law School

and also to the formula

(x) (Hx. Fx)


A typical E proposition such as "No
human are fallible." can be stated successively as


Given any individual thing whatever, if It
is human then it is not fallible
Given any x, x
is human implies x is not fallible
and finally as


(x) (Hx implies Fx)

We know
already that an E proposition is contradiction by
the | proposition. 50 "No humans are fallible" is
denied by "Some humans are fallible." Translating
the latter into our symbolic notation we have
successively.

There is at least one thing
that is human and fallible
There is at least
once x such that x is human x is fallible
and
the process ends with

(x) (Hx. Fx)


Logicians utilize the Greek letters phi
(phi) and psi (psi) to represent whatever
predicates that may occur in the traditional
subject - predicate propositions. With phi
replacing the predicate that occurs before the
logical operations. With phi replacing the
replacing the predicate that occurs before the
logical operations. 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) (phi x implies
psi x)
0 ........ (x) (phi x. psi)
E ...... (x) (phi
x implies psi x)
| ...... (x) (phi x) (phi x. Psi x)


Arguments Containing Quantified
Propositions

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.
|n this section, we
shall introduce these additional rules one by one,
and exemplify how they are applied in the
appropriate sleogistic arguments.
The first of
the rules to be discussed is the principle of Universal Instantiation,abbreviated as UI. The principle asserts that any substitution

Solution:


1. (x) (Mx implies /K)

2. Mc /:/c
3. Mc implies /c 1, Ul
4. /c
3, 2, M. P.


The second rule is called
Universal Generalization, often rendered as UG.

Unlike U| where the Inference proceeds from a
universally quantified propositional function to its
Instantiation, the Inference involved in UG
proceeds from an arbitrary selected individual to a
universally quantified propositional function. We
state the principle of U6 thus: From the
substitution instance of

https://www.imgonline.com.ua/result_img/imgonline-com-ua-demotivatorGJ9ljiXjGVX.jpg

https://shorturl.at/V9hRA

:(x)(phix) (where y reprsents any arbitrarily chosen individual)

An argument Which can be proved valid by using the principles of UI

and UG in the following:

All politicians are hypocrites
No martyrs are hypocrites
Therefore, no martyrs are politicians

We may construct its formal proof of validity as:

l. (x) (Px c Hx)

1. (x)(Mx c ~Hx) /.'. (x)(Mx c ~Px)
2. Py c Hy 1,UI
3. My c ~Py 2,UI
4. ~Hy c ~Py 3,Trans
5. My c ~Py 4,5,H.S.
6. (x)(Mx c ~Px) 6. UG.

Before we discuss the next quantification principle,a few explanatory remarks on the proper application of UI and UG are necessary. The principle of UG allows us to infer (x) phi x from any arbitrarily selected individual phi y. How, you may ask, is this inference justified? Recall that if a particular characteristic or attribute is true of all members of a given class, it is true for each and every of that same class. Now, assuming that k is a member of a certain class n and it is shown that the attribute h belongs to k solely by virtue of the fact that k has the class defining attribute for n, it follows too that if each and every member of n has h, then from any arbitrarily selected member of n one can validly infer that all members of n as a whole have the attribute h. This kind of reasoning is regularly utilized by mathematicians, scientists, and logicians. Thus a mathematician can prove that all parallelograms have a particular property on the ground that an arbitrarily chosen parallelogram PQRS, because it is a parallelogram, has that very property. One might say that the above reasoning is based on the "parallelogramness" of PQRS.
In order to construct a formal proof of validity for an argument containing a universally quantified propositional function as the conclusion, the principle of UI must be applied in order to get a line in the proof containing the symbol y. For it is only when the symbol for the name of any arbitrarily selected individual appears in a proof that the desired conclusion can be inferred through the principle of UG. The argument "All drunkards are neurotics, No neurotics are philosophers. So no philosophers are drunkards", cannot be proved valid unless the symbol y appears in the proof:

1. (x)(Dx c Nx)
2. (x)(Nx c ~Px) /: (x)(Px c ~Dx)
3. Dy c Ny 1,UI
4. Ny c ~Ny 2,UI
5. Py c Ny 4,Trans
6. ~Ny c ~Dy 3,Trans
7. Py ~Dy 5,6.H.S.
8. (x)(Px c ~Dx) 7,UG Inspecting the proof we have just constructed it can be observed that y occurred for the first time in line 3, and again in line 4. It has to be so because unless a line containing y occurs in our proof, the argument's conclusion can not be inferred from its premisses. Additionally, the proposition (x)(Dx c Nx) is a non compound proposition whereas the statement Dy c Ny is compound, since it is a conditional. No logical principle allows us to infer anything from non-compound propositions (x)(Dx c Nx) and (x)(Nx c ~Ix). However, from the compound propositions Iy c ~Ny and ~Ny c Dy the proposition Iy c Dy follows by the rule of hypothetical syllogism. It is only when the latter proposition has been deduced that the conclusion itself could be inferred using the principle of UG. The principle of UI permits the logician to reason from noncompound proposition to compound ones to which the rules of inference can be meaningfully applied. In any proof of formal validity for an argument whose conclusion is a universally quantified proposition, the principle of UG last so that the conclusion may be deduced from the penultimate line containing y. Not all arguments contain universal propositions as premisses and conclusion. The argument "All Pentecostal pastors are demagogues. Some Pentecostal pastors are charlatans. Therefore, Some charlatans are demagogues", can be used to introduce the next rule of inference. It is symbolical expressed as: (x)(Px c Dx) (Ex)(Px . Cx) : (Ex)(Cx . Dx) The principle of Existencial Instantiation asserts that: From the existential quantification of a propositional function one can deduced the truth of its substitution instance with regard to any individual constant apart from y that does not have any previous occurence in the proof. The principle permits us to reason validly from (Ex)(phi x) (where v stands for any individual apart from y that has no previous occurence in the context of the proof) to phi v Our argument about pastors, charlatans and demagogues can be proved valid without any difficulty: