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Oparaugo Michael — Tue, 02 Jun 2026 15:23:57 +0000

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Oparaugo Michael — Tue, 02 Jun 2026 13:36:38 +0000

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TEENCONNECT: YOUR HEALINGS

AUGUST 27, 2024

Healed of anorexia

Name Withheld
₦&nbsp&nbspψ&nbsp&nbsp&nbspω&nbsp&nbsp&nbsp&nbsp&nbsp&nbspϑ
In high school, I was really focused on body image and beauty. I spent a lot of time comparing myself to other girls. This led me to feel that I needed to lose weight to become what I thought was attractive. I started restricting my food intake, exercising to burn calories, and weighing myself to check my “progress.”

Even though I grew severely thin and was obviously endangering myself, I didn’t care. I still had a negative view of myself and continued to look for beauty in my ever-thinning image in the mirror.

Before this, I’d always loved reading the Bible and Science and Health with Key to the Scriptures by Mary Baker Eddy and attending Christian Science Sunday School. But as I slipped further into these disordered behaviors, my love for spiritual growth diminished. I felt more and more apathetic about God, Christian Science, and church. It seemed like I was stuck in a downward spiral.

Over time, my parents noticed that my eating habits had changed and that my body looked unhealthy. Although I never told them about the anorexic thoughts I was dealing with, I knew that they were supporting me by praying for me daily. They also provided the types of food I was willing to eat, took me to Sunday School, and encouraged me to turn to God for healing.

I continued to look for beauty in my ever-thinning image in the mirror.

After engaging in this unhealthy behavior for a while, I finally felt led to dive back into reading the Bible and studying Science and Health along with it. What I read in Science and Health explained the spiritual meaning of the Bible and helped connect the ideas in the Bible to my everyday life.

I loved reading the first chapter of Genesis in the Bible. It describes who we really are—our spiritual identity—this way: “And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth. So God created man in his own image, in the image of God created he him; male and female created he them. And God blessed them” (verses 26–28).

As I studied these verses, I realized that I had been hoping my body would tell me who I am, or confirm something about my identity. But this hope was empty from the outset because we are made in God’s image. The Bible tells us that God is Spirit, so we are Spirit’s own image. That means that our true identity is spiritual and can never be known or assessed by what’s in the mirror.

I started to see that this spiritual image is the true and only representation of who I am. Then one day this passage from Science and Health grabbed my attention and helped me understand this idea even more deeply: “Now compare man before the mirror to his divine Principle, God. Call the mirror divine Science, and call man the reflection. Then note how true, according to Christian Science, is the reflection to its original” (pp. 515–516). Spirit is my original, so as Spirit’s image or reflection, I am and do only what Spirit is and does. I am pure because Spirit is pure. I am good because God is only good. Although I wasn’t able to put this into words at the time, I can see now that I was beginning to understand this new concept of myself as the image of divine Spirit rather than a physical body.

I started to identify beautiful spiritual qualities of God that constitute who I am.

I started to identify beautiful spiritual qualities of God that constitute who I am. Some of these are boundless joy, compassion, trust, and intelligence. I also realized that these spiritual qualities are eternally consistent—unlike changing standards of physical beauty. And although an expression of joy may be noticed through a smile, spiritual qualities themselves can’t be measured by looking in a mirror.

Through my prayers and study, I started to feel satisfied and

Rules of Inference.

Consider this argument:

If either Bill Clinton should be tried for misconduct or kingsley Starr was lying, then the US congress should investigate the Clinton/Lewinsky affair thoroughly. If either the US congress investigates the Clinton/Lewinsky affair thoroughly or a serious misdemeanour will go unpunished, then the American democratic system is on trial. Either serious misdemeanour will go unpunished or Bill Cliton should be tried for misconduct. Serious misdemeanour will not go unpunished. Therefore the American democratic system is on trial.

We may translate it into

(BvK)⊃U

(UvS)⊃D

SvB

~S

:/D

Our argument has five variables, and to test it with the truth table method will require thirty two rows. We can use the truth table to test whether the given argument is valid or invalid. It is equally true that we can use the method to test the arguments analysed. However the truth table technique becomes difficult and inconvenient to use as the number of component statements increase. As we observed before the argument with which we began requires a truth table of thirty two rows. The actual construction of such lenghty truth tables is definitely cumbersome and confusing. Thus, logicians have devised a less cumbersome method of proving the validity of an argument by inferring its conclusion from its premisses through a succession of smaller arguments every one of which is known to be valid in advance. This method is anchored on the availability of valid elementary argument forms which serve as explanation or justification for each line of deduction after the conclusion of the argument have been written down. Put differently, a formal proof of validity of an argument is constructed by writing down the premisses and the subconclusions we infer from them in a line and stating the justification for the deductions to the right of the sub-conclusions. The elementary argument forms that undergird this procedure are called Rules of Inference and nine of them are listed below:

Addition (Add)

P

:/Pvq

Simplification (Simp)

P.q

:/P

Absorption (Abs)

P⊃q

:/P⊃(P.q)

Conjunction

P

q

:/P.q

Disjunctive Syllogism (DS)

Pvq

~P

:/q

Hypothetical Syllogism (HS)

P⊃q

q⊃r

:/P⊃r

Modus Ponens (MP)

P⊃q

P

:/q

Modus Tollens

P⊃q

~q

:/~P

Constructive Dilema (CD)

(P⊃q).(r⊃S)

Pvr

qvs

If we refer back to the argument set forth at the beginning of this section, the use of the rule of inference can be illustrated by proving its validity.The proof goes thus:

1. (PvQ)⊃R

2. (RvS)⊃T

3. SvP

4. ~S &nbsp /:T

5. P. &nbsp 4,3 D.S.

6. PvQ. &nbsp 5. Add.

7. R. &nbsp 1, 6 M.P.

8. RvS. &nbsp 7, Add.

9. T &nbsp 2, 8 M.P.

This method of proof is obvioudly more straight-forward and shorter than proving the argument valid by using the truth table method, provided that the reader is conversant with the rules of inference.

