawyer and possibly a trust officer.

“Speaking with a wealth planner may provide a helpful overview of the options available for estate planning,” Lau says. “Your wealth planner or advisor might also share their experiences based on what they’ve seen in other cases – whether it’s planning or asset distribution – to provide different perspectives.”

Lau adds that a wealth planner will also be able to highlight certain legal and tax issues – and, from there, can guide you toward the professionals you should consult with to form a comprehensive estate plan.

Conduct periodic reviews You should always be vigilant and mindful that changes in laws or in your financial and personal situations may require revisions to your overall estate plan.

Periodic reviews are a must to ensure that your estate plan is still achieving your objectives.

“People may change; their thoughts may change, too. Family situations may also be different, so it is useful to continue to review your plan from time to time,” Liu advises.

Lau agrees, adding that family events such as a birth or a child getting married could be triggers for an estate plan review.

While a periodic review is important, Liu stresses that ensuring the revisions are valid in the presence of witnesses such as tax advisors or lawyers is equally important.

The customary land tenure system in Nigeria is an indigenous practice where land ownership is vested collectively in communities, families, or villages rather than individuals
.
Key Features of Customary Land Tenure
Collective Ownership: Land belongs to the group (community, village, or family). Absolute ownership by a single person is traditionally unknown.
Caretakership: Community or family heads hold the land in trust and manage it for the collective benefit of all members.
User Rights: Individual members only have rights to use, occupy, and enjoy specific portions of land allocated to them for farming or building.
Restriction on Alienation: Land cannot be sold or permanently transferred outside the family or community without the collective consent of key members.
Impact of the Land Use Act of 1978
State Control: The Land Use Act vested all land in a state's urban areas in the State Governor, and rural land in local government chairmen, held in trust for the public.
Coexistence: The statutory law did not completely destroy customary ownership; instead, it created a dual system where traditional rights of occupancy continue to exist alongside statutory titles.

A land tenure system is a set of rules and regulations that govern how land is owned, used, managed, and transferred within a society.

These systems act as the foundation for regulating land ownership rights, establishing who can use the land, for how long, and under what conditions.

The land tenure system in Nigeria is a global subject matter and with it, has come the need to discuss how to go about owning land in Nigeria.

This article will open a conversation about the regulations of land ownership in Nigeria so that we know the various forms of land ownership behavior in the country.

What is the meaning of Land Tenure System?

About Land Tenure System in Nigeria

7 types of Land Tenure System in Nigeria

Freehold Tenure System:
Inheritance Tenure System:

3.Communal Land Tenure system:

4.Leasehold Tenure System:

5.Gift Tenure System:

Rent Tenure System:
Tenants at Government Will:

How you can become a landowner

What are the features of land tenure system in Nigeria

Advantages of land tenure system in Nigeria

Disadvantages of land tenure system in Nigeria

Conclusion:

What is the Land Tenure System?
The Land Tenure System in Nigeria is the process of granting ownership of land to individuals, legal bodies, corporate bodies, and natural bodies based on their use of these lands.

This statutory instrument is used to ensure that human habitats are safe and sustainable.

Furthermore, the Land tenure or Land administration system in Nigeria is an institution of laws that can regulate the use, management, and transfer of land.

About Land Tenure System in Nigeria
Land tenure system in Nigeria is an important issue of discussion, search, and review. The basic land law regimes and determinants of relevant rights and obligations concerning land property.

Land tenure in Nigeria is a complex series of relationships between multiple institutions concerning the uses of land.

It also includes institutions that are significant in determining the pattern of land ownership and use. These institutions include the government, customary laws, etc.

More so, the history of land administration in Nigeria can be traced back to the 1800s while some statistics show that only a smaller percentage of the nation’s urban residents live in officially designated urban areas.

Other people seek ways to own or buy land in their vicinity.

7 types of Land Tenure System in Nigeria
To strengthen property rights and eliminate systemic sources of wealth inequality by changing the classification, distribution, and administration of its land resources, the country came up with types of Land Tenure systems. They are:

Freehold Tenure System:
Individuals who subscribe to the Freehold tenure system pay a predetermined amount for the right to own a plot of land. Upon obtaining it, you can use it as collateral for a loan. The larger the land, the greater the payment. The land is surveyed and closed by signing backup documents.
Inheritance Tenure System:
In this case, land ownership is transferred to the next of kin. Thus, lands are provided for both born and unborn children. Some villages transfer lands to the children upon their parents’ death.
Communal Land Tenure system:
The community becomes the ruling power of the land under this system. The head of the community determines the sharing ratio. Farming on a large scale is often encouraged, but a single individual cannot claim ownership of the land or even use it as security.
Leasehold Tenure System:
An individual is granted temporary ownership of a plot of land by some form of a title from the owner. During the lease period, an individual may have temporary access to the land, but cannot use it as collateral for loans.
Gift Tenure System:
This type of land ownership is when the landowner gives up his or her land voluntarily and without being coerced by anybody. Since the new owner now owns the full and permanent title to the land, he can use it as collateral for a loan.
Rent Tenure System:
The tenants pay a rent amount to the landlord for the time that they use the property. Depending on the agreement and terms, the rent period could be one to two years.
Tenants at Government Will:
According to this system, land is leased by the Nigerian government to farmers for cultivating. The land is mostly used for large-scale farming and crop production. The land is relatively inexpensive to acquire.

Advantages of land tenure by Inheritance in Nigeria
Here are the advantages of land tenure by inheritance:

Security and Stability: Inheritance tenure provides a sense of security for families. Families can keep their land for future generations.

Reduced Land Disputes: Clear inheritance rights help families and communities avoid arguments.

Promotes Long-Term Investment: Landowners may be more likely to invest in and improve the land if they know it will be passed on to their heirs.

It is backed up by “Acts” and these acts provided for all issues at or about lands in Nigeria, its ownership, descriptions, partition use, inheritance, mortgage purchase/ selling transfer, etc.
For more information about the land system in Nigeria, check out my guides:

Proof of Land Ownership in Nigeria

Excision of Land

Land Use Charges in Nigeria

Disadvantages of Land Tenure System by Inheritance
While inheritance offers a sense of security and stability for families, there are also some drawbacks to consider:

Land Fragmentation: When land is divided amongst multiple heirs, it can become fragmented into smaller and smaller parcels over generations. This can make it difficult to use the land for large-scale agriculture or development projects.
Disputes and Delays: Inheritance can sometimes lead to disagreements among family members about who should inherit the land or how it should be divided. This can result in lengthy legal battles and delays.
Limited Access to Credit: Land inherited through inheritance may not be readily usable as collateral for loans since ownership can be unclear or disputed. This can hinder investment and development on the land.
How you can become a landowner
You can become a landowner by inheritance or by other existing types of land tenure systems discussed in this article.

More so, Mixta Africa, a team of innovative minds working towards building interconnected African cities and spaces can also help you out.

What are the features of land tenure system in Nigeria
The things that make up the Land Tenure system in Nigeria include:

Property right by an individual
Use of land within a community
Control of communal piece of land and
Transfer of allocated piece of land legitimately.
Conclusion:
Land tenure is a system of laws that determines who is considered to be the owner of land in a given jurisdiction.

Nigeria has a federal system of government and land tenure laws are generally made by the various States. This topic often creates attention in the country the “Land Tenure System”.

Therefore, in an attempt to explore further, we have conducted some research on land tenure in Nigeria.

Our findings were elaborated on in this article so that it will contribute to the numerous debates about the land tenure system in Nigeria.

Understanding The Land Use Act In Nigeria: A Comprehensive Guide
The Land Use Act of 1978 is one of the most important legal frameworks governing land ownership and management in Nigeria. The act vests control of all land within a state’s territory in the governor of that state, effectively centralizing the authority to grant land rights. Understanding this law is crucial for anyone involved in land acquisition, property development, or real estate investment in Nigeria.

Historical Background: Land Ownership Before the Act
Before 1978, land ownership in Nigeria was primarily governed by a customary land tenure system. This system allowed traditional rulers, families, and communities to control land according to local customs and traditions. While this system worked in many rural areas, it created significant problems in urban and developing regions, especially as the country industrialized.

In urban areas, there was often confusion and inconsistency in land transactions, with multiple claims to ownership on the same piece of land. Moreover, the variety of customary laws across Nigeria’s ethnic groups made it difficult to standardize land ownership rules. The Nigerian government recognized that without reform, this fragmented system would hinder economic growth, real estate development, and equitable access to land.

The Colonial Influence
During the colonial era, the British introduced laws to govern land ownership, but these laws largely coexisted with indigenous customs. For instance, the Public Lands Ordinance of 1903 allowed the government to take control of lands for public purposes, while still acknowledging customary ownership in many cases. This dual system of colonial and customary land laws persisted into post-independence Nigeria, creating a legal patchwork that was ripe for reform.

Urbanization and Industrialization
By the late 1960s and early 1970s, Nigeria was experiencing rapid urbanization and industrial growth, especially in cities like Lagos, Ibadan, and Kano. With this expansion came a surge in land disputes, land speculation, and unequal access to land. Wealthy individuals and corporations were able to amass large tracts of land, while ordinary Nigerians faced barriers to ownership, further entrenching inequality.

It became clear that the government needed to create a unified framework that would standardize land ownership and make land accessible for development, agriculture, and residential purposes. This laid the groundwork for the creation of the Land Use Act.

Objectives of the Land Use Act
The Land Use Act of 1978 was introduced with several key objectives in mind. The Nigerian government intended to centralize land administration, ensure that land was used for the public good, and simplify the complex land tenure systems that had previously existed.

