Saturday, November 20, 2010

Lesson3-Special forms, syllogisms


LESSON 3 E
INDUCTION AND OTHER FORMS OF SYLLOGISMS
Induction is a method of inference that proceeds from the relationships of particular truths towards a universal truth.
In reality, human knowledge started from the inductive process since what man experiences are particular realities and man will have to grasp first the particular truths of reality before he can abstract these to universal truths. From these universal truths man is able to further improve his knowledge of the particular as deduced from universal truths and principles.
The conclusions of induction are less probable and are a far cry from the certainty of the conclusions of deduction. However, the synthesis of the different human experiences and the body of human knowledge that has evolved and survived the test of time lends credit to the validity of the inductive process. This was further reinforced by the conclusions of deductive inferential thinking that validated and was often intertwined and has complemented the conclusions of inductive inferential thinking.

Type of Analogy Tests

  1. Single Approach. Example: Fish: Shoal :: Swine:________
  2. Paired Approach. Example: Gynophobia: Women::_________:___________
  3. Elimination Approach. Example: Eyes, Ears, Mouth, Nose, Feet
  4. Number Series or Completion Approach. Example: Three, Twelve, Forty-eight, _________
  5. Abstract Reasoning Approach. Example:  ABCD, BCDE, CDEF, DEFG,__________
Note: The Key in answering analogy test is to determine the relationships that exist between and among the items and to complete the sentence.
Examples: A group of fish is a shoal as a group of swine is a drift. The answer to #1 is Drift
              Eyes, ears, mouth and nose are found in the head while feet is not. Feet is the answer to # 3.

Types of Induction
1 Essential Induction when the mind grasps in experience the necessary link between a subject and its property. It makes use of formal principles, which are so clear that they do not need any proof for they are self-explanatory and self-evident truths. These principles are:
1. The principle of Identity. Everything is itself.
     Ex.: A school is a School.
2. The principle of excluded middle. A thing either is or is not.
     Ex.: A school is either a school or not a school
                        3.  The principle of non-contradiction. Nothing can be and not be at the same
time or respect.
     Ex.: A school cannot be a school and not a school at 
           the same time.
4. The principle of sufficient reason. Everything that exist has sufficient reason 
            for its existence.
      Ex.: A school is a place of learning.

2. Empirical Induction is the generalization of the connection between the subject and the predicate based not on the essential link between them but on the repeated occurrence of the observed phenomenon.
2 kinds of Empirical Induction
1.      Complete or perfect induction. The generalization rest on the knowledge of each instance covered. This is otherwise known as the induction by simple enumeration because it is simply the summation of all individual cases observed.
Example: Since Dr. Marin, Dr. Garcia , Prof. Manad, Prof. Gonzales and the rest of the faculty members of PNU are Master’s degree holders, We can conclude that all PNU teachers are master’s degree holders.
2.      Incomplete induction. The conclusion takes the instances as a sample of the class and generalizes from the properties of the sample to the properties of the class.
Example:  Since 80 % of survey respondents said that they prefer LRT to Jeepney’s as mode of transportation, we conclude that LRT is preferred as a mode of transportation by the public.

Examples of Incomplete Induction

·        Analogy is a form of induction, which seeks to establish a conclusion on something that is yet unknown to a thing based on similarities. Example: Peso is to Philippines as dollar is to United States of America.

·        Generalization is a form of induction that seeks to establish a conclusion about a whole group or population based on some samples. Example: We conclude that Filipinos are hospitable based on the result of the survey.

·        Causal Relation is a form of induction, which seeks to establish a conclusion based on the connection between cause and effect. Example: Based on our past experiences, Metro Manila becomes flooded whenever it is hit by a typhoon.

OTHER FORMS OF SYLLOGISMS
The five special forms of syllogisms are Enthememes, Epichiremes, Polysyllogisms, Sorites and Dilemmas.
1.ENTHYMEME
            Enthymemes are shortened forms of syllogism in which one of the premises or the conclusion is not expressed but implied.
            3 kinds of Enthymeme
1.      Enthymeme of the first order- the major premise is omitted
Example: Peter is free because he has a will
The syllogism is as follows:
 (All beings that have a will are free)
 But Peter has a will
Therefore, Peter is free
2.      Enthymeme of the second order- where the minor premise is omitted.
Example: Since all training programs are contributors to
human development then care-giving  is a contributor to human development.
                   