2. Example:
If we refer back to the argument set forth at the beginning of this section, the use of the rule of inference can be illustrated by proving its validity.

1. (PvQ)⊃R

2. (RvS)⊃T

3. SvP

4. ~S /∴T

5. P 4,3 D.S

6. PvQ 5 Add

7. R 1,6 M.P

8. RvS 7.Add

9. T 2,8 M.P.

3.1.A⊃B

2.(A.B)⊃C

3.(A.C)⊃D
/∴A⊃D

4.A⊃(A.B) 1. Abs.

5. A⊃C 4,2.H.S.

6. A⊃(A.C) 5. Abs.

7. A⊃D 6,3.H.S.

4.

1.F⊃G

2. G⊃H

3. ~H /∴~F.~G

4. ~G 2,3. M.T

5. ~F 1,4 M.T

6. ~F.~G 5,4. Conj.

5.

1. M⊃N

2. Mv(O.P)

3. ~N.~Q /∴O

4. ~N. 3, simp

5. ~M. 1,4.M.T

6. O.P. 2,5.D.S

7. O. 6. Simp.

Rule of Replacement.

There are some tautologies which are important because they complement the nine rules of inference just discussed. Such logical equivalences serve as additional principles of inference for the handling of problems which may not be solved by applying the rules of inference alone. The rule of replacement, as these logical equivalences are called, permits us to infer from any proposition, the result of substituting any component of that proposition by another statement which is logically equivalent to the component substituted.

Before we start applying the rule of replaement, it is expedient to state the tautologies first.

1.&nbsp&nbspDouble negation (D.N.): P&#x2261~~P

2.&nbsp&nbspTautology (Taut): P&#x2261PvP
: P≡P.P

3.&nbsp&nbspCommutation (Com.):(PvQ)≡(QvP)

&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp

(P.Q)≡(Q.P)

4.&nbsp&nbspMaterial implication &nbsp&nbsp&nbsp&nbsp(Impl):&nbsp&nbsp&nbsp(P⊃Q)≡(~PvQ)

5.&nbsp&nbspTransposition &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp(Trans):&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp(P⊃Q)≡(~Q⊃~P)

6.&nbsp&nbspExportation &nbsp&nbsp&nbsp&nbsp&nbsp(Exp):&nbsp&nbsp&nbsp[(P.Q)⊃r]≡[P⊃(Q⊃r)]

7.&nbsp&nbspDe Morgan's Theorem (Dem):

&nbsp&nbsp&nbsp&nbsp&nbsp
~(P.Q)≡(~Pv~Q)

&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp~(PvQ)≡(~P.~Q)

8.&nbsp&nbspAssociation (Assoc): &nbsp&nbsp&nbsp&nbsp&nbsp[Pv(Qvr)]≡[(PvQ)vr]

&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp[P.(Q.r)]≡[(P.Q).r]

9.&nbsp&nbspDistribution (Dist):

&nbsp&nbsp&nbsp&nbsp&nbsp[P.(Qvr)]≡[(P.Q).(P.r)]

&nbsp&nbsp&nbsp&nbsp&nbsp[Pv(Q.r)≡[(PvQ).(Pvr)]

10.Material Equivalence &nbsp&nbsp(Equiv):

&nbsp&nbsp&nbsp&nbsp&nbsp(P≡Q)≡[(P⊃Q).(Q⊃P)]

&nbsp&nbsp&nbsp&nbsp&nbsp(P≡Q)≡[(P.Q)v(~P.~Q)]

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.

Example 2.

(DvE)⊃(F.G)

~F

∴~D

Proof

1.(DvE)⊃(F.G)

2.~F/∴

3.~Fv~G 2,Add.

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

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

6.~D.~E5,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 /&#8756G.

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. /&#8756L

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 /&#8756W

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. /&#8756~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) /&#8756(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 /&#8756A⊃D

3.A /&#8756D(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 &nbsp&nbsp&nbsp&nbsp/&#8756P⊃(Q⊃S)

3.P &nbsp&nbsp&nbsp&nbsp/&#8756Q⊃S(C.P.)

4.Q &nbsp&nbsp&nbsp&nbsp/&#8756S(C.P.)

5.P.Q. &nbsp&nbsp&nbsp&nbsp3,4, Conj.

6.R &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp1,5,M.P.

7.Q.R. &nbsp&nbsp&nbsp&nbsp4,6,Conj.

8.S &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp2,7, 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 /&#8756P⊃(C⊃T)

Solution:

1.P⊃(C⊃N)

2.(N.R)⊃E

3.(R⊃E)⊃T /&#8756P⊃(C⊃T)

4.P /&#8756C⊃T (C.P.)

5.C /&#8756T (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

/&#8756A⊃C

Solution:

1.A⊃(BvC)

2.B⊃C

3.A&nbsp&nbsp&nbsp/&#8756C(C.P.)

4.BvC&nbsp&nbsp&nbsp1,3,M.P.

5.~B⊃C&nbsp&nbsp&nbsp4.Impl

6.~C⊃~B&nbsp&nbsp&nbsp2, Trans

7.~C⊃C&nbsp&nbsp&nbsp6,5,H.S.

8.~~CvC&nbsp&nbsp&nbsp7,Impl

9.CvC&nbsp&nbsp&nbsp8,D.N.

10.C&nbsp&nbsp&nbsp&nbsp9,Taut.

Example 3

F⊃W
&#8756(F.S)⊃(WvX)

Solution:

1.F⊃W&nbsp&nbsp&nbsp/&#8756(F.S)⊃(WvX)

2.F.S.&nbsp&nbsp&nbsp/&#8756WvX(C.P.)

3.F&nbsp&nbsp&nbsp2,Simp.

4.W&nbsp&nbsp&nbsp1,3,M.P.

