Logic

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Subalterns
Are A and I as well as E and O. They have the same quality but differ either in being partial instead of total or in being contingent instead of necessary.
Oppositions
Have the same matter (subject and predicate) but differ in form (quality, quantity, modality, or in two of these).
Contradictories
A (all S is/must be P) and O (some S is not/may not be P) as well as E (no S is/may be P) and I (some S is/may be P). They differ in quality and either quantity or modality. There is no middle ground between them.
Rule for contradictories
One must be true and the other false.
Contraries
Are A (all S is/must be P) and E (no S is/can be P). They differ in quality and are either total in quantity or necessary in modality. There is a middle ground between them.
Rule for contraries
Both cannot be true, but both may be false.
Subcontraries
Are I (some S is/may be P) and O (some S is not/may not be P). They differ in quality and are either partial in quantity or contingent in modality.
Rule for subcontraries
Both cannot be false, but both may be true.
Subalterns
Are A (all S is/must be P) and I (some S is/may be P) as well as E (no S is/can be P) and O (some S is not/may not be P). They have the same quality but differ in being partial instead of total or in being contingent instead of necessary.
Rule for subalterns
If the total or necessary proposition is true, the partial or contingent must be true; but if the former is known to be false, the value of the latter is unknown.

And if the partial or contingent proposition is false, the total or necessary must be false; but if the former is known to be true, the value of the latter is unknown.

A
Total or necessary affirmation.
S a P
All S is P
S must be P
E
Total or necessary negation.
S e P
No S is P
S cannot be P
I
Partial or contingent affirmation.
S i P
Some S is P
S may be P
O
Partial or contingent negation.
S o P
Some S is not P
S may not be P
Distribution of
S a P
S is distributed while P is not.
Distribution of
S e P
Both are distributed.
Distribution of
S i P
Neither is distributed.
Distribution of
S o P
S is not distributed, but P is.
Distribution of Predicates
Occurs in negative propositions.
Categorematic words
From the 10 categories of being:

Substantives (nouns and pronouns)

Primary Attributives ( verbs, verbals, adjectives)

Secondary Attributives (attributes of attributes: adverbs)

Syncategorematic Words
Words significant only with other words:

Definitives (articles and pronomials)

Connectives (prepositions and conjunctions)

Pure Copula

10 Categories
Substance
Quantity
Quality
Relation
Action
Passion
When
Where
Posture
Habiliment
Contradictories
A and O; E and I
Contraries
A and E
Subcontraries
I and O
Subalterns
A and I; E and O
Opposition of singular empirical propositions for categorical forms
Is achieved through a difference of quality alone and is restricted to contradiction:

A Mary is tall
E Mary is not tall

Opposition of singular empirical propositions in modal forms
Includes all four relations:

A Mary must be nice
E Mary cannot be nice
I Mary may be nice
O Mary may not be nice

Given E is true
Then I is false,
A is false,
O is true
Given E is false
Then I is true,
A and O are unknown
Given I is true
Then E is false,
A and O are unknown
Given I is false
Then E is true,
A is false,
O is true
Given O is true
Then A is false,
E and I are unknown
Given O is false
Then A is true,
E is false
Obversion
Changes a proposition’s quality (from affirmation to negation or negation to affirmation) and predicate (from P to P’ = non-P) but not the meaning.
Obversion of A
S a P > S e P’

All voters are citizens>No voters are noncitizens

Obversion of E
S e P>S a P’
No Jews are Muslims>All Jews are non-Muslims
Obversion of I
S i P>S o P’

Some chairs are comfy>Some chairs are not comfy

Obversion of O
S o P>S i P’
Some pupils listen>Some pupils don’t listen
Conversion
Reverse the subject and predicate, change the quantity if need be, but don’t change the quality.

No term may be distributed in the converse that was undistributed in the original proposition.

Converting S a P
To P i S

To P a S, but only when P is the definition or property of S, i.e., when it is distributed through the matter.

E.g. Man is a rational animal. A rational animal is man.

Otherwise beware of distributing P in affirmative propositions.

Converting S e P
To P e S
Converting S i P
To P i S
Converting S o P
It may be true to convert to P o S, but the original S is not distributed and yet becomes a P in a negative proposition (and so is then distributed incorrectly).

The truth of P o S is an accident of matter and the proposition is formally invalid.

Contrapositive
The quality of a proposition is changed and the predicate is converted to its contradictory
Inverse
A proposition that uses the contradictory of the subject and the predicate of the original proposition.
Eductions of S a P:
All voters are citizens.
Obverse – S e P’:
No voters are noncitizens.