Equitable Distribution of Land One of the primary goals of the act was to ensure that land was available to all Nigerians, regardless of social or economic status. Before the act, wealthy individuals and corporations could easily acquire vast areas of land, making it difficult for small-scale farmers, rural dwellers, and ordinary citizens to access land. By vesting control of all land in the hands of the state governor, the act aimed to provide more equitable access to land resources.

For instance, under the Land Use Act, all land within a state’s boundaries is now held “in trust” by the state governor, who allocates it based on the needs of the people. This move was intended to prevent land hoarding and speculation, making land available for more productive uses, such as agriculture, industry, and housing.

Centralized Land Control The act also sought to eliminate the fragmented land tenure systems by centralizing land control under the state government. By doing so, it was hoped that land transactions would become more transparent, standardized, and easier to regulate.

Instead of individuals obtaining land directly from families or traditional rulers, they would now apply to the state government for rights of occupancy. This centralization aimed to reduce land disputes and ensure that land use aligned with the state’s economic development plans.

Simplification of Land Transactions
Before the Land Use Act, land transactions were often cumbersome, requiring negotiations with multiple parties, including traditional leaders, families, and local officials. The act simplified this process by introducing the Certificate of Occupancy (C of O), a legal document issued by the state governor that grants the holder the right to use a parcel of land for a specified period (usually 99 years). This document serves as proof of legal land ownership or tenancy, making it easier for individuals and businesses to buy, sell, lease, or transfer land.
Regulating Land Use for Economic Development
Another important objective of the act was to ensure that land was used in ways that contributed to the country’s economic growth. The Nigerian government recognized that land is a finite resource, and its improper allocation could stifle industrial and agricultural development. The Land Use Act allows the government to control land allocation in a manner that supports national development goals, such as urban planning, infrastructure development, and environmental conservation.

Also Read:

How to Buy Land in Nigeria: A Legal Guide for First-time Buyers

Understanding the Land Use Act in Nigeria: A Comprehensive Guide

How to Avoid Real Estate Fraud in Nigeria: Legal Guide

How to Verify Property Titles in Nigeria: A Comprehensive Guide

How to Obtain Certificate of Occupancy in Nigeria: A Comprehensive Guide

Conclusion
The Land Use Act plays a pivotal role in shaping land ownership and real estate transactions in Nigeria. While it was introduced to streamline land allocation and eliminate disparities in land access, it has also introduced several challenges, particularly in terms of bureaucracy and the high cost of transactions. Understanding the act’s provisions is essential for property owners and investors to navigate Nigeria’s real estate landscape effectively.

Mortgage Transactions In Nigeria: Borrower And Lender Rights Explained
A mortgage is a conveyance or lien over a property for the security of the repayment of a loan facility, which is discharged after the liquidation of the loan facility. It should be noted that there is always a provision for redemption on repayment of the loan or discharge of such other obligation.[1]

Mortgage transactions in Nigeria are a key means of securing credit, allowing borrowers to access funds while protecting lenders through security over property. The legal framework, guided by laws such as the Land Use Act 1978, the Property and Conveyancing Law, and judicial precedents, defines the rights and obligations of both parties.

Understanding these rights is essential, as disputes often arise around consent, title perfection, and enforcement.

DEFINITION OF BASIC TERMS IN A MORTGAGE TRANSACTION

MORTGAGOR: A party to a mortgage that uses his property as security to collect a loan facility or discharge an obligation. A mortgagor is also known as a “Borrower.”

MORTGAGEE: A party in whose favour a mortgage is created and is entitled to the payment of the money secured to him by the mortgage. Also known as a “Lender.”

DEED OF MORTGAGE: A mortgage deed is a legal document in which the mortgagor transfers an interest in real estate to a mortgagee for the purpose of providing a mortgage loan. It can also be TRIPARTITE, meaning a party bringing in a Surety who usually owns the Mortgage property used as Security.

TYPES OF MORTGAGE

There are two broad types of mortgages, namely, Legal and Equitable

LEGAL MORTGAGE: This mortgage is created pursuant to statutory provisions. It is usually by Deed. There are three operative laws regulating the creation of legal mortgages in Nigeria, namely, The Conveyancing Act, 1881 (for States created from the old Northern and Eastern regions and some parts of Lagos), Property & Conveyancing Law, 1959 (for the States created from the old Western and Midwestern regions,) and the Registration of Titles Law, Cap R4 Laws of Lagos state 2003, (for some parts of Lagos, especially Victoria Island, Ikoyi and Surulere, Lagos Island, Yaba, Bariga, Somolu, Apapa, Oyingbo, Badagry).
The form and contents of the instrument creating such a mortgage are prescribed by law, and non-compliance with the law may be fatal to the entire transaction. Subject to the relevant laws and due execution, a legal mortgage conveys the legal estate of the mortgagor over the property to the mortgagee as security for a loan on the condition that the mortgagor is entitled to redeem the mortgaged property upon fulfillment of his obligations under the mortgage.

EQUITABLE MORTGAGE: An equitable mortgage is a type of mortgage created under the rules of equity. It confers equitable interest on the mortgagee. An equitable mortgage is more suitable for short-term loans. An equitable mortgage is not as secure as a legal mortgage, but in practice, the mortgagee protects his/herself by requesting that the mortgagor at the time of creating the equitable mortgage, sign a legal mortgage and Consent Form.
BORROWER AND LENDER RIGHTS IN A MORTGAGE TRANSACTIONS

a. MORTGAGOR’S RIGHT TO REDEEM: This is the right the Mortgagor has to recover his property upon the fulfillment of his obligations. Thus court of equity will not allow the mortgagee to take any undue advantage of the mortgagor; equity will not give effect to any clause in a mortgage deed that is a clog to the mortgagor’s right to redeem; this principle has been extended to include any clause that delays redemption.[2] Upon creation of a valid mortgage, legal or equitable, a mortgagor possesses three distinct potential rights to redeem the mortgaged property. One of these rights is in law, while the other two are rights in equity. The rights are:

Legal right to redeem; and
Equitable right to redeem;
Equity of redemption.
LEGAL RIGHT TO REDEEM: This is the right specifically reserved for the mortgagor to recover his property as the owner upon discharging his obligations under the mortgage. The mortgagor to be entitled to exercise this right must comply punctiliously with the proviso for redemption at a fixed date; repayment must be made precisely on that date for the mortgagor to be entitled to exercise this very right. However, in practice, the date for redemption is usually short because it is an advantage to the mortgagee to place the mortgagor in default as soon as possible.

EQUITABLE RIGHT TO REDEEM: This is the right that arises after the legal date for redemption has passed. The mortgage agreement will provide a legal date by which the mortgagor should have paid. If he fails to pay on or before the legal due date, his legal right to redeem will be extinguished on that date, and his equitable right kicks off and extinguishes upon foreclosure. Equity will allow redemption on a date later than the contractual date.

EQUITY OF REDEMPTION: Equity of redemption is different from the equitable right to redeem. Equity of redemption is the equitable interest which a mortgagor has in the land as the owner. The mortgagor can redeem his property by paying to the mortgagee the principal money and the interest that has accumulated on the principal money. Where the mortgagor has paid to the mortgagee the amount that is due, the mortgagee shall re-convey the property to the mortgagor.

A Deed of Release is usually prepared, and the particulars of the document of title of the property that is being re-conveyed to the mortgagor shall be stated in the Deed of Release. The deed of release shall be registered in the Land Registry. Equity of redemption arises in favour of the mortgagor as soon as the mortgage is created, and continues until the property is sold or foreclosure occurs. Equity from the onset treats the mortgagor as continuing to be the owner of the property, subject only to the mortgagee’s interest, which is not a right to the mortgaged property but to the mortgage debt.[3]

b. MORTGAGEE’S RIGHT OF REDEMPTION: These are means by which the mortgagee may enforce the security so as to recover the loan facility granted to the Mortgagor. There are basically three such rights:

· Statutory power of sale

· Foreclosure

· Appointment of Receivers[4]

STATUTORY POWER OF SALE: Under sections 19 (1) of the Conveyancing Act and 123 (1) of the Property & Conveyancing Law, every mortgagee (legal or equitable) whose mortgage is created by deed may enforce its/his security after the legal due date by sale of the mortgaged property. Power of sale here is automatic; the mortgagee does not require a court order before he/it can sell. However, for the mortgagee to be entitled to exercise its power of sale, the power must have arisen and become exercisable. For the power of sale to arise, the following three conditions must exist:

i) The mortgage must have been created by a deed.

ii) There must be no contrary intention against sale in the mortgage deed; and

iii) The legal due date, which is the date of redemption of the mortgage, must have passed.

FORECLOSURE: Foreclosure is a judicial process through which the mortgagor’s equity of redemption is terminated, and all the interests in the mortgagor property become vested in the mortgagee, subject to the right of other mortgages that rank in priority above him. An interim order called “a foreclosure nisi” is first decreed, giving the mortgagor six months within which to redeem the mortgaged debt. At the expiry of the six months, the order is made absolute.

In the case of a successive mortgage, all subsequent mortgagees and the mortgagor should be made parties to the action.

APPOINTMENT OF RECEIVERS: A mortgagee can take possession of a mortgaged property without selling it entirely in the event of a borrower’s default by designating a receiver. Property management, income collection, and debt repayment are all handled by the receiver. By doing this, the borrower’s right to redeem the property when the debt is paid off is maintained while the lender recovers money.