5.WvX&nbsp&nbsp&nbsp&nbsp4,Add

Example 4

(I⊃J).(IvK)

(K⊃L).(KvI)

&#8756~J⊃L

Solution:

1.(I⊃J).(IvK)

2.(K⊃L).(KvI)&nbsp&nbsp&nbsp/&#8756~J⊃L

3.~J&nbsp&nbsp&nbsp&nbsp/&#8756L(C.P)

4.I⊃J&nbsp&nbsp&nbsp&nbsp1,Simp.

5.~I&nbsp&nbsp&nbsp&nbsp&nbsp4,3,M.T.

6.(KvI).(K⊃L)&nbsp&nbsp&nbsp2, Com.

7.KvI&nbsp&nbsp&nbsp6,Simp.

8.IvK&nbsp&nbsp&nbsp&nbsp7.Com.

9.K&nbsp&nbsp&nbsp&nbsp&nbsp8,5,D.S.

10.K⊃L&nbsp&nbsp&nbsp&nbsp2,Simp.

11.L&nbsp&nbsp&nbsp&nbsp10,9,M.P.

Example 5

(D⊃E).(F⊃H)/&#8756(DvF)⊃(HvE)

Solution:

1.(D⊃E).(F⊃H)/&#8756(DvF)⊃(HvE)

2.DvF&nbsp&nbsp&nbsp&nbsp/&#8756HvE&nbsp&nbsp&nbsp(C.P.)

3.EvH&nbsp&nbsp&nbsp&nbsp1,2,C.D.

4.HvE&nbsp&nbsp&nbsp&nbsp&nbsp3,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&nbsp&nbsp&nbsp&nbsp/&#8756O 3.~O&nbsp&nbsp&nbsp&nbspI.P. 4.~M&nbsp&nbsp&nbsp&nbsp2,3,M.T.

N.O.&nbsp&nbsp&nbsp&nbsp1,4,DS

~Ov~N&nbsp&nbsp&nbsp&nbsp3,Add

~Nv~O&nbsp&nbsp&nbsp&nbsp6. Com

~(N.O)&nbsp&nbsp&nbsp&nbsp7,De M 9.(N.O).~(N.O)&nbsp&nbsp&nbsp&nbsp5,8,Conj.

O.N&nbsp&nbsp&nbsp&nbsp&nbsp5.Com

O&nbsp&nbsp&nbsp&nbsp10, Simp. Example 2 D /&#8756Ev(E⊃F) Solution

D/&#8756Ev(E⊃F)

~[Ev(E⊃)] &nbsp&nbsp&nbsp&nbspI.P.

~[Ev(~EvF)] & &nbsp&nbsp2.Impl.

~[(Ev~E)vF] &nbsp&nbsp&nbsp&nbsp3,Assoc.

~(Ev~E).~F&sup4. De M 6.~(Ev~E)&nbsp&nbsp&nbsp5,Simp

~E.~~E&nbsp&nbsp&nbsp6,De M

~E.E&nbsp&nbsp&nbsp7,D.N.

E.~E&nbsp&nbsp&nbsp8,Com

E&nbsp&nbsp&nbsp9, Simp.

Ev(E⊃F) &nbsp&nbsp&nbsp&nbsp10,AddIn every formal proof that demands the rule of Indirect Proof, the logical structure of the Inference is from p/&#8756q to p. ~q/&#8756. 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&#8756(2)p~qThis 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./&#8756R constitutes a Conditional Proof of validity for another argument A.I&#8756Q⊃R. Similarly, a formal proof of validity for (2) constitutes a Conditional Proof of validity for a third argumenty.(3) p&#8756~q⊃qThe 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&nbsp&nbsp&nbsp/&#8756Q

Solution

1.&nbsp&nbspW⊃(X.Y)
2.&nbsp&nbsp(XvZ)⊃Q
3.&nbsp&nbspZvW&nbsp&nbsp&nbsp&nbsp/&#8756Q
4.&nbsp&nbsp~Q&nbsp&nbsp&nbsp&nbspI.P.
5.&nbsp&nbsp~(XvZ)&nbsp&nbsp&nbsp2,M.T.
6.&nbsp&nbsp~X~Z&nbsp&nbsp&nbsp5,D.M.
7.&nbsp&nbsp~Z.~X&nbsp&nbsp&nbsp6,Com
8.&nbsp&nbsp~Z&nbsp&nbsp&nbsp&nbsp7,Simp9.&nbsp&nbspW&nbsp&nbsp&nbsp&nbsp 3,8,D.S.
10.X.Y&nbsp&nbsp&nbsp&nbsp1,9,M.P.
11.&nbsp&nbspX&nbsp&nbsp&nbsp&nbsp10,Simp
12.&nbsp&nbsp~X&nbsp&nbsp&nbsp&nbsp6,Simp
13.&nbsp&nbspX.~X&nbsp&nbsp&nbsp&nbsp11,12,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.&nbsp~Z&nbsp&nbsp&nbspI.P 4.&nbsp~ZvU&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
REGISTRATION FORM
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TEENCONNECT: YOUR HEALINGS

AUGUST 27, 2024

Healed of anorexia

Name Withheld
₦&nbsp&nbspψ&nbsp&nbsp&nbspω&nbsp&nbsp&nbsp&nbsp&nbsp&nbspϑ
In high school, I was really focused on body image and beauty. I spent a lot of time comparing myself to other girls. This led me to feel that I needed to lose weight to become what I thought was attractive. I started restricting my food intake, exercising to burn calories, and weighing myself to check my “progress.”

Even though I grew severely thin and was obviously endangering myself, I didn’t care. I still had a negative view of myself and continued to look for beauty in my ever-thinning image in the mirror.

Before this, I’d always loved reading the Bible and Science and Health with Key to the Scriptures by Mary Baker Eddy and attending Christian Science Sunday School. But as I slipped further into these disordered behaviors, my love for spiritual growth diminished. I felt more and more apathetic about God, Christian Science, and church. It seemed like I was stuck in a downward spiral.

Over time, my parents noticed that my eating habits had changed and that my body looked unhealthy. Although I never told them about the anorexic thoughts I was dealing with, I knew that they were supporting me by praying for me daily. They also provided the types of food I was willing to eat, took me to Sunday School, and encouraged me to turn to God for healing.