The syllogism is as follows:
All training programs are contributors to human development
                    (But care giving is a training program)
  Therefore, care giving is a contributor to human  
  development
3.      Enthymeme of the third order- where the conclusion is omitted
Example: All intelligent beings are beings that have
  morality and man is an intelligent   being
The complete syllogism is as follows:
All intelligent beings are beings that have morality
But man is an intelligent being;
 (Therefore, man is a being that has morality)
2. EPICHIREME.
Epichireme is another form of syllogism wherein a proof is joined to one or both premises. The proof is an explanation of the given premise. It is normally connected to the premise by such causal clauses as for, because, since, and insofar as. An epichireme makes use of an enthymeme for its premise thus there are actually more than one argument in an epichireme.
Two types of Epichireme
1.      Simple Epichireme- where one of the premises is accompanied by a proof or an explanation
Example: All human beings are rational
 because all men have a soul
               But doctors are human beings;
               Therefore, doctors are rational.
2.      Compound Epichireme- where both of the premises are accompanied by proofs or explanations. Actually the syllogism is made up of three arguments; the main syllogism and the two enthymemes that act as premises.
Example: All research works are beneficial to students
 because students learn to look for primary   sources of information
But all things beneficial to students are the   tasks of education
 since education is for total human development of 
 the students
                               therefore, some tasks of education are research
       works.



3.  POLYSYLLOGISM
             A Polysyllogism is a series of syllogism that are so related that the conclusion of the first becomes the major premise of the second syllogism and so on. When there are only two syllogism that are connected as such, it is known as Episyllogism.
            Example: 1. Every human being is a rational being;
                               But students are human beings;
                           2. therefore, students are rational beings
                               But All rational beings are free
                           3.Therefore, some free beings are students;
                              No slaves are free beings;
                              Therefore, some slaves are not students.
4. SORITES
            A Sorite is a shortened polysyllogism wherein the conclusions are suspended or not stated except the last.
            2 kinds of Sorites
1.      Aristotelian Sorites- a sylogism that uses the predicate of the preceding premise as the subject of the next premise and so on and so forth.
Example: All good students are good learners
                  But all good learners are rational beings
                  But all rational beings are responsible beings
                  But all responsible beings perform are good
performers of their civic duties;
                        But all good performers of their civic duties are
good citizens;
                        Therefore, all good students are good citizens.
2.      Goclenian Sorites. This employs the opposite form of the Aristotilian sorite. The subject of the preceding premise becomes the predicate of the next premise and so forth.
Example:    All responsible teachers are role models;
                  But effective teachers are responsible teachers;
                  But all efficient teachers are effective teachers;
                  But all awardees are efficient teachers;
                  Therefore, all awardees are role models.
Rules for the validity of Sorites
1.      All the premises must be universal except for the first. If the first premise is particular the conclusion must be particular.
2.      All the premises must be affirmative except for the last. If the last premise is negative, the conclusion should be negative.
3.      There must be sequence between and among the premises.

5. DILEMMA
A dilemma is a horned argument. This is an argument that gives two alternatives or choices both of which are detrimental to the one who will make the choice. The structure of the dilemma is so arranged that both alternatives will result into a conclusion that is both unfavorable to the one choosing thus pinning him down.
The major premise is a conditional hypothetical proposition and the minor premise is a disjunctive proposition or vice versa. The conclusion is either a categorical or a hypothetical proposition.
Kinds of Dilemma
1.      Simple constructive dilemma- where the conclusion is a simple and an affirmative proposition.
Example: Either, you give a report or you improve your
extra- curricular involvement;
              if  you give a report you’ll gain additional credit.
              if  you improve your extra- curricular activities
       you’ll gain additional credit.
                               Therefore, in any case, you’ll earn additional credit.

2.      Compound constructive dilemma-where the conclusion is a compound and an affirmative proposition.
Example:
    If I continue my studies then I will finish my degree.
                             and If I work then I will earn money to support my family.
   But either I continue my studies or work;
  Therefore, either I will finish my studies or I will earn money to support my family.

3.      Simple destructive dilemma- where the conclusion is a simple and negative proposition.
Example: The man trapped at the 20th floor of a burning
                                   building will either not jump or he will not stay.
             If the man doesn’t jump, he will not survive because of the fire;
             If the man doesn’t stays, he will not survive because of the fall;
            Therefore, the man will not survive.
4.      Compound destructive dilemma-where the conclusion is a compound and negative proposition.
Example:
      In politic, if I’ll be honest then my colleagues will not like me
                        If I’ll be corrupt then people will not vote for me again;
                        But in politics, either I’ll be honest or I’ll be corrupt
                        Therefore, either my colleagues will not like me or the
people will not vote for me again.

Rules for a Valid Dilemma
1.       All the possible alternatives of the disjunction must be stated.
2.       The consequent of the conditional proposition must necessarily follow from the antecedents. There must be a valid sequence.
3.       Present alternatives that cannot be taken in two different respects, that is, it may not be interpreted both in its positive and negative perspective.



Ways to Defeat a Dilemma
1.      Going between the horns. Point out that there is another alternative not presented in the disjunction and its consequent is different or opposite the stated consequent. Or point out that there is another possible consequent for the given antecedent.
Example: Either you concentrate on relationship or on task
 If you concentrate on relationship then your efficiency will suffer.
      If you concentrate on task then your social life will suffer.
Thus, either your efficiency will suffer or your social life will suffer

This dilemma may be countered by pointing out that a balance between relationship and task will improve both your efficiency and social life.