Mortgage transactions in Nigeria balance the borrower’s right to redeem property with the lender’s right to enforce repayment. While laws like the Land Use Act 1978, Property and Conveyancing Law 1959, and judicial precedents provide guidance, issues such as delays in title perfection and enforcement hinder efficiency. Strengthening the system through reforms and modernization will better protect both parties and enhance confidence in Nigeria’s credit and property markets.

A life policy (or life insurance) is a legal contract where an insurance company pays a set amount of money to your named beneficiaries when you die, in exchange for regular payments called premiums.
Main Types of Life Policies
Term Life Insurance: Provides coverage for a specific number of years (like 10 or 20) and pays out only if you pass away during that time. It is usually the most affordable option.
Permanent Life Insurance: Covers your entire life and builds up a cash value that you can borrow from or withdraw while you are still alive. Common types include whole life and universal life.
How It Works
Premiums: You pay the insurer money monthly or yearly to keep the policy active.
Beneficiaries: You choose people (like a spouse or children) or organizations to receive the payout.
Payout: When you die, your beneficiaries receive a tax-free lump sum or installments to help cover expenses, debts, or living costs.
If you'd like, let me know:Are you looking for term or permanent coverage?What financial goals are you trying to protect (like family support or debt coverage)?I can help you figure out which policy type fits your needs.

Life insurance (or life assurance, especially in the Commonwealth of Nations) is a contract between an insurance policy holder and an insurer or assurer, where the insurer promises to pay a designated beneficiary a sum of money upon the death of an insured person. Depending on the contract, other events such as physical complications or illness can also trigger payment. The policyholder typically pays a premium, either regularly or as one lump sum. The benefits may include other expenses, such as funeral expenses.

Life policies are legal contracts and the terms of each contract describe the limitations of the insured events. Often, specific exclusions written into the contract limit the liability of the insurer; common examples include claims relating to suicide, fraud, war, riot, and civil commotion. Difficulties may arise where an event is not clearly defined, for example, the insured knowingly incurred a risk by consenting to an experimental medical procedure or by taking medication resulting in injury or death.

Modern life insurance bears some similarity to the asset-management industry,[1][failed verification] and life insurers have diversified their product offerings into retirement products such as annuities.[2]

Life-based contracts tend to fall into two major categories:

Protection policies: designed to provide a benefit, typically a lump-sum payment, in the event of a specified occurrence. A common form of a protection-policy design is term insurance.
Investment policies: the main objective of these policies is to facilitate the growth of capital by regular or single premiums. Common forms (in the United States) are whole life, universal life, and variable life policies.

Stocks (also capital stock, or sometimes interchangeably, shares) consist of all the shares[a] by which ownership of a corporation or company is divided.[1] A single share of the stock means fractional ownership of the corporation in proportion to the total number of shares. This typically entitles the shareholder (stockholder) to that fraction of the company's earnings, proceeds from liquidation of assets (after discharge of all senior claims such as secured and unsecured debt),[3] or voting power, often dividing these up in proportion to the number of like shares each stockholder owns. Not all stock is necessarily equal, as certain classes of stock may be issued, for example, without voting rights, with enhanced voting rights, or with a certain priority to receive profits or liquidation proceeds before or after other classes of shareholders.

Stock can be bought and sold privately or on stock exchanges. Transactions of the former are closely overseen by governments and regulatory bodies to prevent fraud, protect investors, and benefit the larger economy. As new shares are issued by a company, the ownership and rights of existing shareholders are diluted in return for cash to sustain or grow the business. Companies can also buy back stock, which often lets investors recoup the initial investment plus capital gains from subsequent rises in stock price. Stock options issued by many companies as part of employee compensation do not represent ownership, but represent the right to buy ownership at a future time at a specified price. This would represent a windfall to the employees if the option were exercised when the market price is higher than the promised price, since if they immediately sold the stock they would keep the difference (minus taxes).

Stock bought and sold in private markets fall within the private equity realm of finance.

Shares
Main article: share (finance)
A person who owns a percentage of the stock has the ownership of the corporation proportional to their share. The shares form a stock; the stock of a corporation is partitioned into shares, the total of which are stated at the time of business formation. Additional shares may subsequently be authorized by the existing shareholders and issued by the company. In some jurisdictions, each share of stock has a certain declared par value, which is a nominal accounting value used to represent the equity on the balance sheet of the corporation. In other jurisdictions, however, shares of stock may be issued without associated par value.[4]

Shares represent a fraction of ownership in a business. A business may declare different types (or classes) of shares, each having distinctive ownership rules, privileges, or share values. Ownership of shares may be documented by issuance of a stock certificate. A stock certificate is a legal document that specifies the number of shares owned by the shareholder, and other specifics of the shares, such as the par value, if any, or the class of the shares.

In the United Kingdom, Republic of Ireland, South Africa, and Australia, stock can also refer, less commonly, to all kinds of marketable securities.[5]

Types
Stock typically takes the form of shares of either common stock or preferred stock. As a unit of ownership, common stock typically carries voting rights that can be exercised in corporate decisions. Preferred stock differs from common stock in that it typically does not carry voting rights but is legally entitled to receive a certain level of dividend payments before any dividends can be issued to other shareholders.[6][7][page needed] Convertible preferred stock is preferred stock that includes the ability of the holder to convert the preferred shares into a fixed number of common shares, usually any time after a predetermined date. Shares of such stock are called "convertible preferred shares" (or "convertible preference shares" in the UK).

New equity issue may have specific legal clauses attached that differentiate them from previous issues of the issuer. Some shares of common stock may be issued without the typical voting rights, for instance, or some shares may have special rights unique to them and issued only to certain parties. Often, new issues that have not been registered with a securities governing body may be restricted from resale for certain periods of time.[8]

Preferred stock may be hybrid by having the qualities of bonds of fixed returns and common stock voting rights. They also have preference in the payment of dividends over common stock and also have been given preference at the time of liquidation over common stock. They have other features of accumulation in dividend. In addition, preferred stock usually comes with a letter designation at the end of the security; for example, Berkshire-Hathaway Class "B" shares sell under stock ticker BRK.B, whereas Class "A" shares of ORION DHC, Inc will sell under ticker OODHA until the company drops the "A" creating ticker OODH for its "Common" shares only designation. This extra letter does not mean that any exclusive rights exist for the shareholders but it does let investors know that the shares are considered for such, however, these rights or privileges may change based on the decisions made by the underlying company.

Rule 144 stock
"Rule 144 Stock" is an American term given to shares of stock subject to SEC Rule 144: Selling Restricted and Control Securities.[9] Under Rule 144, restricted and controlled securities are acquired in unregistered form. Investors either purchase or take ownership of these securities through private sales (or other means such as via ESOPs or in exchange for seed money) from the issuing company (as in the case with Restricted Securities) or from an affiliate of the issuer (as in the case with Control Securities). Investors wishing to sell these securities are subject to different rules than those selling traditional common or preferred stock. These individuals will only be allowed to liquidate their securities after meeting the specific conditions set forth by SEC Rule 144. Rule 144 allows public re-sale of restricted securities if a number of different conditions are met.

Stock derivatives
Further information: Equity derivative
A stock derivative is any financial instrument for which the underlying asset is the price of an equity. Futures and options are the main types of derivatives on stocks. The underlying security may be a stock index or an individual firm's stock, e.g. single-stock futures.

Stock futures are contracts where the buyer is long, i.e., takes on the obligation to buy on the contract maturity date, and the seller is short, i.e., takes on the obligation to sell. Stock index futures are generally delivered by cash settlement.

A stock option is a class of option. Specifically, a call option is the right (not obligation) to buy stock in the future at a fixed price and a put option is the right (not obligation) to sell stock in the future at a fixed price. Thus, the value of a stock option changes in reaction to the underlying stock of which it is a derivative. The most popular method of valuing stock options is the Black–Scholes model.[10] Apart from call options granted to employees, most stock options are transferable.

Intellectual property (IP) is a category of property that includes intangible creations of the human intellect.[1][2] There are many types of intellectual property, and some countries recognize more than others.[3][4][5] The best-known types are patents, copyrights, trademarks, and trade secrets. The modern concept of intellectual property was developed in England in the 17th and 18th centuries. The term "intellectual property" began to be used in the 19th century, though it was not until the late 20th century that intellectual property became commonplace in most of the world's legal systems.[6]

Supporters of intellectual property laws often describe their main purpose as encouraging the creation of a wide variety of intellectual goods.[7] To achieve this, the law gives people and businesses property rights to certain information and intellectual goods they create, usually for a limited period of time. Supporters argue that because IP laws allow people to protect their original ideas and prevent unauthorized copying, creators derive greater individual economic benefit from the information and intellectual goods they create, and thus have more economic incentives to create them in the first place.[7] Advocates of IP believe that these economic incentives and legal protections stimulate innovation and contribute to technological progress of certain kinds.[8]

The intangible nature of intellectual property presents difficulties when compared with traditional property like land or goods. Unlike traditional property, intellectual property is "indivisible", since an unlimited number of people can in theory "consume" an intellectual good without its being depleted.[9] Additionally, investments in intellectual goods suffer from appropriation problems: Landowners can surround their land with a robust fence and hire armed guards to protect it, but producers of information or literature can usually do little to stop their first buyer from replicating it and selling it at a lower price. Balancing rights so that they are strong enough to encourage the creation of intellectual goods but not so strong that they prevent the goods' wide use is the primary focus of modern intellectual property law.[10]

Industrial property is one of two subsets of intellectual property (the other being copyright), it takes a range of forms, including patents for inventions, industrial designs (aesthetic creations related to the appearance of industrial products), trademarks, service marks, layout-designs of integrated circuits, commercial names and designations, geographical indications and protection against unfair competition.[1][2][3] In some cases, aspects of intellectual creation, although present, are less clearly defined. The object of industrial property consists of signs conveying information, in particular to consumers, regarding products and services offered on the market. Protection is directed against unauthorized use of such signs that could mislead consumers, and against misleading practices in general.[4][5]