I continued to look for beauty in my ever-thinning image in the mirror.

After engaging in this unhealthy behavior for a while, I finally felt led to dive back into reading the Bible and studying Science and Health along with it. What I read in Science and Health explained the spiritual meaning of the Bible and helped connect the ideas in the Bible to my everyday life.

I loved reading the first chapter of Genesis in the Bible. It describes who we really are—our spiritual identity—this way: “And God said, Let us make man in our image, after our likeness: and let them have dominion over the fish of the sea, and over the fowl of the air, and over the cattle, and over all the earth, and over every creeping thing that creepeth upon the earth. So God created man in his own image, in the image of God created he him; male and female created he them. And God blessed them” (verses 26–28).

As I studied these verses, I realized that I had been hoping my body would tell me who I am, or confirm something about my identity. But this hope was empty from the outset because we are made in God’s image. The Bible tells us that God is Spirit, so we are Spirit’s own image. That means that our true identity is spiritual and can never be known or assessed by what’s in the mirror.

I started to see that this spiritual image is the true and only representation of who I am. Then one day this passage from Science and Health grabbed my attention and helped me understand this idea even more deeply: “Now compare man before the mirror to his divine Principle, God. Call the mirror divine Science, and call man the reflection. Then note how true, according to Christian Science, is the reflection to its original” (pp. 515–516). Spirit is my original, so as Spirit’s image or reflection, I am and do only what Spirit is and does. I am pure because Spirit is pure. I am good because God is only good. Although I wasn’t able to put this into words at the time, I can see now that I was beginning to understand this new concept of myself as the image of divine Spirit rather than a physical body.

I started to identify beautiful spiritual qualities of God that constitute who I am.

I started to identify beautiful spiritual qualities of God that constitute who I am. Some of these are boundless joy, compassion, trust, and intelligence. I also realized that these spiritual qualities are eternally consistent—unlike changing standards of physical beauty. And although an expression of joy may be noticed through a smile, spiritual qualities themselves can’t be measured by looking in a mirror.

Through my prayers and study, I started to feel satisfied and

Rules of Inference.

Consider this argument:

If either Bill Clinton should be tried for misconduct or kingsley Starr was lying, then the US congress should investigate the Clinton/Lewinsky affair thoroughly. If either the US congress investigates the Clinton/Lewinsky affair thoroughly or a serious misdemeanour will go unpunished, then the American democratic system is on trial. Either serious misdemeanour will go unpunished or Bill Cliton should be tried for misconduct. Serious misdemeanour will not go unpunished. Therefore the American democratic system is on trial.

We may translate it into

(BvK)⊃U

(UvS)⊃D

SvB

~S

:/D

Our argument has five variables, and to test it with the truth table method will require thirty two rows. We can use the truth table to test whether the given argument is valid or invalid. It is equally true that we can use the method to test the arguments analysed. However the truth table technique becomes difficult and inconvenient to use as the number of component statements increase. As we observed before the argument with which we began requires a truth table of thirty two rows. The actual construction of such lenghty truth tables is definitely cumbersome and confusing. Thus, logicians have devised a less cumbersome method of proving the validity of an argument by inferring its conclusion from its premisses through a succession of smaller arguments every one of which is known to be valid in advance. This method is anchored on the availability of valid elementary argument forms which serve as explanation or justification for each line of deduction after the conclusion of the argument have been written down. Put differently, a formal proof of validity of an argument is constructed by writing down the premisses and the subconclusions we infer from them in a line and stating the justification for the deductions to the right of the sub-conclusions. The elementary argument forms that undergird this procedure are called Rules of Inference and nine of them are listed below:

Addition (Add)

P

:/Pvq

Simplification (Simp)

P.q

:/P

Absorption (Abs)

P⊃q

:/P⊃(P.q)

Conjunction

P

q

:/P.q

Disjunctive Syllogism (DS)

Pvq

~P

:/q

Hypothetical Syllogism (HS)

P⊃q

q⊃r

:/P⊃r

Modus Ponens (MP)

P⊃q

P

:/q

Modus Tollens

P⊃q

~q

:/~P

Constructive Dilema (CD)

(P⊃q).(r⊃S)

Pvr

qvs

If we refer back to the argument set forth at the beginning of this section, the use of the rule of inference can be illustrated by proving its validity.The proof goes thus:

1. (PvQ)⊃R

2. (RvS)⊃T

3. SvP

4. ~S &nbsp /:T

5. P. &nbsp 4,3 D.S.

6. PvQ. &nbsp 5. Add.

7. R. &nbsp 1, 6 M.P.

8. RvS. &nbsp 7, Add.

9. T &nbsp 2, 8 M.P.

This method of proof is obvioudly more straight-forward and shorter than proving the argument valid by using the truth table method, provided that the reader is conversant with the rules of inference.

2. Example:
If we refer back to the argument set forth at the beginning of this section, the use of the rule of inference can be illustrated by proving its validity.

1. (PvQ)⊃R

2. (RvS)⊃T

3. SvP

4. ~S /∴T

5. P 4,3 D.S

6. PvQ 5 Add

7. R 1,6 M.P

8. RvS 7.Add

9. T 2,8 M.P.

3.1.A⊃B

2.(A.B)⊃C

3.(A.C)⊃D
/∴A⊃D

4.A⊃(A.B) 1. Abs.

5. A⊃C 4,2.H.S.

6. A⊃(A.C) 5. Abs.

7. A⊃D 6,3.H.S.

4.

1.F⊃G

2. G⊃H

3. ~H /∴~F.~G

4. ~G 2,3. M.T

5. ~F 1,4 M.T

6. ~F.~G 5,4. Conj.

5.

1. M⊃N

2. Mv(O.P)

3. ~N.~Q /∴O

4. ~N. 3, simp

5. ~M. 1,4.M.T

6. O.P. 2,5.D.S

7. O. 6. Simp.

Rule of Replacement.