2.      Grasping the dilemma by the horn. Point out that there is no sequence between the antecedent and the consequent. Or point out that the opposite of the consequent can also be the consequent of the given antecedent.
Example: Either I give the students a lecture or give them activities
              But If I give the students a lecture, I will be wasting
 time since they normally get bored and don’t listen.
                                If I give them activities, I will be wasting time for
they normally don’t cooperate and participate.
                                Therefore, in either case, I will be wasting time.
This dilemma may be countered by pointing out that “If I give a lecture then my students will learn and if I give activities, students will learn. So, in any case it will not be wasting time because students will learn”.
3.      Rebutting the dilemma. Point out an opposite conclusion in a counter-dilemma by reversing the original dilemma. If the perspective is negative, use a positive perspective and vice versa in a counter -dilemma.

Example:  If I tell a lie my teachers will be angry with me.
                If I tell the truth my classmates will be angry with me
                                  But either I tell a lie or l tell the truth
                                  In any case people will be angry with me.
                   Rebuttal: If I tell a lie my classmates will not be angry with me
                                If I tell the truth, my teachers will not be angry with me
                                But either I tell a lie or I tell the truth;
                                In any case, people will not be angry with me.
                       

Lesson3- interference

LESSON 3 d
INFERENCES
Inference is a process by which a conclusion is drawn from a valid sequence and relationship of premises.

Kinds of inference
1.      Inductive inference. A form of reasoning that proceeds from particular premises to a general conclusion.
2.      Deductive Inference. A form of reasoning that proceeds from universal /general premises to a particular conclusion.
Types of Induction
1.      Essential Induction when the mind grasps in experience the necessary link between a subject and its property. This makes use of formal principles, which are so clear that they do not need any proof for they are self-explanatory and self-evident truths. These principles are:
1.  The principle of Identity. Everything is itself.
     Ex.: A school is a School.
2. The principle of excluded middle. A thing either is or is not.
     Ex.: A school is either a school or not a school
            3. The principle of non-contradiction. Nothing can be and not
                     be at the same time or respect.
    Ex.: A school cannot be a school and not a school at 
           the same time.
4. The principle of sufficient reason. Everything that exist has sufficient reason for its existence.
     Ex.: A school is a place of learning.

2. Empirical Induction is the generalization of the connection between the subject and the predicate based not on the essential link between them but on the repeated occurrence of the observed phenomenon.

2 kinds of Empirical Induction
1.   Complete or perfect induction. The generalization rest on the knowledge of each instance covered. This is otherwise known as the induction by simple enumeration because it is simply the summation of all individual cases observed.
Example: Since Dr. Marin, Dr. Garcia, Prof. Manad, Prof. Gonzales and the rest of the faculty members of PNU are Master’s degree holders, We can conclude that all PNU teachers are master’s degree holders.
2.   Incomplete induction. The conclusion takes the instances as a sample of the class and generalizes from the properties of the sample to the properties of the class.
Example:  Since 80 % of survey respondents said that they preferred LRT to Jeepny as mode of transportation, we conclude that LRT is preferred as a mode of transportation by the public.

1.      Deductive Inference. A form of reasoning that proceeds from universal /general premises to a particular conclusion.

1. Immediate Inference is a kind of inference by which the mind directly draws the implication of one proposition to arrive at a new proposition without the use of a medium.
    It is the process of drawing conclusions from the implication of a statement and its Opposing categorical statements. It does not need a third statement to mediate or to connect the statements and infer from them. It is also the process of restating the same judgment in different forms.

Kinds of Immediate Inferences
1.      Eduction or logical Equivalence- the process of creating a new proposition that conveys the same meaning.
2.      Oppositional Inference- the process of establishing the relationship of propositions having the same subject and predicate but different qualities and or quantities. The truth or falsity of an opposite proposition is inferred from the truth or falsity of a given proposition.
 
2. Mediate Inference is the process of drawing new proposition from the relationship of two propositions that are related through a medium or a middle term. Inferences are externally manifested through a syllogism.
Syllogisms are verbal expressions of arguments that are products of reasoning or inferential thinking.
2 kinds of Syllogisms
1.      Categorical Syllogism is a syllogism that is made up of categorical propositions
Example: All books are tools for learning;
                    But the dictionary is a book;
                    Therefore,the dictionary is a tool for learning.

2.      Hypothetical Syllogism is a syllogism wherein at least one proposition is a hypothetical proposition.
Example: If all books are tools for learning then they must be valued.
                                        But all books are tools for learning;
                           Therefore, they must be valued.