In United States legal terminology, industrial property refers to patented goods, trademarks, copyrights, and industrial designs that are owned by a business, and that the business may exclude others from using.[6]

MPL

Oparaugo Michael — Wed, 09 Sep 2026 03:38:19 +0000

C.P.F

Oparaugo Michael — Mon, 07 Sep 2026 13:30:55 +0000

Forever Living was founded in 1978 in Tempe, Arizona by Rex Maughan. In the 1990s, Rex Maughan had purchased the Texas company Aloe Vera of America, with Aloe Vera of America selling its products to Forever Living for distribution. Some journalists have likened the Forever Living Products distribution system.
click 1 to view flp online store >[http://github.com/mikekm02/flpmo]Click to view image >
[https://iili.io/CpEvbhx.jpg]
[https://iili.io/CpEvbhx.jpg/body spray][Forever Living was founded in 1978 in Tempe, Arizona by Rex Maughan. In the 1990s, Rex Maughan had purchased the Texas company Aloe Vera of America, with Aloe Vera of America selling its products to Forever Living for distribution. Some journalists have likened the Forever Living Products distribution system.
click 1 to view flp online store >[http://github.com/mikekm02/flpmo]Click to view image >
[https://iili.io/CpEvbhx.jpg]
[https://iili.io/CpEvbhx.jpg/body spray][https://dev.to/oparaugo_michael_f02c4c0d/mkflp-42ok/flp]
user:Ppflp

Examples 1.

A ⊃ B

B ⊃ [A ⊃ (CvD)]

C ≡ D

~(C.D) /∴~A

Solution:

1.A ⊃ B

2.B ⊃ [A ⊃ (CvD)]

3.C ≡ D

4.~(C.D)/∴~A

5.(C.D)v(~C.~D) 3,Equiv.

6.(~C.~D) 5,4, D.S.

7.~(CvD) 6, DeM.

8.A ⊃ [A ⊃ (CvD) 8,Exp.

9.(A.A) ⊃ (CvD) 8,Exp.

10.A ⊃ (CvD) 9, Taut.

11.~A 10,7, M.T.
br>
Example 2.

(DvE) ⊃ (F.G)

~F

/∴~D

Proof

1.(DvE) ⊃ (F.G)

2.~F /: ~D

3.~Fv~G 2,Add.

4.~(F.G)3,De M.

5.~(DvE)1,4,M.T.

6.~D.~E 5,De M.

7.~D. 6,Simp.

Example3.

M ⊃ (N.O)

(NvO) ⊃ P

/∴M ⊃ P

Solution.

1.M ⊃ (N.O)

2.(NvO) ⊃ P /: M ⊃ P

3.~Mv(N.O) 1,Impl.

4.(~MvN).(~MvO)3,Dist.

5.~MvN 4,Simp.

6.~(NvO)vP 2,Impl.

7.(~N.~O)vP 6,De M.

8.Pv(~N.~O) 7,Com.

9.(Pv~N).(Pv~O) 8,Dist.

10. Pv~N 9,Simp.

11. ~NvP 10,Com.

12. N ⊃ P 11,Impl.

13. M ⊃ N 5,Impl.

14. M ⊃ P 13,12, H.S.

Example 4.

Iv(J.~k)

(IvJ) ⊃ (Lv~k)

/∴K ⊃ L

Proof.

1.Iv(J.~k)

2.(IvJ) ⊃ (Lv~k) /∴K ⊃ L

3.(IvJ).(Iv~k) 1,Dist.

4. IvJ. 3,Simp.

5. Lv~k. 2,4,M.P.

6. ~kvL. 5,Com.

7. k ⊃ L. 6,Impl.

Example 5.

If the legislators are wealthy, then poverty was not the reason for the bribes they collected. But either poverty or greed was the reason for the bribes they collected. The legislators are wealthy. Hence greed must have been the reason for the bribes they collected. (L,P,G).

Proof.

1. L ⊃ ~P

2. PvG

3. L /: G.

4. ~P 1,3, M.P.

5. G 2,4, D.S.

Example 6.
If it is not the case that the National Electric Power Authority is efficient and electricity consumers pay their bills promptly, frequent power cuts will not be eliminated. If prompt payment of electricity bills by consumers implies that frequent power cuts will be eliminated, then our electronic gadgets could still be damaged by voltage fluctuations. The National Electric Power Authority is not efficient. Therefore our electronic gadgets could still be damaged by voltage fluctuations (N,B,F,E)

Proof.

1. ~(N.B) ⊃ ~F

2. (B ⊃ F) ⊃ E

3. ~N /:E

4. ~Nv~B 3, Add.

5. ~(N.B) 4, De M.

6. ~F 1,5, M.P.

7. ~FvB 6, Add.

8. F ⊃ B 7,Impl.

9. E 2,8, M.P.

Example.

If the Super Eagles is good and the players are complaining, then the Nigerian Football Association must be making some mistakes. If the complaint of the players implies that the Nigerian Football Association is making some mistakes, then Nigeria will lose some mmatches in the world cup tournament. The Super Eagles is good. Therefore Nigeria will lose some matches in the world cup tournament. (E,P,N,L.).

Solution.

1. (E.P) ⊃ N

2. (P ⊃ N) ⊃ L

3. E. /:L

4. (E.P) ⊃ N 1,Exp.

5. P ⊃ N. 4,3,M.P.

6. L 2,5,M.P.

Example.

If the crisis in the Niger Delta continues, the oil installations in that area will not be safe again, The oil companies will continue to lift petroleum products only if the oil installations in that area are safe again. Business in Warri will reduce drastically unless oil companies continue to lift petroleum products. But if the crisis will not be happy and if those who benefit from the crisis are not happy then oil companies will not continue to lift petroleum products. The crisis in the crisis Niger Delta must either continue or does not continue. Therefore business in Warri will reduce drastically.

Solution.

1.C ⊃ ~O

2.L ⊃ O

3.WvL

4.(~C ⊃ ~H).(~H ⊃ ~L)

5.Cv~C /:W

6.(~H ⊃ ~L).(~C ⊃ ~H) 4,Com.

7.~C ⊃ ~H 4,Simp.

8.~H ⊃ ~L. 6,Simp.

9.~C ⊃ ~L 7,8,H.S.

10.~O ⊃ ~L 2,Trans.

11.C ⊃ ~L 1,10,H.S.

12.(C ⊃ ~L).(~C ⊃ ~L) 11,9,Conj.

13.~Lv~L. 12,5,C.D.

14.~L. 13,Taut.

15.LvW 3.Com.

16.W 15,14,D.S.

If Abacha or Babangida had allowed the conclusion of the transition programme, Abiola would have become president and democratic government would have been installed in 1993. Had democratic government been installed in 1993, the dictatorial rule of Abacha would have been avoided. Of course, the dictatorial rule of Abacha was not avoided. Therefore Abacha did not allow the conclusion of the transition programme. (A,B,P,D,R).

Solution.

1.(AvB) ⊃ (P.D)

2.D ⊃ R

3.~R. /:~A

4.~D 2,3,M.T.

5.~Dv~P 4.Add.

6.~Pv~D 5,Com.

7.~(P.D) 6,De M.

8.~(AvB) 1,7,M.T.

9.~A.~B 8,De M.

10.~A 9,Simp.

Example.

If you are a student of philosophy, then you have a lot of reading to do and if you are a mother then you have a lot of responsibilities at home. Thus, if you are both a student of philosophy and a mother, then you have a lot of reading to do and a lot of responsibilities at home. (S,R,M,H)

Solution.

1.(S ⊃ R).(M ⊃ H) /: (S.M) ⊃ (R.H)

2.S ⊃ R 1,Simp.

3.~SvR 2,Impl.

4.(~SvR)v~M 3,Add.

5.~Sv(Rv~M) 4,Assoc.

6.~Sv(~MvR) 5,Com.

7.(~Sv~M)vR 6,Assoc.

8.~(S.M)vR 7,De M.

9.(M ⊃ H).(S ⊃ R) 1,Com.

10.M ⊃ H 9,Simp.

11.~MvH 10,Impl.

12.(~MvH)v~S 11,Add.

13.~Mv(Hv~S) 12,Assoc.

14.(Hv~S)v~M 13,Com.

15.Hv(~Sv~M) 14,Assoc.

16.(~Sv~M)vH 15,Com.

17.~(S.M)vH 16,De M.

18.{[~(S.M)vR].[~(S.M)vH]} 8,17,Conj.

19.~(S.M)v(R.H) 18, Dist.

20.(S.M) ⊃ (R.H) 19,Impl.

Conditional Proof, Indirect Proof and Quantification.

The rule of Conditional Proof, it must be pointed out at the outset, allows us to construct shorter proofs of validity for arguments which could be established as valid by the application of the relevant nineteen rules considered earlier on. It also makes it possible for one to prove some arguments valid whose validity cannot be demonstrated by suing the original nineteen rules of inference.

The fundamental concept that underpins the rule of Conditional Proof is the idea that every deductive argument has a corresponding conditional statement whose antecedent is the conjunction of the argument's premises and whose consequent is the conclusion of that argument. Now, the rule of Conditional Proof is applicable to arguments whose conclusions are conditional statements. To construct such a proof for an argument, we assume the antecedent of its conclusion as an additional premiss and then infer the consequent of the same conclusion by applying the relevant rules of inference.