There are some tautologies which are important because they complement the nine rules of inference just discussed. Such logical equivalences serve as additional principles of inference for the handling of problems which may not be solved by applying the rules of inference alone. The rule of replacement, as these logical equivalences are called, permits us to infer from any proposition, the result of substituting any component of that proposition by another statement which is logically equivalent to the component substituted.

Before we start applying the rule of replaement, it is expedient to state the tautologies first.

1.&nbsp&nbspDouble negation (D.N.): P&#x2261~~P

2.&nbsp&nbspTautology (Taut): P&#x2261PvP
: P≡P.P

3.&nbsp&nbspCommutation (Com.):(PvQ)≡(QvP)

&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp

(P.Q)≡(Q.P)

4.&nbsp&nbspMaterial implication &nbsp&nbsp&nbsp&nbsp(Impl):&nbsp&nbsp&nbsp(P⊃Q)≡(~PvQ)

5.&nbsp&nbspTransposition &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp(Trans):&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp(P⊃Q)≡(~Q⊃~P)

6.&nbsp&nbspExportation &nbsp&nbsp&nbsp&nbsp&nbsp(Exp):&nbsp&nbsp&nbsp[(P.Q)⊃r]≡[P⊃(Q⊃r)]

7.&nbsp&nbspDe Morgan's Theorem (Dem):

&nbsp&nbsp&nbsp&nbsp&nbsp
~(P.Q)≡(~Pv~Q)

&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp~(PvQ)≡(~P.~Q)

8.&nbsp&nbspAssociation (Assoc): &nbsp&nbsp&nbsp&nbsp&nbsp[Pv(Qvr)]≡[(PvQ)vr]

&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp[P.(Q.r)]≡[(P.Q).r]

9.&nbsp&nbspDistribution (Dist):

&nbsp&nbsp&nbsp&nbsp&nbsp[P.(Qvr)]≡[(P.Q).(P.r)]

&nbsp&nbsp&nbsp&nbsp&nbsp[Pv(Q.r)≡[(PvQ).(Pvr)]

10.Material Equivalence &nbsp&nbsp(Equiv):

&nbsp&nbsp&nbsp&nbsp&nbsp(P≡Q)≡[(P⊃Q).(Q⊃P)]

&nbsp&nbsp&nbsp&nbsp&nbsp(P≡Q)≡[(P.Q)v(~P.~Q)]

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.

Example 2.

(DvE)⊃(F.G)

~F

∴~D

Proof

1.(DvE)⊃(F.G)

2.~F/∴

3.~Fv~G 2,Add.

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

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

6.~D.~E5,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 /&#8756G.

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. /&#8756L

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 /&#8756W

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. /&#8756~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) /&#8756(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 /&#8756A⊃D

3.A /&#8756D(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 &nbsp&nbsp&nbsp&nbsp/&#8756P⊃(Q⊃S)

3.P &nbsp&nbsp&nbsp&nbsp/&#8756Q⊃S(C.P.)

4.Q &nbsp&nbsp&nbsp&nbsp/&#8756S(C.P.)

5.P.Q. &nbsp&nbsp&nbsp&nbsp3,4, Conj.

6.R &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp1,5,M.P.

7.Q.R. &nbsp&nbsp&nbsp&nbsp4,6,Conj.

8.S &nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp&nbsp2,7, 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 /&#8756P⊃(C⊃T)

Solution:

1.P⊃(C⊃N)

2.(N.R)⊃E

3.(R⊃E)⊃T /&#8756P⊃(C⊃T)

4.P /&#8756C⊃T (C.P.)

5.C /&#8756T (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

/&#8756A⊃C

Solution:

1.A⊃(BvC)

2.B⊃C

3.A&nbsp&nbsp&nbsp/&#8756C(C.P.)

4.BvC&nbsp&nbsp&nbsp1,3,M.P.

5.~B⊃C&nbsp&nbsp&nbsp4.Impl

6.~C⊃~B&nbsp&nbsp&nbsp2, Trans

7.~C⊃C&nbsp&nbsp&nbsp6,5,H.S.

8.~~CvC&nbsp&nbsp&nbsp7,Impl

9.CvC&nbsp&nbsp&nbsp8,D.N.

10.C&nbsp&nbsp&nbsp&nbsp9,Taut.

Example 3

F⊃W
&#8756(F.S)⊃(WvX)

Solution:

1.F⊃W&nbsp&nbsp&nbsp/&#8756(F.S)⊃(WvX)

2.F.S.&nbsp&nbsp&nbsp/&#8756WvX(C.P.)

3.F&nbsp&nbsp&nbsp2,Simp.

4.W&nbsp&nbsp&nbsp1,3,M.P.

5.WvX&nbsp&nbsp&nbsp&nbsp4,Add

Example 4

(I⊃J).(IvK)

(K⊃L).(KvI)

&#8756~J⊃L

Solution:

1.(I⊃J).(IvK)

2.(K⊃L).(KvI)&nbsp&nbsp&nbsp/&#8756~J⊃L

3.~J&nbsp&nbsp&nbsp&nbsp/&#8756L(C.P)

4.I⊃J&nbsp&nbsp&nbsp&nbsp1,Simp.

5.~I&nbsp&nbsp&nbsp&nbsp&nbsp4,3,M.T.

6.(KvI).(K⊃L)&nbsp&nbsp&nbsp2, Com.

7.KvI&nbsp&nbsp&nbsp6,Simp.

8.IvK&nbsp&nbsp&nbsp&nbsp7.Com.

9.K&nbsp&nbsp&nbsp&nbsp&nbsp8,5,D.S.

10.K⊃L&nbsp&nbsp&nbsp&nbsp2,Simp.

11.L&nbsp&nbsp&nbsp&nbsp10,9,M.P.

Example 5

(D⊃E).(F⊃H)/&#8756(DvF)⊃(HvE)

Solution:

1.(D⊃E).(F⊃H)/&#8756(DvF)⊃(HvE)

2.DvF&nbsp&nbsp&nbsp&nbsp/&#8756HvE&nbsp&nbsp&nbsp(C.P.)