Validity of Inference
Let us analyze how we arrived at these. To illustrate all these, let us consider the table below:
 
The distinction between truth and validity is the fundamental distinction of formal logic. You cannot understand inference unless this distinction is clear and familiar to you.
The seven sample syllogisms above show the general principles of inferential thinking:
·        True premises do not guarantee validity. As proven by examples #1 and #3 in the table above.
·        A true conclusion does not guarantee validity. As proven by examples #3 and #7.
·        True premises and a true conclusion together do not guarantee validity. As proven by example #3.
·        Valid reasoning does not guarantee a true conclusion. As proven by example #4.
·        False premises do not guarantee invalidity. As proven by examples #4 and #6.
·        A false conclusion does not guarantee invalidity. As proven by example #4.
·        False premises and a false conclusion together do not guarantee invalidity. As proven by example #4.
·        Invalid reasoning does not guarantee a false conclusion. As proven by examples  #3 and #5.
Therefore, while the truth of propositions and the validity of reasoning are distinct, the relationship between them is not entirely straightforward. We cannot say that truth and validity are utterly independent because the impossibility of "case zero" (a valid argument with true premises and false conclusion) shows that one combination of truth-values is an absolute bar to validity. When an argument has true premises and a false conclusion, it must be invalid. In fact, this is how invalidity is defined..
Thus an argument is sound if (and only if) all its premises are true and its reasoning is valid; all others are unsound. It follows that all sound arguments have true conclusions.
The Validity and soundness of categorical syllogisms and of Hypothetical Syllogisms will be further discussed in the succeeding chapters.

OPPOSITION AND EDUCTION: IMMEDIATE INFERENCES
Immediate Inference is a kind of inference by which the mind directly draws the implication of one proposition to arrive at a new proposition without the use of a medium or a third idea.
Types of Immediate Inference
1. Oppositional inference is a type of immediate inference that proceeds from the relationship between propositions with the same subject and predicate but different quantities and/or qualities. Oppositional inference is the process of determining the truth or falsity of an opposite statement from the truth or falsity of a given statement.
2. Eduction or Logical Equivalence is the process of reformulating one proposition to another kind without changing the meaning. . The resulting proposition may have different quantities, qualities, subjects and predicates but the meaning is essentially the same.
The Square of Oppositions
The square of opposition is a table in which the four opposing propositions are fitted to each other. By using this table one can easily see and determine the truth-value of opposing propositions. It is therefore, a useful tool in oppositional inference.
Types of Oppositions of Categorical Propositions
            Categorical Propositions are statements of direct claim of relationship or non-relationship. They are either true or false. They are classified into four types, namely A, E, I, O  propositions. In chapter 4, we have learned the following:
·        Those propositions that are Universal/Singular in quantity and affirmative in quality are known as A Propositions
ex. All men are rational; Cathy is a student; Every citizen is free to own properties
·        Those that are Universal/ Singular in quantity and negative in quality are known as E propositions.
Ex.   No man is an island;   All students are not teachers;  Every dog is not a cat.
·        Those that are particular in quantity and affirmative in quality are known as I propositions.
Ex.  Some politicians are liars; Many countries are for peace; Filipinos are Christians.
·        Those that are particular in quantity and negative in quality are known as O propositions.
Ex.  Some civilians are not responsible people; Many are not in favor of war; Not all Filipinos are not rich. 
These four types of categorical propositions are actually opposed to each other either as contradictories, contraries, sub-contraries, or sub-alternates.
  1. Contradictory Opposition exists between two statements of different qualities and quantities. Contradictory propositions cannot be both true and cannot be both false. If one is true then the other is false and if one is false the other is true. A & O Propositions and E & I Propositions are contradictories.
  2. Contrary Opposition exists between two statements of both universal quantities but of different qualities, that is, one is affirmative and the other is negative. Contrary propositions may be both false but only one can be false. So if one is true the other is false but if one is false the other is doubtful. . A and E propositions are contraries
  3. Sub-Contrary Opposition exists between two statements that are both Particular in quantities but of different qualities. Sub-Contrary propositions may be both true but only one can be false. So if one is true the other is doubtful but if one is false the other is true. I and O Propositions are sub-contraries.
  4. Sub- Alternate Opposition exists between two propositions that are of the same quality but different in quantities, that is, one is universal and the other is particular. Sub- alternates may be both true and maybe both false. If the Universal proposition is true the particular proposition is true and if the particular proposition is false the universal proposition is false but if the particular proposition is true the universal proposition is doubtful and if the universal proposition is false the particular proposition is doubtful. A & I propositions and E and O Propositions are sub-alternates.

If one is to apply these rules in analyzing the relationships that exist between and among the four types of categorical propositions, then one can immediately infer that:

·        if  A is true, then E is false, I is true, and O is false
·         if A is false, then E is doubtful, I is doubtful and O is true
·        If E is true then A is false, I is false and O is true.
·        If E is false then A is doubtful, I is true and O is doubtful
·        If I is true then A is doubtful, E is false and O is doubtful.
·        If I is false then A is false, E is true and O is true.
·        If O is true then A is false, E is doubtful and I is doubtful.
·        If O is false then A is true, E is false, and I is true.

To have a visual illustration of these inferences, please remember the square of opposition.  Recall that on the upper left hand corner is found the A proposition while on the upper right hand corner is found the E proposition. Recall further that on the lower left hand corner is found the I proposition while on the lower right hand corner is found the O proposition. If we now apply the rules of opposition, we will have the table below 
Truth-Value Table of Opposing Propositions

 
Oppositions of Modal Propositions
Modal propositions are propositions that tell how or in what manner does the predicate affirms or denies the subject.
           