Partial contrapositive – P’ e S:
No noncitizens are voters.

Full contrapositive – P’ a S’:
All noncitizens are nonvoters.

Full inverse – S’ i P’:
Some nonvoters sre noncitizens.

Partial inverse – S’ o P:
Some nonvoters are not citizens.

Converse – P i S:
Some citizens are voters.

Obverted converse – P o S’:
Some citizens are not nonvoters.

Eductions of S e P:
No Jews are Muslims.
Obverse – S a P:
All Muslims are non-Jews.

Partial contrapositive – P’ i S:
Some non-Jews are Muslims.

Full contrapositive – P’ o S’:
Some non-Jews are not non-Muslims.

Converse – P e S:
No Jews are Muslims.

Obverted converse – P a S’:
All Jews are non-Muslims.

Partial inverse – S’ i P:
Some non-Muslims are Jews.

Full inverse – S’ o P’:
Some non-Muslims are not non-Jews.

Eduction of forms
Alternately and successively use obversion (change quality and substitute P with P’) and conversion (reverse S and P and don’t change quality).
Once you arrive at S o P, return to the original proposition and begin with the opposite process of eduction. E.g., if you began with obversion, then use conversion.
Eduction of S i P:
Some chairs are uncomfy.
Obverse – S o P’:
Some chairs are not uncomfy.

Coverse – P i S:
Some comfy things are chairs.

Obverted coverse – P o S’:
Some comfy things are not nonchairs.

Eduction of S o P:
Some kids are not attentive.
Obverse – S i P’:
Some kids are inattentive.

Partial contrapositive – P’ i S:
Some inattentive beings are kids.

Full contrapositive – P’ o S’:
Some inattentive beings are not nonkids.

Eduction of S a P with P fully distributed, i. e., when P is either the definition or a property of S:

All men are rational animals.

Converse – P a S:
All rational animals are men.

Obverted converse- P e S’:
No rational animals are nonmen.

Partial inverse – S’ e P:
No nonmen are rational animals.

Full inverse – S’ a P’:
All nonmen are nonrational animals.

Full contrapositive – P’ a S’:
All nonrational animals are nonmen.

Partial contrapositive – P’ e S:
No nonrational animals are men.

Obverse – S e P’:
No men are nonrational animals.

Original – S a P:
All men are rational animals.

Eduction by added determinants
S is P; therefore Sa is Pa.
“a” must affect S and P to the same degree and respect.

Of degree:
Original: An ant (S) is an animal (P).
Invalid: a large (a) ant (S) is a large (a) animal (P).
Valid: a small (a) ant (S) is a small (a) animal (P).

Of respect:
Original: A contralto is a woman.
Invalid: a low contralto is a low woman.
Valid: a blonde contralto is a blonde woman.

Eduction by omitted determinants
S is Pa; therefore S is P. A subject included in a more determined predicate is necessarily included in that predicate when it is less determined.

E.g., Socrates is a rational animal; therefore Socrates is an animal.

Eduction by converse relation
S r1 P; therefore P r2 S.
(r1 and r2=copulas with correlative modifiers rather than simple copulas.)

Relative terms necessarily imply their correlatives (e.g., genus and species), so the subject and predicate with a relative copula may be transposed if the relative copula is supplanted by its correlative.

Original: Aristotle (S) taught (r1) Alexander the Great (P).
Valid inference: Alexander the Great (P) was taught (r2) by Aristotle (S).

Original: A (S) is greater (r1) than B (P).
Valid inference: B (P) is less (r2) than A (S).

Original: Lily (S) is a species (r1) of flower (P).
Valid inference: Flower (P) is a genus (r2) of lily (S).

Predicables
Classification of relations a predicate may be affirmed to have to a subject
List of predicables
Species
Genus
Differentia
Definition
Property
Accident
The four relations of propositions
Conjunction
Opposition
Eduction
Syllogism
Rules of conjunctional propositions
True only when every proposition conjoined is true

False if any one of the propositions conjoined is false

Probable if at least one of the propositions conjoined is propable and none are false

Syllogism
Two propositions (called premises) having one term in common leading to a third proposition (called the conclusion) in which the common term (called the middle term, M) does not appear.