3.EvH&nbsp&nbsp&nbsp&nbsp1,2,C.D.

4.HvE&nbsp&nbsp&nbsp&nbsp&nbsp3,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&nbsp&nbsp&nbsp&nbsp/&#8756O 3.~O&nbsp&nbsp&nbsp&nbspI.P. 4.~M&nbsp&nbsp&nbsp&nbsp2,3,M.T.

N.O.&nbsp&nbsp&nbsp&nbsp1,4,DS

~Ov~N&nbsp&nbsp&nbsp&nbsp3,Add

~Nv~O&nbsp&nbsp&nbsp&nbsp6. Com

~(N.O)&nbsp&nbsp&nbsp&nbsp7,De M 9.(N.O).~(N.O)&nbsp&nbsp&nbsp&nbsp5,8,Conj.

O.N&nbsp&nbsp&nbsp&nbsp&nbsp5.Com

O&nbsp&nbsp&nbsp&nbsp10, Add Example 2 D /&#8756Ev(E⊃F)

<a href="http://www.philosophypages.com/lg/e11a.htm">Philosophy</a> <a href="https://www.html.am/html-codes/#google_vignette">HTML</a> (PvQ)

∴ ∴∴

(a) he is a good man.

&nbsp&nbsp&nbsp&nbsp&nbsp&nbsphow do you know him?

<>mike<>

(i'm in the light dom)

                                      <a href="http://www.philosophypages.com/lg/e11a.htm">Philosophy</a><br><br>
                                      <a href="https://www.html.am/html-codes/#google_vignette">HTML</a><br><br>









                                        <br>(PvQ)

∴ ∴∴

(a) he is a good man.

&nbsp&nbsp&nbsp&nbsp&nbsp&nbsphow do you know him?

<>mike<>

(i'm in the light dom)

U+2200♥
Gift Acknowledgment Letter Template
To make sure you include everything that’s needed in your letter, use this template as a starting point:

Dear [Donor’s Name],

Thank you so much for your generous donation of $[amount] on [date]. This letter is to officially acknowledge the receipt of your donation, which we have designated to support our [specific program]. Your support is vital to our efforts and makes a significant impact.

Please keep this letter for your records as it may be used as official documentation for tax purposes. No goods or services were provided in exchange for your donation.

We are deeply grateful for your generosity and commitment to our cause. Your support not only helps us achieve [specific goals], but it also strengthens our community of advocates and changemakers. We would love to see you at [upcoming event] or work with you on our volunteer team. Please visit our website [website URL] or contact us directly for more information on how you can greatly enhance our collective impact.

Thank you once again for your commitment to [cause].

Sincerely,

[Your Name]

[Your Position]

[Organization’s Name]
Acknowledgement Letters for Receipt of Payment

Dear Sir,

This letter acknowledges the receipt of your payment for the products supplied. We have received the complete payment for order number 345, which was supplied last week. Thank you for your continued trust in our company.

We look forward to working with you in the future.

Yours sincerely,

Nikita Singh

Manager Accounts

Dear Manager,

On behalf of our company Infinity Pvt. Ltd., I would like to acknowledge that we have received a payment of Rs. 50,000, which was due for order number 576. We look forward to collaborating with you in the future.

Respectfully yours,

Malini Malhotra

Store Manager

Acknowledgement Letter for Cheque Received

1st Format

Mr Rohan Gupta

Club President

Subject: Acknowledgement of Cheque Received for Membership Renewal

Dear Mr Rohan,

Concerning your letter regarding the Quarterly Fee Payment of Our club, it is to acknowledge that we have received your Cheque No. XXXXXX dated ______, amounted Rs.20,000/- as a quarterly membership renewal fee. We will renew your membership within 8-9 business days, and you will receive your membership card by the first week of May.

Looking forward to serving you in the future.

Regards,

Sweta Acharya

2nd Format

Ms Shruti Sen

Manager Sales

Galaxy Supplies

Dear Ms Shruti,

With this letter, we hereby acknowledge the invoice of Rs. 17,000 (seventeen thousand rupees only) as payment for office supplies from (date) to (date). I have attached our official invoice no—7890 acknowledging your payment for your reference. We thank you for your continued trust in our company and look forward to serving you.

Best Regards,

Sonia Mahapatra

Manager Accounts

Acknowledgement Letter for Payment Received

Ms Neha Kaur

Manager Sales

Anglewise Pvt. Ltd.

72/C Mukherjee Park Extension, Delhi – 110081

Subject: Acknowledgement Letter for Receiving Pending Amount

Dear Ms Neha,

This letter acknowledges the receipt of all our pending amounts, Cheque No. XXXXXX and Amount Rs.40,000. The invoice and receipt are attached with this mail for your reference.

We respect your kind gesture of clearing all our pending payments. Your payment has immensely helped manage our audit reports and clear all constraints. We look forward to getting together on more projects and ideas with your esteemed organisation.

Warm Regards,

Mrs Sneha Chowdhury

Manager Finance

Galaxy Enterprises
Sample Acknowledgment Letter
(Date)

(Name of recipient)
(Position)

Dear (name of recipient)

We would like to acknowledge the receipt of ______________ (document name). We are presently examining it and if there are no issues to be addressed, the signed (document) will be returned to you within ______ business days. If there are any further concerns regarding the (document), we will contact you by (date).

Thank you for your continued trust in our company and we look forward to doing business with you in the future.

Respectfully yours.

(Name of sender)
(Position in company,
Bonus professional acknowledgment email templates

If you've read our guide this far but are still stuck on how to send an acknowledgment email, this template is for you.

We've broken down the email into blocks. Work your way through from top to bottom, adding information, editing sections, and deleting unwanted blocks.

By the end, you'll have an acknowledgment email suitable for your specific situation.

General acknowledgment email template

To
sample@mail.com
Hi (Recipient's name),

Thank you for your email. I acknowledge that I/we received it on (you can add a date here).

(You can add details about what you are going to do here...)

The next stage in the process is as follows:

(You may want to list what's happening in a job application process, for example. Bullet points are great for this purpose!)

You can contact me at (insert details) if you want more information.

Kind regards,

(Your name)

Acknowledgment of receipt with follow-up

Use this template when you need to acknowledge receipt and indicate a follow-up action.

To
sample@mail.com
Confirmation of Receipt and Follow-up
Dear (Recipient's name),

Thank you for your email dated [Date]. I am writing to confirm that I have received it.

I will review the information provided and will get back to you with my feedback by [Specific Date].

If you need any additional information in the meantime, please do not hesitate to reach out.

Warm regards,

(Your name)

Acknowledgment of receipt from a boss

Use this template to acknowledge an email from your boss.

To
sample@mail.com
Acknowledgment of Your Email
Dear (Boss’ name),

I hope you are doing well. I wanted to let you know that I have received your email dated [Date].

Thank you for the detailed instructions and guidance. I will ensure that the tasks are completed as per your expectations and will keep you updated on the progress.

Best regards,

(Your name)

Acknowledgment of receipt for a document

Use this template when acknowledging the receipt of a specific document.

To
sample@mail.com
Confirmation of Document Receipt
Dear (Recipient's name),

I am writing to confirm that I have received the [Document Name] on [Date].

Thank you for sending this document. I will review it thoroughly and will get back to you if any further information is required.

Kind regards,

(Your name)

Acknowledgment of receipt of a meeting request

Use this template to acknowledge the receipt of a meeting request.

To
sample@mail.com
Meeting Request Acknowledgment
Dear (Recipient's name),

Thank you for your email regarding the meeting request for [Date and Time].

I am pleased to confirm my availability for the proposed schedule.

Please let me know if there are any specific preparations needed from my side.

Looking forward to our discussion.

Best regards,

(Your name)

Acknowledgment of receipt for job application

Use this template to acknowledge the receipt of a job application.

To
sample@mail.com
Job Application Receipt
Dear (Applicant’s name),

Thank you for submitting your application for the position of [Job Title] on [Date].

I wanted to confirm that we have received your application and our team will be reviewing it shortly.

We appreciate your interest in joining our company and will be in touch with you regarding the next steps.

Best regards,

(Your name)

Acknowledgment of receipt for payment

Use this template to acknowledge the receipt of a payment.

To
sample@mail.com
Payment Receipt Confirmation
Dear (Recipient's name),

I hope you are well. I am writing to confirm that we have received your payment of [Amount] on [Date].

Thank you for your prompt payment.

If you require a formal receipt or have any further questions, please let us know.

Kind regards,

(Your name)

Acknowledgment of receipt for an event invitation

Use this template to acknowledge the receipt of an event invitation.

To
sample@mail.com
Event Invitation Acknowledgment
Dear (Recipient's name),

Thank you for inviting me to [Event Name] scheduled for [Event Date].

I am writing to acknowledge receipt of your invitation and express my appreciation.

I will confirm my attendance as soon as possible.

Warm regards,

(Your name)

Acknowledgment of receipt for feedback

Use this template to acknowledge the receipt of feedback.

To
sample@mail.com
Feedback Receipt Confirmation
Dear (Recipient's name),

I hope this message finds you well. I wanted to thank you for providing your feedback on [Date].

I acknowledge the receipt of your feedback and appreciate you taking the time to share your insights with us. Your input is valuable, and we will consider it carefully.

Best regards,

(Your name)
Acknowledgment email to boss sample

If your boss emails you something important, you should acknowledge and reply to your boss ASAP. Don't waste time or wait; get on with crafting a message!

So, how to acknowledge an email from a boss? Check out our sample here...

To
sample@mail.com
Hi (Recipient's name),

I can confirm that I have received (whatever your boss has sent you!). Thank you for sending it to me so quickly; it's appreciated.

I'll get to work... (outline what you'll do with the information or documents your boss has sent you!)

Thanks,

(Your name)

How to acknowledge payment received via email sample

If someone has sent you cash (in a professional context or a personal one), you have to acknowledge that as soon as possible.

There's always a risk with sending money online, so this email will provide peace of mind that it's reached the right person or place.

This is how we recommend acknowledging receipt of payment.

To
sample@mail.com
Hi (Recipient's name),

I'm contacting you to acknowledge your payment of (amount) for (the product or service they have purchased).

We'll send you a copy of your receipt separately.

Now that we have received your payment, we can (set out the next steps).

Thanks for paying so promptly. It's appreciated!

(Your name)

Job application acknowledgment email sample

Acknowledging receipt of a job application is essential, as someone's career could rest upon it. A simple one-line message works, but enthusiastic applicants will often email you with questions.

We recommend adding as much detail as possible in your acknowledgment emails to counteract that happening (and cut out future unwanted emails!). Set out the next steps, including dates and future correspondence.

To
sample@mail.com
Hi (Recipient's name),

Thank you for applying for the (job role) post at (company). We acknowledge receipt of your application.

The deadline for applications closes at (insert date). After this, we will review all applications and contact those who have successfully secured an interview.

We will be in touch as soon as we can.

Many thanks,

(Your name)

School

Oparaugo Michael — Fri, 29 May 2026 06:02:47 +0000

Writing a Great Post Title
Think of your post title as a super short (but compelling!) description — like an overview of the actual post in one short sentence.
Use keywords where appropriate to help ensure people can find your
post by search.

compelling!) description — like an overview of the actual post in one short sentence.
Use keywords where appropriate to help ensure people can find your post by search.

Ever wondered what an LLM is doing when it's "thinking"?

In this episode of Release Notes Explained, we cover the fundamentals of how thinking and reasoning models work including concepts like:

Scaling laws
Test-time compute
Reinforcement learning from verifiable rewardsHope you enjoy!