There are four kinds of modal propositions: Necessary, contingent, possible, and impossible.
1.      Necessary. Necessary relationship is not possible to deny because it
cannot be otherwise.
            Examples: A square has four sides
                               The square root of four is two
                               A surgeon is a doctor
2.      Contingent. When the relationship expressed by the proposition is    
        one that is but not necessarily so. 
            Example:  The square is blue
                               The students are wearing uniforms.
                               The principal is a Doctor of Philosophy
3.      Possible. The relationship in this proposition signify something that is not, but may eventually be.
            Example: The student may be the head nurse by 2010.
                            You may be successful someday.
                              It’s possible to have women priest in the Philippines.
4.      Impossible. The relationship in this proposition is something that
        cannot be and will never be.
            Example: A square cannot have six sides
                              A man can never be a woman.
                             It is impossible for a one year old baby to be a mother
             
The rules applied in these oppositions are the same as the rules that govern the oppositions of the four types of propositions, A- proposition is to necessary, I- proposition is to possible, E- proposition is to impossible and O- proposition is to contingent. It follows therefore that:

·        If Necessary is true, then Possible is true, Impossible is false and Contingent is false
·        If Necessary is false, then Possible is doubtful, Impossible is doubtful and contingent is true
·        If Impossible is true then necessary is false, possible is false and contingent is true,
·        If Impossible is false then Necessary is doubtful, possible is true and contingent is doubtful.
·        If Possible is true then Necessary is doubtful, Impossible is false and contingent is doubtful.
·        If possible is false then Necessary is false, Impossible is true and contingent is true
·        If contingent is true then Necessary is false, Impossible is doubtful and possible is doubtful.
·        If contingent is false then Necessary is true, Impossible is false and possible is true.

Oppositions of Circumstantially Quantified Propositions

Circumstantially quantified propositions are those propositions affected by some contingency of time, place, or circumstances. They are actually A, E, I, O propositions quantified by the use of always, under
all circumstances, in all instance, everywhere, by all means in the A propositions. The E propositions are quantified by never, nowhere, by no means, under no circumstance. The I propositions are quantified by occasionally, sometimes, somewhere, under some circumstances and the O propositions are quantified by sometimes not, not always, occasionally not. The rules applied in these oppositions are the same as the rules applied in the oppositions of the four types of propositions and that of modal propositions

Example: If it is necessary for the patient to be operated on in order to survive. Then the operation is possible but not contingent and impossible.

 
2. Eduction or Logical Equivalence is the process of restating the same meaning of proposition using different forms. The result of the inference is a new proposition but it has the same meaning and truth- value as the given.
Types of Eduction or Logical Equivalence
1. Conversion is the process of interchanging the position of the subject and predicate terms without extending any term and without changing the quality of the proposition. The original proposition is reformulated by interchanging the subject and the predicate while maintaining the quantities of the terms. The original proposition is called the convertend while the resulting proposition is called the converse.
Ex. All computers are expensive =Some expensive things are computers.
 A to I. Partial Conversion
      Man is a rational Animal     =A rational animal is a man. A to A full conversion
     No terrorist is  a peace lover= No peace lover is a terrorist. E to E conversion


Note : An O proposition cannot be validly converted because it will extend a term from  a particular to a universal extension or  quantity


Practice your conversion skill:
1.      No man is an Island =_____________________________________
2.      Some patients are restless=_________________________________
3.      All operating rooms are hygienic=___________________________

2. Obversion is the process of expressing an affirmative proposition in a negative manner or a negative proposition in the affirmative manner. It involves the changing of the copula and contradicting the original predicate. It is the process in which a new proposition is formulated by changing the quality of the original proposition and contradicting the original predicate.
Ex.: All trees are useful = All trees are not non –useful. A to E obversion
       Some actions are not good = Some actions are bad. O to I obversion
 
Practice your obversion skill
1.      All babies are special =________________________________
2.      Some children are malnourish =_________________________
3.      All rooms are not occupied =___________________________


3. Contraposition is the process of interchanging and contradicting the original subject and predicate terms. It obverts, then converts then obverts the original proposition. It is the process of formulating a new proposition by getting the obverse of the converse of the obverse. The original subject and predicate are contradicted and interchanged but the quality is retained.

Contraposition Process
Contraponend      =  All nurses are health providers
Step 1 : Obvert    =  All nurses are not non-health providers
Step 2 : Convert  =  All non-health providers are not nurses  = Partial Contraposit
Step 3 : Obvert    =  All non-health providers are non-nurses= Full Contraposit

Ex. All Nurses are role models = All non- role models are non-nurses
      Every virus is not a bacterium = Some non-bacteria are not non-virus.

Practice your contraposition skill
1.      Many information were not validated=____________________________
2.      Any bleeding is a symptom=_____________________________________
3.     No late comers were admitted=__________________________________


4. Inversion is the process of changing the quantity of a proposition and contradicting the original subject and predicate terms. It proceeds by obverting and converting the contraposit of a universal proposition. It is the process of changing the quantity of the original proposition and contradicting both subject and predicate and retaining the quality of the original proposition.

