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[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.
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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 (∴). 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 /∴(AvC)
4. ~~(AvC)   I.P
5. AvC   4,D.N.
BvD   1,5,C.D.
7. E   2,6,M.P.
8. E.~E   7,3,Conj
Example 5
1. (WvX)⊃(~Z⊃Y)
2. (ZvU)⊃(W.Y)/∴Z
3. Z   I.P
4. ZvU   3,Add
5. W.Y   2,4,M.P.
6. (WvX)⊃(Y⊃Z)  Trans
7. W    5,Simp
8. WvX   7,Add.
9. Y⊃Z   6,8,MP.
10. Y.W   5,Com.
11. Y   10,Simp.
12. Z   9,11,M.P.
13. Z.~Z   12,3,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    B    
An individual with different predicates & nbsp  Attribute 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
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.
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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 (∴). 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 /∴(AvC)
4. ~~(AvC)   I.P
5. AvC   4,D.N.
BvD   1,5,C.D.
7. E   2,6,M.P.
8. E.~E   7,3,Conj
Example 5
1. (WvX)⊃(~Z⊃Y)
2. (ZvU)⊃(W.Y)/∴Z
3. Z   I.P
4. ZvU   3,Add
5. W.Y   2,4,M.P.
6. (WvX)⊃(Y⊃Z)  Trans
7. W    5,Simp
8. WvX   7,Add.
9. Y⊃Z   6,8,MP.
10. Y.W   5,Com.
11. Y   10,Simp.
12. Z   9,11,M.P.
13. Z.~Z   12,3,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    B    
An individual with different predicates & nbsp  Attribute 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:
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