1.A ⊃ (B.C)

2.(BvC) ⊃ D /:A ⊃ D

3.A /: D(C.P)

4.B.C. 1,3 M.P.

5.B 4,Simp.

6.BvC 5,Add.

7.D 2,6. M.P.

Notice that line 3 of the proof is the antecedent of the conclusion A⊃D. Line 4 typifies the way in which the method of C.P. is applied in a proof. Like other rules of inference, the rule of Conditional Proof can be used up to two three times in the course of the same proof, depending, of course, on the the nature of the conclusion of the given argument.

Consider the following argument.

Here, the rule of Conditional Proof was used twice to arrive at S. Conventionally, each successive application of the principle is to be denoted by a diagonal separating the premises from the new conclusion, followed by the therefore sign (&#8756). Finally the acronym C.P. must be written to the right of the conclusion. The final demonstration can be penned down:

1.(P.Q) ⊃ R

2.(Q.R) ⊃ S /:P ⊃ (Q ⊃ S)

3.P /: Q ⊃ S(C.P.)

4.Q /:S(C.P.)

5.P.Q. 3,4, Conj.

6.R 1,5,M.P.

7.Q.R. 4,6,Conj.

8.S 2,7, M.P.

We can now restate precisely the application of the rule of Conditional Proof. The rule is applied to arguments whose conclusions are conditional statements. To prove such an argument valid, assume the antecedent of its conclusion as an additional premiss and then deduce the consequent of its conclusion through a succession of elementary valid arguments.

Some examples would facilitate our understanding of the rule of Conditional Proof.

Example1

P ⊃ (C ⊃ N)

(N.R) ⊃ E

(R ⊃ E) ⊃ T /:P ⊃ (C ⊃T)

Solution:

1.P ⊃ (C ⊃ N)

2.(N.R) ⊃ E

3.(R ⊃ E) ⊃ T /: P ⊃ (C ⊃ T)

4.P /: C ⊃ T (C.P.)

5.C /:T (C.P.)

6.(P.C.) ⊃ N 1, Exp.

7.P.C. 4,5, Conj.

8.N 6,7, M.P.

9.N ⊃ (R ⊃ E) 2, Exp.

10.R ⊃ E 9,8, M.P.

11.T 3,10, M.P.

Example 2.

A ⊃ (BvC)

B ⊃ C

/:A ⊃ C

Solution:

1.A ⊃ (BvC)

2.B ⊃ C /:A ⊃ C

3.A /:C(C.P.)

4.BvC 1,3,M.P.

5.~B ⊃ C 4.Impl

6.~C ⊃ ~B 2, Trans

7.~C ⊃ C 6,5,H.S.

8.~~CvC 7,Impl

9.CvC 8,D.N.

10.C 9,Taut.

Example 3

F ⊃ W

/:(F.S) ⊃ (WvX)

Solution:

1.F ⊃ W /:(F.S) ⊃ (WvX)

2.F.S. /: WvX(C.P.)

3.F 2,Simp.

4.W 1,3,M.P.

5.WvX 4,Add

Example 4

(I ⊃ J).(IvK)

(K ⊃ L).(KvI)

/: ~J ⊃ L

Solution:

1.(I ⊃ J).(IvK)

2.(K ⊃ L).(KvI) /: ~J ⊃ L

3.~J /: L(C.P)

4.I ⊃ J 1,Simp.

5.~I 4,3,M.T.

6.(KvI).(K ⊃ L) 2, Com.

7.KvI 6,Simp.

8.IvK 7.Com.

9.K 8,5,D.S.

10.K ⊃ L 2,Simp.

11.L 10,9,M.P.

Example 5

(D ⊃ E).(F ⊃ H) /:(DvF) ⊃ (HvE)

Solution:

1.(D ⊃ E).(F ⊃ H) /:(DvF) ⊃ (HvE)

2.DvF /:HvE (C.P.)

3.EvH 1,2,C.D.

4.HvE 3,Com

Indirect Proof

The rule of Conditional Proof aside we turn our attention now to Indirect Proof method. An Indirect Proof of validity for a given argument is constructed by assuming the negation of its conclusion as an additional premiss and then deriving an explicit contradiction from the increased set of premisses. One may eventually go beyond the contradiction itself to deduce the conclusion of the original argument. The whole process can be made more explicit with the help of an example:

Example 1

Mv(N.O)

M ⊃ O /: O

3.~O I.P.

4.~M 2,3,M.T.

N.O. 1,4,DS

OvN 3,Add

NvO 6,Com

(N.O) 7,De M

9.(N.O).(N.O) 5,8,Conj.

O.N 5.Com

O 10, Simp.

Example 2

D

/: Ev(E ⊃ F)

Solution

D /: Ev(E ⊃ F)

~[Ev(E ⊃ F)] I.P.

~[Ev(~EvF)] 2.Impl.

[(EvE)vF] 3,Assoc.

(EvE).F ⊃ 4. De M

6.(Ev~E) 5,Simp

~E.~~E 6,De M

~E.E 7,D.N.

E.~E 8,Com

E 9, Simp.

Ev(E ⊃ F) 10,Add

In every formal proof that demands the rule of Indirect Proof, the logical structure of the Inference is from p/:q to p. q/:. In otherwords, if p symbolizes the premisses of such an argument and q its conclusion, an Indirect Proof of validity for the argument.

(1) p

/:q.

Can be accomplished through the formal proof of validity of

(2)p

q

/:q

This connection is possible if one calls to mind What we said in the preceding section about the rule of Conditions Proof. There we stated that a formal proof of validity for the argument A.Q./:R constitutes a Conditional Proof of validity for another argument A.IR. Similarly, a formal proof of validity for (2) constitutes a Conditional Proof of validity for a third argumenty.

(3) p

/:q ⊃ q

The conclusion of argument (3) is precisely the same thing as the conclusion of argument (1). Three logical steps can establish this fact. First, by the rule of Implication. q⊃q is Logical equivalent to ~~qvq which, second, is logically equivalent to qvq by the principle of Double Negation. Third, qvq is exactly the same as q by the principle of tuatology. The reader can easily verify that (1) and (3) have identical premisses and logically equivalent conclusions, which means that any proof of validity for (1) is a proof of validity for (3), and vice-versa. A connection between (1) and (3) is made possible by (2) because a proof of validity for (2) is simultaneously a Conditional Proof of validity for (3) and an Indirect Proof of (1). And since we have shown that (1) and (3) are logically equivalent and that the proof of validity or (2) is a Conditional Proof for (3) it follows also that there is an intimate connection between (1) and (2).

We shall deal with more problems to further illustrate the rule of Indirect Proof.

Example 3

W⊃(X.Y)

(XvZ)⊃Q

ZvW /∴Q

Solution

1.W⊃(X.Y)

2.(XvZ)⊃Q

3.ZvW /∴Q

4.~Q I.P.
5.~(XvZ) 2,M.T.
6.~X.~Z 5,D.M.
7.~Z.~X 6,Com
8.~Z 7,Simp
9.W 3,8,D.S.

10.X.Y 1,9,M.P.

11.X 10,Simp

12.~X 6,Simp
13.X.~X 11,12,Conj

Example 4

(A⊃B).(C⊃D)
(BvD)⊃E
E&nbsp/&#8756(AvC)
4.&nbsp~~(AvC)&nbsp&nbsp&nbspI.P
5.&nbspAvC&nbsp&nbsp&nbsp4,D.N.
BvD&nbsp&nbsp&nbsp1,5,C.D.
7.&nbspE&nbsp&nbsp&nbsp2,6,M.P.
8.&nbspE.~E&nbsp&nbsp&nbsp7,3,Conj
Example 5

1.&nbsp(WvX)⊃(~Z⊃Y)
2.&nbsp(ZvU)⊃(W.Y)/&#8756Z
3.&nbspZ&nbsp&nbsp&nbspI.P
4.&nbspZvU&nbsp&nbsp&nbsp3,Add
5.&nbspW.Y&nbsp&nbsp&nbsp2,4,M.P.
6.&nbsp(WvX)⊃(Y⊃Z)&nbsp&nbspTrans
7.&nbspW&nbsp&nbsp&nbsp&nbsp5,Simp
8.&nbspWvX&nbsp&nbsp&nbsp7,Add.
9.&nbspY⊃Z&nbsp&nbsp&nbsp6,8,MP.
10.&nbspY.W&nbsp&nbsp&nbsp5,Com.
11.&nbspY&nbsp&nbsp&nbsp10,Simp.
12.&nbspZ&nbsp&nbsp&nbsp9,11,M.P.
13.&nbspZ.~Z&nbsp&nbsp&nbsp12,3,Conj.

It is legitimate, when dealing with Indirect Proof, that the demonstration should end in the line which contains an explicit contradiction. For in proving that the premisses of an argument together with the contradictory of its conclusion lead to an inconsistent proposition, we have demonstrated indirectly that the argument in question is valid. At any rate, we can interpret the Indirect Proof of the validity of a particular argument as the process of deducing the argument's conclusion from the inconsistent or contradictory proposition itself. This procedure is justified by the fact that from a contradictory proposition any proposition whatsoever can be deduced from It. From the proposition:

P. ~p

We can infer Q or whatever proposition we choose.
The rule of Contraditional Proof and Indirect Proof are closely related methods of proof. But while the former is based on the principle that every valid argument has a corresponding tautologous conditional, the latter is anchored on the idea that if the negation of the conclusion to be proved leads to a contradiction then from that contradiction the conclusion itself can be inferred.