     Invertend     = No X are Y                                       = All X are Y
       Step 1: Obvert  =E Propositions start with step 2   = All X are not non-Y
       Step 2: Convert= No Y is X or All Y are not X       = All non-Y are not X
       Step 3: Obvert  = All Y are non-X                          = All non-Y are non-X
       Step 4: Convert= Some non-X are Y =Partial        = Some non-X are non-Y =Full
       Step 5: Obvert = Some non-X are not non-Y =Full = Some non-X are not Y 
=Partial

Ex. No vice is virtue     =  Some non-vices are non-virtues. E to O Inversion
All knights are brave   = Some non- knights are non-brave. A to I inversion



Practice your inversion skill
1. No nurse is a liar=______________________­­­­­­­­­­­­­____________________________

2. Any hospital administrator is a management expert________________________________________________________
              
3. All dengue victims are to be admitted immediately_________________________________________________________



Through logical equivalence or eduction one may express a proposition in different forms without significantly altering its meaning. An A-proposition may be expressed as an E or an I or an O proposition. Great care, however, should be exercised when logical equivalence is combined with opposition for the meaning may not be that so evident. The table above may be of help to distinguish the truth value of any statement which is the logical equivalent of the opposite of a given statement

Example: If the given statement “ All nurses are registered professionals” is true then;

1.   Some registered professionals are nurses” is likewise true because it is the Converse of the original statement- a kind of logical equivalence
2. “All registered professionals are nurses” is doubtful because it is the subaltern of the converse of the original statement- a combination of opposition and logical equivalence
3. “All nurses are not unregistered professionals” is true because it is the Obverse of the original statement- a kind of logical equivalence.
4. “All nurses are unregistered” is false because it is the contrary of the obverse, of the original statement- a combination of opposition and logical equivalence
           
Another way of using the combination of opposition and logical equivalence is when one is asked to give a statement, which is logically equivalent of the opposite of any given statement.

Example:
1. Give the obverse of the sub-contrary of the contradictory of “All technicians are trainees.”
                        To answer this, one begins with the given statement and proceeds backward, thus;
                        1. The given statement is an A Proposition
2. The contradictory of A is an O-Proposition- Some technicians are not trainees
3. The sub-contrary of O is an I-Proposition- Some technicians are trainees
4. The obverse of I is an O- Some technicians are not non-trainees


Legend:
 Prop=   Type of Categorical Proposition         Con =     Converse                      Obv =   Obverse                
PC  =      Partial Contraposit                            FC   =   Full Contraposit            P I  =    Partial Inverse  
FI    =   Full Inverse   

            Thus if it is true that all technicians are trainees then some technicians are trainees will likewise be true, because it is the sub-altern of the original statement. And the obverse of the sub altern, some technicians are not non-trainees, will likewise be true because it is simply the logical equivalent of the sub-altern of the original statement.


CATEGORICAL AND HYPOTHETICAL SYLLOGISMS: MEDIATE INFERENCES
a. Categorical Syllogism
A Categorical Syllogism is a mediate inference that is made up of categorical propositions. The two propositions, called premises, are related through a middle term. These result into a third proposition, called conclusion, which flows necessarily from such relationship.


Basic Components of a Categorical Proposition
1. The Three Categorical Propositions
1.      Major Premise-the proposition of greater extension that is made up of the major term and the middle term. It is usually written as the first premise.
2.      Minor Premise- The proposition of lesser extension that is made up of the minor term and the middle term. It is usually written as the second premise.
3.      Conclusion- The proposition that flowed from the relationship of the two premises. This contains the inferred truth, which is a necessary implication of the two premises. It is made up of the minor and the major term and is usually written last.
Example:
                       Major Premise: All rational animals are mortal;
                       Minor Premise: But all men are rational animals;
                      Conclusion:     Therefore, all men are mortal.
2. The Three Terms
1.      The Major Term- is one of the terms found in the major premise and the predicate of the conclusion.
2.      The Minor Term- is one of the terms found in the minor premise and the subject of the conclusion.
3.      The Middle Term- is found in both major and minor premises but not in the conclusion. It is the term that mediates and connects the two premises. The common idea that allows the truth to flow from the premises to the conclusion.
Example:
          All health workers are professionals;
                But, some public servants are health workers;
                Therefore, some public servants are professionals.
                                    Major Term:     professionals
                                    Minor Term:     public servant
                        Middle Term:   health workers
Rules on the Validity of Categorical Syllogism 
1.      There must be three terms only; the major term, the minor term, and the middle term.
2.      Each of the terms must be used twice univocally.
3.      The middle term must be used in both premises and not in the conclusion.
4.      The middle term must be universal at least once.
5.      The Major and the minor terms may only be universal in the conclusion if they are universal in the premises
6.      Two affirmative proposition results into an affirmative conclusion.
7.      Two negative premises cannot have a valid conclusion
8.      One affirmative and one negative premise results into a negative conclusion.
9.      The premises must be universal at least once
10.   One universal and one particular premise results into a particular conclusion.
11.   Two particular premises cannot have a valid conclusion.
12.   Never claim something in the conclusion something that was not claimed and proven in the premises
Valid Moods of Categorical Syllogisms
Figure 1 : Barbara, Celarent, Darii, Ferio   or  AAA,EAE,AII,EIO.
Figure 2: Cesare, Camestres, Festino, Baroco  or EAE, AEE, EIO, AOO
Figure 3: Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison  or AAI, IAI, AII, EAO, OAO, EIO
Figure 4: Bramantip, Camenes, Dimaris, Fesapo, Fresison or AAI, AEE, IAI, EAO, EIO
(Note: The 1st vowel refers to the major premise, the 2nd to the minor premise and the 3rd to the conclusion)
Examples of the valid moods
            Figure 1: Subject: Predicate Middle Terms
1. BARBARA:
A- All X are Y;                                   A- All spiritual are immortal;
A- But all W are X;                           A- but all human souls are spiritual;
A- therefore, all W are Y                A- thus, all human souls are immortal