FLP ARGI

Oparaugo Michael — Thu, 28 May 2026 17:15:39 +0000

FLP ARGI

Health

Oparaugo Michael — Tue, 26 May 2026 19:50:19 +0000

NURITION
1. Carbohydrates: They are the main
source of energy in our body
sugar, potatos,bread, rice
yam, puris, roti.
2. Fat: They provide more energy
than carbohydrate, because of
higher percentage of carbon and
hydrogen and less oxygen
than that of carbohydrate - oils,
groundnuts, nuts, meat, milk,
ghee, butter and soyabin.
3.Vitamins:They are necessary for
normal growth, good health,
good vision, proper digestion,
healthy teeth, bones and gums.
Almost all the foods contain one
or the other vitamins required by
our body, like whole grain cereals,
milk, fish, meat and vegetables.
Classification of Vitamins:All
vitamins can be divided into
two groups or
classes according to
their solubility in water
and oils.
Class 1: Vitamin A, D,E and K are
soluble in oil, also called fat
soluble vitamins.
Class 2: Vitamin B-complex and
vitamin Care soluble in water,
where vitamin B-complex
represents a group of
eleven vitamins.
Vitamin A:It is good eye-sight
(specially the night vision),
normal skin and hair.
Milk, eggs, fish, butter,ghee
is also called ratinal.
Vitamin B1:It is essential for
growth, metabolism,
carbohydrates, nerves
functioning of heart
and muscles.
Sources are milk, sea-food,
grain cereals and soyabeans.
Chemical name is thianmane.

Forever Living Products.

Oparaugo Michael — Tue, 26 May 2026 19:23:51 +0000

Forever Living was founded in 1978 in Tempe, Arizona by Rex Maughan.[1] By 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.[4] Some journalists have likened the multi-level marketing business model of Forever Living's distribution system to that of a pyramid scheme.[5][6][7]

Forever Living Products headquarters in McCormick Ranch in Arizona in 2007. The company's resort division owned several similar properties.
In 1983, the company was named No. 6 on Inc. Magazine's annual Inc. 500 list of the fastest-growing private companies in the United States.[8]

According to Arthur Andersen's Top 100, as of 1993, Forever Living Products International was Arizona's second-largest private company.[9] As of August 1995, Forbes reported the company's product line included "deodorants, toothpaste, laundry detergent and three dozen other products, nearly all of which contain extract of aloe."[10]

A three-part special report by the Manila Times in 2003 discussed similarities between FLPI's business model and an illegal pyramid scheme, noting that FLPI participants are said to be rewarded primarily for recruiting new members to the organization, rather than for selling products to genuine end-users.[11]

Forever Living reported unaudited annual revenue exceeding $1.15 billion in 2005[2] and ended the year with around 150,000 distributors[12] and 55 employees.[13] The following year, Forever Living was listed at No. 340 on the Forbes 400 list, which ranks the largest private companies in the United States. At the time, the company was described as having 4,100 employees and sold its product in 100 countries.[2]

In 2010, the company reported unaudited revenue of $1.7 billion and a network of 9.3 million distributors.[3] In 2013 the publication New Vision reported that Forever Living had over 20,000 distributors in Uganda, of which only 83 had reached a managerial level and begun to recoup expenses; their investigation concluded that Forever Living Products' "distribution system does not guarantee profits and majority of members drop out along the way, after losing millions."[14] The company was active in over 165 countries as of 2018.[1] In February 2015, the company announced they had appointed a new management team to "oversee the affairs of the company in Nigeria."[15]

In January 2023, the company named Aidan O'Hare as President with Gregg Maughan, son of founder Rex Maughan, continuing as CEO.[16]

Legal
In 1996, upon suggestion of the American authorities, the Internal Revenue Service (IRS) and the National Tax Agency of Japan (NTA) initiated a joint audit of Rex and Ruth Maughan and related entities Aloe Vera of America (AVA), Selective Art Inc., FLP International, and FLP Japan for the period of 1991 to 1995.[17] In 1997, the NTA imposed a penalty tax of 3.5 billion yen on Forever Living's Japan division for concealing income of 7.7 billion yen over the five-year period.[18] Later that year, AVA, Rex and Ruth Maughan, Maughan Holdings, Gene Yamagata, and Yamagata Holdings[19] sued the IRS for unauthorized disclosure of tax return information.[18] In the midst of the lawsuit, The IRS asked the NTA to drop its decision against Forever Living, and in 2002, the agency "grudgingly complied with the IRS's request", announcing that the penalty tax had been effectively withdrawn.[18] In February 2015, a US district court ruled that the IRS knowingly provided some false information about AVA to the NTA, in violation of the United States' tax treaty with Japan[20] and awarded three of the plaintiffs one thousand dollars each in statutory damages.[20]

In 2004, claims made about Forever Living products were found to be in violation of several laws in Hungary related to advertising, registration of nutritional products, and the use of cosmetics as medicinal agents. As a result, the company was fined 60 million HUF (approximately US$280,000).[21]

In 2007, author Richard Bach made claims against the company for copyright infringement and trademark infringement.[22] The lawsuit stated that for over 20 years Forever Living had used the character, storyline, and copyrighted excerpts from the novel Jonathan Livingston Seagull to promote its marketing plan, and also used the motion picture and novel as its corporate logo.[23][24] The claim was satisfied through arbitration,[25] and shortly after, Forever Living changed its company logo[26] from a seagull to an eagle.[27]

In 2015, Forever Living was criticized by the UK Advertising Standards Authority for making false claims about the health benefits of its products, which were sold as a cure for various diseases ranging from diabetes to Crohn's disease. The company was also warned not to use health professionals in its promotional materials. Subsequently, the UK Medicines And Healthcare Products Regulatory Agency launched an investigation after it was revealed that NHS staff were moonlighting as sales people.[28][29][30]

Between 2012 and 2016, several lawsuits were initiated by the Environmental Research Center (ERC) against Forever Living Products over alleged violations of Proposition 65 (the Safe Drinking Water and Toxic Exposure Act).[31] The ERC accused Forever Living Products of having unsafe levels of lead in products manufactured by Aloe Vera of America and distributed by Forever Living.[32] The claims were determined to be unfounded and the case was dismissed in the California Supreme Court in October, 2016.[33]

contact: 09025086481
E-mail: mikeoparaugo@yahoo.com