Quantification

There are some types of argument whose validity cannot be demonstrated by the principles we have examined so far. These arguments contain noncompound prepositions and require different methods for symbolizing and testing them. An obviously valid argument such as "A goat is an animal; therefore a goat's head is the head of an animal", cannot be proved valid by the nineteen rules of Inference and by the methods of conditional and Indirect Proofs.
Propositions such as "Enwerem is human", "kanu is tall" and "Buhari is audacious" are called singular Propositions. In each of these statements, a predicate or attribute is ascribed to a subject. The subject term of singular Propositions could be the name of a person, place, idea or thing; It could, that is, be a noun or noun phrase. A predicate term could be a noun as in "Awojobi is a mortal", or an adjective as in "Achebe is creative". It could even be a verb as in the proposition "Mbakwe weeps".
Just as an individual can have many attributes, an attribute can be predicated of many individuals. Below is a short list of different attributes of an individual and different attributes of an individual and different individuals with the same predicate:

An&nbsp&nbsp&nbsp&nbspB&nbsp&nbsp&nbsp&nbsp

An individual with different predicates & nbsp&nbsp&nbspAttribute Predicated of several individuals

A look at A and B reveals that some propositions in each are true, while some are false. In A, the first, second and fifth proposition s are true whereas the remainder are false. The first, fourth and fifth propositions in B are false but the second and third propositions are true.
In quantificational logic, small letters from a down to w are used to represent individuals. These are called individual constants. For instance, the letter "a" can be used to represent the individual "Achebe" in any argument in which that name occurs, Of course, It is customary to represent an individual with the first letter of his or her (or its) name. Thus Beko, Chukwu, Douglas etc can be denoted by b, c, d respectively throughout the context which they occur. To differentiate individual constants from attribute symbols, logicians designate the latter with capital letters. For Example, the first letters of the predicates beautiful, charming, decent, and educated, that is, B,C,D and E represent these attributes.
A singular proposition such as "Russell was brilliant" is symbolized as Br: the attribute symbol is written immediately to the left of the individual constant. In the table above, under B, we represented all those propositions as Bt, Bb, Bi, Bs and Ba respectively. The changing individual constants can be replaced by the small letter x, which is an individual variable, some logicians prefer to call It free variable. The common pattern that emerges in all this is symbolized as Bx. Bx is a propositional function representing the common structure of the singular propositions that have the same predicate, Beautiful, attributed to several individuals. A propositional function, then, is a symbol that has an individual variable and becomes a proposition the moment an individual or free variable is replaced by an individual constant. It follows that Bt, Bb, Bu etc are propositions which are derivable from the propositional function Bx. It is obvious that the number of propositions that can be derived from a propositional function is indefinite since the number of things to which a particular attribute can be meaningfully predicated cannot be determined in advance.
Moreover, any replacement of a free variable with an individual constant results in a proposition which is a substitution instance of that very propositional function. Of course, a propositional function may have true substitution instances and false ones as well. Under B, for example, Bb and Bi are true substitution instances of the propositional function Bx whilst Bt. Bs and Ba are false substitution instances of the same propositional function. All the propositional functions we have considered upto now are given the technical name simple predicates to demarcate them from the more complicated predicates of quantificational logic. We then say that a simple predicate occurs as a singular propositional logic. We then say that a simple predicate occurs as a singular propositional function with true and false substitution instances.
Translating Propositions into the Symbols of Quantification Theory

Apart from singular propositions, there are also quantified or generalized propositions. Quantified propositions contain predicate terms which are not asserted of any definite individual. The propositions "Everything is transient" and "Something is attractive" are propositions of this kind.

In interpreting quantified propositions a stepwise approach is deemed appropriate by logicians. To begin with, "Everything is transient" can be rendered into the logically equivalent proposition "All things are transient", or into "Given any individual thing whatever, It is transient."

With the notation for individual variable, x, we can translate the last statement into

Given any x, x is transient

And using the technique introduced earlier for symbolizing propositional functions "Given any x" is customarily symbolized as the universal quantifier "(x)". Thus, the original proposition with which we started is completely symbolized as:

(x) Tx

The other general proposition in our example, that is, "something is attractive", can be symbolized with the help of the existential quantifier "(∃x)". "There is area least one x such that". The expression (∃x) is the symbolic representation of the phrase "There is at least one x such that." Now, the statement under consideration asserts that there is at least one thing which is attractive. It does not name the object, but merely ascribes an attribute to It. We symbolize the statement thus:

(∃x) Ax.

It is obvious that a proposition such as "Everything is transient" is true if and only if each and every individual thing whatsoever is transient: a single counter-instance (a permanent object, for instance) is enough to falsify It. This means that a universally quantified propositional function is true on condition that all its substitution instances are true. Furthermore, an existentially quantified propositional function is true if It has at least one true substitution instance. It takes just one attractive entity (an attractive lady, for instance) to prove the statement that "Something is attractive".

Negative proposition can be symbolized also by applying the basic principles which we employed is symbolizing affirmative Propositions. For instance.

(1) Nothing is permanent

can be stated as

(2) Given any individual thing whatsoever, it is not permanent.

Using the capital letter P to designate "permanent", proposition (2) becomes:

(3) (x) ~Px

Again, the assertion "Something is not permanent" means that

There is at least one thing that is not permanent

There is at least one thing that is not permanent.

It also can be rewritten as

There is at least one x such that x is not permanent

or as

There is at least one x such that ~Px

Symbolically, the proposition "something is not permanent" can now be written down completely.

(∃x)~Px

A graphic presentation of the four general Propositions and their logical equivalences, using the Greek letter phi (written as) to represent any predicate whatsoever, is set forth below:

The kinds of general or quantified Propositions we have considered up to this point are really not the only types that fall within the: orbit of Quantification. We can also translate the traditional A, E, I and O Propositions using some of the Ideas that have been highlighted here. Consider, as an illustration, the A proposition "All humans are fallible." It can be restated as Given any x, if x is human then x is fallible We can also rewrite It as follows: Given any x, x is human body x is fallible. Finally, the A proposition with which we began can be completely symbolized as (x) (Hx Fx) The contradictory of the A proposition is the O proposition. "Some humans are not fallible." It is logically equivalent to the following: There is at least one thing that is human and not fallible There is at least one x such that x is human. ~x is fallible and also to the formula (x) (Hx. ~Fx) A typical E proposition such as "No humans are fallible," can be stated successfully as Given any individual thing whatsoever, if It is human then It is not fallible Given any x, x is human x is not fallible and finally as (x) (Hx ~Fx) We know already that an E proposition is contradicted by the I proposition. So "No humans are fallible" is denied by "Some humans are fallible." Translating the latter into our symbolic notation we have succesively. There is at least one thing that is human and fallible There is at least one x such that x is human. x is fallible and the process ends with (x) (Hx. Fx) Logicians utilize the Greek letters phi (O) and psi () to represent whatever predicates that may occur in the traditional subject - predicate Propositions. With phi replacing the predicate that occurs before the logical operator and psi the predicate that occurs after the operator, the four traditional standard - form categorical Propositions can be presented symbolically as follows: A...... (x) (Ox) x) O...... (x) (x. ~x) E...... (x) (x)~x) I....... (x) (x. x) Argument Containing Quantified Propositions Arguments that have Arguments that have quantified Propositions and propositional functions either as premisses or conclusion or both can be tested for validity by having formal proofs constructed for them. In order to do this, four additional rules of Inference are required, bringing the number of rules of Inference to twenty - five. In this section, we shall introduce these additional rules one by one, and examplify how they are applied in the appropriate syllogistic arguments The first of the rules to be discussed is the principle of Universal Instantiation, abbreviated as UI. The principle asserts that any substitution instance of a propositional function can be validly inferred from its Universal quantification. It is evident that the principle of UI follows from What we, said earlier about the condition to be satisfied before a Universally quantified proposition can be accepted as true. The underlying logical principle here is the idea that a Universally quantified propositional function is true if and only if all its substitution instances are true. Therefore from (x) (x) we can deduce v (where v stands for any individual symbol) Through the principle of University Instantiation. For instance, the argument: All mathematicians are intelligent Chike Obi is a mathematician Therefore Chike Obi is intelligent can be proved valid by first symbolizing the premisses and conclusion, followed by the application of UI to infer "If Chike Obi is a mathematician then he is intelligent". The conclusion follows automatically from the rule of Modus Ponens: Our proof for the argument proceeds as follows:
https://punchng.com

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Examples 1.

A ⊃ B

B ⊃ [A ⊃ (CvD)]

C ≡ D

~(C.D) /∴~A

Solution:

1.A ⊃ B

2.B ⊃ [A ⊃ (CvD)]

3.C ≡ D

4.~(C.D)/∴~A

5.(C.D)v(~C.~D) 3,Equiv.

6.(~C.~D) 5,4, D.S.

7.~(CvD) 6, DeM.

8.A ⊃ [A ⊃ (CvD) 8,Exp.

9.(A.A) ⊃ (CvD) 8,Exp.

10.A ⊃ (CvD) 9, Taut.

11.~A 10,7, M.T.
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Example 2.

(DvE) ⊃ (F.G)

~F

/∴~D

Proof

1.(DvE) ⊃ (F.G)

2.~F /: ~D

3.~Fv~G 2,Add.

4.~(F.G)3,De M.

5.~(DvE)1,4,M.T.

6.~D.~E 5,De M.

7.~D. 6,Simp.

Example3.

M ⊃ (N.O)

(NvO) ⊃ P

/∴M ⊃ P

Solution.

1.M ⊃ (N.O)

2.(NvO) ⊃ P /: M ⊃ P

3.~Mv(N.O) 1,Impl.

4.(~MvN).(~MvO)3,Dist.