2. CELARENT
E- All X are not Y                                           E- All spiritual are not immortal
A- but all W are X                            A- but all human souls are spiritual;
E- All W are not Y                            E- Thus, all human souls are not immortal

3. DARII
A- All X are Y                                    A- All spiritual are immortal;
I- but some W are X                                        I- but some human souls are spiritual;
I- thus, some W are Y                                      I- thus, some human souls are immortal

4. FERIO
E- No X is Y;                                     E- No spiritual being is immortal;
I- but some W is X;                           I- but some human souls are spiritual;
O- thus, some W are Y                                  O- thus, some human souls are immortal
              
Figure II: Predicate: Predicate Middle Terms
1. CESARE
E- All Y  are not X                           E- Every Manual is not a newspaper
A- but all W are X                            A- but all Bulletin Today are newspapers
E- thus, All W are not Y                                 E-Thus, all Bulletin today are not manuals.

2. CAMESTRES
A- All Y are X                                    A- All manuals are newspapers
E- but no W is an X                          E- but no Bulletin Today is a newspaper
E- thus, no W is an Y                        E- therefore, no Bulletin Today is a manual

3. FESTINO
E- All Y are not X                            E- All manuals are not newspapers
I- but some W are X                         I- but some Bulletin Today are newspapers
O-thus, some W are Y                     O- thus, some Bulletin Today are not manuals


4. BAROCO
A- All Y are X                                    A- All manuals are newspapers
O-but some W are not X                                O- but some Bulletin Today are not newspapers
O- thus some W are not Y               O- thus, some Bulletin Today are not manuals

Figure III: Subject: Subject Middle Terms

1. DARAPTI
A- All X are Y                                    A- All transparencies are plastic
A-but all X are W                                            A-but all transparencies are instructional materials
I- thus, some W are Y                       I- thus, some instructional materials are plastic

2. DISAMIS
I- Some X are Y                                I- Some transparencies are plastic
A- but all X are W                            A- but all transparencies are instructional materials
I- thus, some W are Y                       I- thus, some instructional materials are plastic

3. DATISI
A- All X are Y                                   A- All transparencies are plastic
I- but some X are W                         I-but some transparencies are instructional materials
I- thus, some W are X                        I- thus, some instructional materials are plastic

4. FELAPTON
E- No X is Y                                      E- No transparencies are plastic
A- but all X are W                             A- but all transparencies are instructional materials
O- thus, some W are not Y              O- thus, some instructional materials are not plastic

5. BOCARDO
O- Some X are not Y                        O- Some transparencies are not plastic
A- but all X are W                             A- but all transparencies are instructional materials
O- thus, some W are not Y              O- thus, some instructional materials are not plastic.

6.  FERISON
E- All X are not Y                                           E- All transparencies are not plastic
I- but some X are W                        I- but some transparencies are instructional materials
O- thus, some W are not Y               O- thus, some instructional materials are not plastic

Figure IV: Predicate: Subject Middle Term
1.  BRAMANTIP
A- All Y are X                                    A- Education is an investment of human capital
A- but all X are W                            A- but all investments of human capital are precious
I- thus, some W are Y                       I- thus some precious things are education

2. CAMENES
A- All Y are X                                   A- Education is an investment of human capital
E- but no X is W                               E- but no investment of human capital is precious
E- thus, no W is Y                                           E- thus, no precious thing is education

3. DIMARIS
I- Some Y are X                                 I- Some Education are investment of human capital
A- but all X are W                            A- but all investments of human capital are precious
I- thus some W are Y                        I- thus, some precious things are education

4. FESAPO
E- No Y is X                                       E- No education is an investment of human capital
A- but all X are W                             A- but all investments of human capital are precious
O-thus, some W are not Y                O- thus, some precious things are not education

5. FRESISON
E- No Y is X                                      E- No education is an investment of human capital
I- but some X are W                         I- but some investments of human capital are precious
O- thus, some W are not Y              O- thus some precious things are not education.