5.~MvN 4,Simp.

6.~(NvO)vP 2,Impl.

7.(~N.~O)vP 6,De M.

8.Pv(~N.~O) 7,Com.

9.(Pv~N).(Pv~O) 8,Dist.

10. Pv~N 9,Simp.

11. ~NvP 10,Com.

12. N ⊃ P 11,Impl.

13. M ⊃ N 5,Impl.

14. M ⊃ P 13,12, H.S.

Example 4.

Iv(J.~k)

(IvJ) ⊃ (Lv~k)

/∴K ⊃ L

Proof.

1.Iv(J.~k)

2.(IvJ) ⊃ (Lv~k) /∴K ⊃ L

3.(IvJ).(Iv~k) 1,Dist.

4. IvJ. 3,Simp.

5. Lv~k. 2,4,M.P.

6. ~kvL. 5,Com.

7. k ⊃ L. 6,Impl.

Example 5.

If the legislators are wealthy, then poverty was not the reason for the bribes they collected. But either poverty or greed was the reason for the bribes they collected. The legislators are wealthy. Hence greed must have been the reason for the bribes they collected. (L,P,G).

Proof.

1. L ⊃ ~P

2. PvG

3. L /: G.

4. ~P 1,3, M.P.

5. G 2,4, D.S.

Example 6.
If it is not the case that the National Electric Power Authority is efficient and electricity consumers pay their bills promptly, frequent power cuts will not be eliminated. If prompt payment of electricity bills by consumers implies that frequent power cuts will be eliminated, then our electronic gadgets could still be damaged by voltage fluctuations. The National Electric Power Authority is not efficient. Therefore our electronic gadgets could still be damaged by voltage fluctuations (N,B,F,E)

Proof.

1. ~(N.B) ⊃ ~F

2. (B ⊃ F) ⊃ E

3. ~N /:E

4. ~Nv~B 3, Add.

5. ~(N.B) 4, De M.

6. ~F 1,5, M.P.

7. ~FvB 6, Add.

8. F ⊃ B 7,Impl.

9. E 2,8, M.P.

Example.

If the Super Eagles is good and the players are complaining, then the Nigerian Football Association must be making some mistakes. If the complaint of the players implies that the Nigerian Football Association is making some mistakes, then Nigeria will lose some mmatches in the world cup tournament. The Super Eagles is good. Therefore Nigeria will lose some matches in the world cup tournament. (E,P,N,L.).

Solution.

1. (E.P) ⊃ N

2. (P ⊃ N) ⊃ L

3. E. /:L

4. (E.P) ⊃ N 1,Exp.

5. P ⊃ N. 4,3,M.P.

6. L 2,5,M.P.

Example.

If the crisis in the Niger Delta continues, the oil installations in that area will not be safe again, The oil companies will continue to lift petroleum products only if the oil installations in that area are safe again. Business in Warri will reduce drastically unless oil companies continue to lift petroleum products. But if the crisis will not be happy and if those who benefit from the crisis are not happy then oil companies will not continue to lift petroleum products. The crisis in the crisis Niger Delta must either continue or does not continue. Therefore business in Warri will reduce drastically.

Solution.

1.C ⊃ ~O

2.L ⊃ O

3.WvL

4.(~C ⊃ ~H).(~H ⊃ ~L)

5.Cv~C /:W

6.(~H ⊃ ~L).(~C ⊃ ~H) 4,Com.

7.~C ⊃ ~H 4,Simp.

8.~H ⊃ ~L. 6,Simp.

9.~C ⊃ ~L 7,8,H.S.

10.~O ⊃ ~L 2,Trans.

11.C ⊃ ~L 1,10,H.S.

12.(C ⊃ ~L).(~C ⊃ ~L) 11,9,Conj.

13.~Lv~L. 12,5,C.D.

14.~L. 13,Taut.

15.LvW 3.Com.

16.W 15,14,D.S.

If Abacha or Babangida had allowed the conclusion of the transition programme, Abiola would have become president and democratic government would have been installed in 1993. Had democratic government been installed in 1993, the dictatorial rule of Abacha would have been avoided. Of course, the dictatorial rule of Abacha was not avoided. Therefore Abacha did not allow the conclusion of the transition programme. (A,B,P,D,R).

Solution.

1.(AvB) ⊃ (P.D)

2.D ⊃ R

3.~R. /:~A

4.~D 2,3,M.T.

5.~Dv~P 4.Add.

6.~Pv~D 5,Com.

7.~(P.D) 6,De M.

8.~(AvB) 1,7,M.T.

9.~A.~B 8,De M.

10.~A 9,Simp.

Example.

If you are a student of philosophy, then you have a lot of reading to do and if you are a mother then you have a lot of responsibilities at home. Thus, if you are both a student of philosophy and a mother, then you have a lot of reading to do and a lot of responsibilities at home. (S,R,M,H)

Solution.

1.(S ⊃ R).(M ⊃ H) /: (S.M) ⊃ (R.H)

2.S ⊃ R 1,Simp.

3.~SvR 2,Impl.

4.(~SvR)v~M 3,Add.

5.~Sv(Rv~M) 4,Assoc.

6.~Sv(~MvR) 5,Com.

7.(~Sv~M)vR 6,Assoc.

8.~(S.M)vR 7,De M.

9.(M ⊃ H).(S ⊃ R) 1,Com.

10.M ⊃ H 9,Simp.

11.~MvH 10,Impl.

12.(~MvH)v~S 11,Add.

13.~Mv(Hv~S) 12,Assoc.

14.(Hv~S)v~M 13,Com.

15.Hv(~Sv~M) 14,Assoc.

16.(~Sv~M)vH 15,Com.

17.~(S.M)vH 16,De M.

18.{[~(S.M)vR].[~(S.M)vH]} 8,17,Conj.

19.~(S.M)v(R.H) 18, Dist.

20.(S.M) ⊃ (R.H) 19,Impl.

Conditional Proof, Indirect Proof and Quantification.

The rule of Conditional Proof, it must be pointed out at the outset, allows us to construct shorter proofs of validity for arguments which could be established as valid by the application of the relevant nineteen rules considered earlier on. It also makes it possible for one to prove some arguments valid whose validity cannot be demonstrated by suing the original nineteen rules of inference.

The fundamental concept that underpins the rule of Conditional Proof is the idea that every deductive argument has a corresponding conditional statement whose antecedent is the conjunction of the argument's premises and whose consequent is the conclusion of that argument. Now, the rule of Conditional Proof is applicable to arguments whose conclusions are conditional statements. To construct such a proof for an argument, we assume the antecedent of its conclusion as an additional premiss and then infer the consequent of the same conclusion by applying the relevant rules of inference.

1.A ⊃ (B.C)

2.(BvC) ⊃ D /:A ⊃ D

3.A /: D(C.P)

4.B.C. 1,3 M.P.

5.B 4,Simp.

6.BvC 5,Add.

7.D 2,6. M.P.

Notice that line 3 of the proof is the antecedent of the conclusion A⊃D. Line 4 typifies the way in which the method of C.P. is applied in a proof. Like other rules of inference, the rule of Conditional Proof can be used up to two three times in the course of the same proof, depending, of course, on the the nature of the conclusion of the given argument.

Consider the following argument.

Here, the rule of Conditional Proof was used twice to arrive at S. Conventionally, each successive application of the principle is to be denoted by a diagonal separating the premises from the new conclusion, followed by the therefore sign (&#8756). Finally the acronym C.P. must be written to the right of the conclusion. The final demonstration can be penned down:

1.(P.Q) ⊃ R

2.(Q.R) ⊃ S /:P ⊃ (Q ⊃ S)

3.P /: Q ⊃ S(C.P.)

4.Q /:S(C.P.)

5.P.Q. 3,4, Conj.

6.R 1,5,M.P.

7.Q.R. 4,6,Conj.

8.S 2,7, M.P.

We can now restate precisely the application of the rule of Conditional Proof. The rule is applied to arguments whose conclusions are conditional statements. To prove such an argument valid, assume the antecedent of its conclusion as an additional premiss and then deduce the consequent of its conclusion through a succession of elementary valid arguments.

Some examples would facilitate our understanding of the rule of Conditional Proof.

Example1

P ⊃ (C ⊃ N)

(N.R) ⊃ E

(R ⊃ E) ⊃ T /:P ⊃ (C ⊃T)

Solution:

1.P ⊃ (C ⊃ N)

2.(N.R) ⊃ E

3.(R ⊃ E) ⊃ T /: P ⊃ (C ⊃ T)

4.P /: C ⊃ T (C.P.)

5.C /:T (C.P.)

6.(P.C.) ⊃ N 1, Exp.

7.P.C. 4,5, Conj.

8.N 6,7, M.P.

9.N ⊃ (R ⊃ E) 2, Exp.

10.R ⊃ E 9,8, M.P.

11.T 3,10, M.P.

Example 2.

A ⊃ (BvC)

B ⊃ C

/:A ⊃ C

Solution:

1.A ⊃ (BvC)

2.B ⊃ C /:A ⊃ C

3.A /:C(C.P.)

4.BvC 1,3,M.P.

5.~B ⊃ C 4.Impl

6.~C ⊃ ~B 2, Trans

7.~C ⊃ C 6,5,H.S.

8.~~CvC 7,Impl

9.CvC 8,D.N.

10.C 9,Taut.

Example 3

F ⊃ W

/:(F.S) ⊃ (WvX)

Solution:

1.F ⊃ W /:(F.S) ⊃ (WvX)

2.F.S. /: WvX(C.P.)

3.F 2,Simp.

4.W 1,3,M.P.

5.WvX 4,Add

Example 4

(I ⊃ J).(IvK)

(K ⊃ L).(KvI)

/: ~J ⊃ L

Solution:

Example 5

(D ⊃ E).(F ⊃ H) /:(DvF) ⊃ (HvE)

Solution:

1.(D ⊃ E).(F ⊃ H) /:(DvF) ⊃ (HvE)

2.DvF /:HvE (C.P.)

3.EvH 1,2,C.D.

4.HvE 3,Com

Indirect Proof

Example 1

Mv(N.O)

M ⊃ O /: O

3.~O I.P.

4.~M 2,3,M.T.

N.O. 1,4,DS

OvN 3,Add

NvO 6,Com

(N.O) 7,De M

9.(N.O).(N.O) 5,8,Conj.

O.N 5.Com

O 10, Simp.

Example 2

D

/: Ev(E ⊃ F)

Solution

D /: Ev(E ⊃ F)

~[Ev(E ⊃ F)] I.P.

~[Ev(~EvF)] 2.Impl.

[(EvE)vF] 3,Assoc.

(EvE).F ⊃ 4. De M

6.(Ev~E) 5,Simp

~E.~~E 6,De M

~E.E 7,D.N.

E.~E 8,Com

E 9, Simp.

Ev(E ⊃ F) 10,Add

We shall deal with more problems to further illustrate the rule of Indirect Proof.

Example 3

W⊃(X.Y)

(XvZ)⊃Q

ZvW /∴Q

Solution

1.W⊃(X.Y)

2.(XvZ)⊃Q

3.ZvW /∴Q

4.~Q I.P.
5.~(XvZ) 2,M.T.
6.~X.~Z 5,D.M.
7.~Z.~X 6,Com
8.~Z 7,Simp
9.W 3,8,D.S.

10.X.Y 1,9,M.P.

11.X 10,Simp

12.~X 6,Simp
13.X.~X 11,12,Conj

Example 4

P. ~p

Quantification

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An individual with different predicates & nbsp&nbsp&nbspAttribute Predicated of several individuals

With the notation for individual variable, x, we can translate the last statement into

Given any x, x is transient

(x) Tx

(∃x) Ax.

Negative proposition can be symbolized also by applying the basic principles which we employed is symbolizing affirmative Propositions. For instance.

(1) Nothing is permanent

can be stated as

(2) Given any individual thing whatsoever, it is not permanent.

Using the capital letter P to designate "permanent", proposition (2) becomes:

(3) (x) ~Px

Again, the assertion "Something is not permanent" means that

There is at least one thing that is not permanent

There is at least one thing that is not permanent.

It also can be rewritten as

There is at least one x such that x is not permanent

or as

There is at least one x such that ~Px

Symbolically, the proposition "something is not permanent" can now be written down completely.

(∃x)~Px

A graphic presentation of the four general Propositions and their logical equivalences, using the Greek letter phi (written as) to represent any predicate whatsoever, is set forth below:

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Oparaugo Michael — Mon, 07 Sep 2026 08:02:51 +0000

Contact: 08185051292.

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Oparaugo Michael — Sat, 05 Sep 2026 07:39:14 +0000

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Oparaugo Michael — Thu, 20 Aug 2026 12:20:11 +0000

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Oparaugo Michael — Thu, 30 Jul 2026 10:52:19 +0000

Introduction to Logic
Venn Diagrams
Categorical Syllogisms

Abstract: The Venn Diagram technique is shown for typical as well as unusual syllogisms. The problem of existential import is introduced by means of these diagrams.

I. One good method to test syllogisms rapidly is the Venn Diagram technique. This class assumes you are already familiar with diagramming categorical propositions. If not, you might wish to review how to diagram standard form statements here: Venn Diagrams.

A. The syllogism is a two premiss argument having three terms, each of which is used twice in the argument.

B. Each term (major, minor, and middle terms) can be represented by drawing a circle. Anything inside the circle is designated as a member of the class named by that term and anything outside the circle is something else.

C. Since a syllogism is valid if and only if the premisses entail the conclusion, diagramming the premisses will exhibit the logical geography of the conclusion.

If the syllogism is invalid, diagramming the premisses will not show that conclusion necessarily follows from the premises.

D. Since we have three classes, we expect to have three overlapping circles.

A diagram similar to this one was first used by John Venn [Symbolic Logic (London: Macmillan, 1818), 105]:
Blank Venn Diagram

The area within the denoted circle represents where members of the class would be, and the area outside the circle represents all other individuals (the complementary class). The various areas of the diagram are noted above.

A circle with its interior shaded2. Shading represents the knowledge no individual exists in the class; for example, this diagram shows no “Yeti” exists within the circle.

A circle with its blankEmpty space represents the fact that no information is known whether any thing exists in that area of the class.

A circle with with an “x” in its center3. An “X” in the circle represents “at least one “thing” ” exists in the class of things, and so “X” here corresponds with the word “some.”

II. Some typical examples of syllogisms are shown here by their mood and figure.

A. EAE-1

The syllogism has an E statement for its major premiss, an A statement for its minor premiss, and an E statement for its conclusion. By convention the conclusion is labeled with S (the minor term) being the subject and P (the major term) being the predicate.

The position of the middle term is the “left-hand wing” in this illustration of the syllogistic figures:

A circle with with an “x” in its center

The form of this syllogism written out is

No M is P.
All S is M
No S is P.

We diagram one premiss at a time on the diagram of three interlocking circles. So, as shown on the diagram below, we superimpose the major premiss E statement diagram “No M is P” on the area represented by the interlocking circles P and M.

Next, we superimpose the A minor premiss “All S is M” on the interlocking circles S and M.

Note, in the diagram below, how the area in common between S and P (called “the lens of the S and P classes) has been completely shaded out indicating that “No S is P.”

No individual X can exist within the SP lens area because our diagram shows that area must be empty. In a sense, the conclusion “automatically” results from diagramming only the two premisses. We do not pencil in the conclusion — we merely read it out by observing the result of sketching the two premises.
Diagram of EAE-1 Syllogism

This diagram illustrates that all syllogisms of the form EAE-1 are valid because the diagram shows that if there are any Ss then they are in a completely separate area than Ps.

B. AAA-1

This syllogism is composed entirely of “A” statements with the M-terms arranged in the “left-hand wing” as was the syllogism above.
Its form is written out as

All M is P.
All S is M.
All S is P.

Note, after diagramming the figure below, how the only unshaded area of S is within the area of all three classes. The important thing to notice is that this area of S is entirely within the P class.

Hence, we can conclude that if there are any Ss they have to be also in the P class.
Diagram of AAA-1 Syllogism

This diagrams shows all AAA-1 syllogisms are always valid. In ordinary language the AAA-1 and the (above) EAE-1 syllogisms are by far the most frequently used everyday syllogistic argument forms.

C. AII-3

The AII-3 syllogism has the M-terms arranged in the subject position — the right side of the orange flying brick diagram illustration above.
This syllogism sets up as

All M is P.
Some M is S.
Some S is P.

When diagramming the minor premise of this syllogism, notice how you are “forced” to put the X from the minor premiss in the area of the diagram shared by all three classes.

The “X” cannot go on the P-line because the shading indicates that part of the SM lens area is empty. This “logical” forcing enables you to read-off the conclusion, that “Some S is P” because the X is in the union of S, P and M.

This syllogism is a good example why … The universal premiss should be diagrammed before a particular premiss is diagrammed. If we were to diagram the particular premiss first, the “X” would go on the line. Then, we would have to move it when we diagram the universal premiss because the universal premiss empties an area where the “X” could have been. Diagram of AII-3 Syllogism

D. AII-2

The AII-2 has the M terms in the predicate of both premisses.
The syllogism is written out as …

All P is M.
Some S is M.
Some S is P.

The diagram below shows that the “X” could be in the SMP area or in the SPM area. Since we do not know exactly which area it is in, we put the “X” on the line as shown to indicate that the X might be in either place, but we can't be sure. Thus, the conclusion can't be read off, so the argument is invalid.

When an X is on a line, we do not know with certainty exactly where it is. So, when we go to read the conclusion, we do not know for sure whether the X is in the P class or not. Since the conclusion cannot be read with certainty, the AII-2 syllogism is judged invalid.
Diagram of AII-2 Syllogism

E. The final syllogism described here, the EAO-4 raises some interesting problems.

Notice that in this syllogism there are universal premisses with a particular conclusion.
Its form is expanded as … No P is M. All M is S. Some S is not P.
And its diagram is rather easily drawn as …
Diagram of EAO-4 Syllogism
When we try to read the conclusion, we see that there is no “X” in the SMP class. We must conclude that the syllogism is invalid because we cannot read-off “Some S is not P.” There is no X anywhere on the diagram.
However, if we independently knew that at least one individual in the the class represented by M exists, then that one M would have to be in the SMP class. That M would have to be a S's as well. Hence, we know that some S's are not P's!

In other words, the EOA-4 syllogism is valid if we knew ahead of time the additional premiss “M exists.”

Most contemporary logicians have concluded that we should not assume any class exists unless we have some form of independent evidence. a. However, we often want to speculate about theoretical entities without assuming their existence. b. For example, in science and mathematics, our logic will apply when talking about circles, points, frictionless planes, and freely falling bodies even though these entities do not physically exist. c. This diagram illustrates the contemporary topic called the problem of existential import. When can we reasonably conclude something exists? How does this conclusion affect our theory of logical validity?

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Oparaugo Michael — Tue, 28 Jul 2026 16:50:15 +0000

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Oparaugo Michael — Thu, 23 Jul 2026 20:09:49 +0000

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