a.      Hypothetical Syllogism

Rules for the Validity of Mixed Conditional Syllogism
1.      There must be sequence. The consequent must necessarily flow from the antecedent
2.      Posit or assert the truth of the antecedent in the minor, posit or assert the truth of the consequent in the conclusion.
3.      Sublate or deny the truth of the consequent in the minor premise, sublate or deny the truth of the antecedent in the conclusion.
4.      Posit or sublate completely, never partially.
5.      It is invalid to sublate the antecedent in the minor premise.
6.      It is invalid to posit the consequent in the minor premise
(Note: Mutually exclusive alternatives or sequential correlatives are exempted from rule 5 and 6)
Rules for the Validity of Disjunctive Syllogism
1.      If the disjunction is a strict disjunction or contradictory disjunction:
a.      If the Disjunction is Posit one alternative in the minor  premise then sublate the other in the conclusion
b.      Sublate one alternative in the minor premise then posit the other in the conclusion.
2.      contrary or a third alternative is implied:
a.      Posit one alternative in the minor premise then sublate the other or the rest in the conclusion.
b.      It is invalid to sublate one alternative in the minor premise then posit another in the conclusion.
3.      If the disjunction is sub-contrary or both alternatives could be true:
a.      Sublate one alternative in the minor premise then posit the other in the conclusion.
b.      It is invalid to posit one alternative in the minor premise then sublate the other in the conclusion.
Rules for the Validity of Conjunctive Syllogism
1.      Posit one alternative in the minor premise then sublate the other in the conclusion.
2.      It is invalid to sublate one alternative in the minor premise then posit the other in the conclusion.
Refuting a Syllogism and Making a Counter-Argument using Figures and Moods of Syllogisms
            Suppose somebody argued that nurses lack the love of country because they tend to prefer working abroad upon completion of their studies, how then will you offer a counter argument that will show that nurses have the love of country. The following steps may be employed
1.      Rewrite the argument into a syllogism of any of the valid moods in any figure
        Example:
 All who prefer working abroad are people who lack love of country
But nurses are people who prefer working abroad
Therefore, nurses are people who lack love of country
The syllogism above is in the 1st figure with an AAA mood or a Barbara, thus it is valid.
2.      Refute the argument by claiming that the syllogism is invalid (not applicable on this case because the sample syllogism is valid) or if it is valid, by showing that the major premise and/ or the minor premise/s  is/are false and thus the conclusion is false or at least doubtful.
 To do this, one must present evidence or show that there are people who prefer working abroad who still have love of country or that there are nurses who do not prefer working abroad.
Example 1:
All possible contributors to national development are not people who lack love of country
But All who prefer working abroad are possible contributors to national development
Therefore, All who prefer working abroad are not people who lack love of country

The conclusion of this syllogism is contrary to the original major premise and if this argument is sound then the major premise of the first argument must be false making the first argument unsound.
Example 2:

Fe, Martha and Andrew are not people who prefer working abroad
But Fe, Martha and Andrew are nurses
Therefore, Some nurses are not people who prefer working abroad

This syllogism is a valid 3rd figure EAO mood or a Felapton and its conclusion refutes the minor premise of the original argument.
3.      Offer a counter-argument with a contrary or contradictory conclusion. Since the conclusion in the first argument above is an  A-proposition, the counter argument that one may offer should have an O or E proposition for its conclusion.
Thus, one may advance the following syllogisms:

 All who prefer working abroad are not people who lack love of country
 But all nurses are people who prefer working abroad
 Therefore, all nurses are not people who lack love of country

The syllogism above is a valid 1st figure EAE mood, or a Celarent and its conclusion is the contrary of the conclusion of the first argument

     All who lack love of country are people who prefer working abroad
     But some nurses are not people who prefer working abroad
     Therefore, some nurses are not people who lack love of country

The syllogism above is a valid  3rd figure OAO mood, or a Bocardo and its conclusion is the contradictory of the first argument

 All who lack love of country are not people who prefer working abroad
 But some nurses are people who prefer working abroad
 Therefore, Some nurses are not people who lack love of country
           
The syllogism above is a valid 4th figure EIO mood or a Fresison and its conclusion is the contradictory of the first argument
4.      Show that the particular sub-alternate conclusion of the original universal conclusion is false. Thus if the argument is an AAA or Barbara, show that Darii is false. If one is able to do this, then he has proven that the conclusion of Barbara is likewise false using oppositional inference.

The conclusion of the first argument is “ All nurses are people who lack love of country” and its sub- alternate is “some nurses are people who lack love of country”. This can be shown false by offering a valid argument that has an E- conclusion that is “ All nurses are not people who lack love of country”

All people who lack love of country are not contributors to national development
      But all nurses are contributors to national development
 Therefore, all nurses are not people who lack love of country

The syllogism above is a valid 2nd figure EAE mood or a Cesare and if the major and the minor premises are true then the conclusion will be true and if this conclusion is true its contradictory “some nurses are people who lack love of country” will be false. Now, through oppositional inference, the conclusion “All nurses are people who lack love of country” can be inferred to be also false.