A bat (S) is a mammal (M)
No bird (P) is a mammal (M)
A bat (S) is not a bird (P)

Matter of syllogism
Start with the conclusion:

Minor term is the subject (S) of the conclusion

Major term is the predicate (P) of the conclusion

Conclusion:
A bat (S) is not a bird (P)

So the Minor Premise is:
A bat (S) is a mammal (M)

The Major Premise:
No bird (P) is a mammal (M)

Rule 1 of syllogisms
It must contain 3 and only 3 terms.
Rule 2 of syllogisms
It must contain 3 and only 3 propositions
Rule 3 of syllogisms
The middle term must be distributed in at least one of the premises
Rule 4 of syllogisms
No term may be distributed in the conclusion which wasn’t distributed in its own premise
Rule 5 of syllogisms
From 2 negative premises no conclusion can be drawn
Rule 6 of syllogisms
If one premise is negative, the conclusion must be negative. Or, in order to prove a negative conclusion, one premise must be negative
Rule 7 of syllogisms
From two partial or singular or contingent premises, no conclusion can be drawn. See rules 3, 5, and 6.
Rule 8 of syllogisms
If one premise is partial, the conclusion must be partial. See rules 3 and 4.
Rule 9 of syllogisms
If one premise is contingent, thr conclusion must be contingent.
Rule 10 of syllogisms
If one or both premises are empirical, the conclusion must be empirical.
Mood of a syllogism
It is designated by the A, E, I or O forms of propositions and their order.

With 4 propositional forms we get 16 possible combos: AA, AE, AI, AO, EA, EE, EI, EO, IA, IE, II, IO, OA, OE, OI, OO.

Rule 5 eliminates EE, EO, OE, and OO.

Rule 7 eliminates II, IO, OI, and (again) OO.

Special rules (tba) eliminate EI.

8 standard combos of premises
AAA (AAI)
AEE (AEO)
AII
AOO
EAE (EAO)
IAI
IEO
OAO
Figure of syllogism
Determined by position of middle term in the premises: is it the subject or predicate of the minor or major term?
Figure 1
The middle term is the predicate of the minor premise and the subject of the major premise:

S__M
M__P
S__P

M__P
S__M
S__P

Figure 2
The middle term is the predicate of both premises:

S__M
P__M
S__P

P__M
S__M
S__P

Figure 3
The middle term is the subject of both:

M__S
M__P
S__P

M__P
M__S
S__P

Figure 4
The middle term is the subject of the minor premise and the predicate of the major:

M__S
P__M
S__P

P__M
M__S
S__P

The mood and figure of:
1.A bat (S) is a mammal (M).
2.No bird (P) is a mammal (M).
>3.A bat (S) is not a bird (P).
1.A
2.E
3.E
Figure 2
Distribution of terms
See if the middle term is distributed in at least one premise (rule 3).

See if P or S is distributed in the conclusion but undistributed in its premises (rule 4).

1. A bat (S) is (A) a mammal (M): S is distributed, M is not.

2. No bird (P) is (E) a mammal (M): both P and M are distributed.

3. A bat (S) is not (E) a bird (P): both are distributed. Valid.

Enthymeme
A syllogism omitting one proposition, either the major premise, the minor premise, or conclusion.
To find the conclusion of an enthymeme
Look for “because,” “for,” or “since.” These introduce a premise and therefore the other proposition is the conclusion.

Look for therefore, consequently, or accordingly, which introduce a conclusion.

Look for and or but, which connect two premises and indicate that the proposition omitted is rhe conclusion.

Validity of an enthymeme
If it is valid in one expansion, it’s valid.

If it is invalid in the first expansion, expand it in the alternate figure to see if it can become valid. For example:

An oak (S-D) is a plant (P-U) because it (S-D) is a tree (M-U).

Expansion B:
An oak (S-D) is a tree (M-U).
All plants (P-D) are trees (M-U).
An oak (S-D) is a plant (P-U).

This is invalid Figure 2 syllogism because of an undistributed (U) middle term (M).

So Expansion A:
An oak (S-D) is a tree (M-U).
A tree (M-D) is a plant (P-U).
An oak (S-D) is a plant (P-U).

This is a valid Figure 1 syllogism.

Typical enthymeme
Has the conclusion (with S and P terms) and one premise stated (with either the S and M or P and M terms).

Either the premise with the minor term S is missing or the premise with the major term P is missing.

Sorites
A chain of enthymemes or abridged syllogisms, in which the conclusion of one syllogism becomes a premise of the next
First type of sorites
That in which the conclusion of one syllogism becomes the major premise of the next:

Goclenian sorites (figure 1)

Second type of sorites
That in which the conclusion of one syllogism becomes the minor premise of the next:

Aristotelian sorites (figure 1)

Aristotelian sorites;

The first proposition is the minor premise of its syllogism and all the rest are major premises except for the last, which is a conclusion.

The omitted conclusion in each syllogism becomes the minor premise of the following syllogism.

Socrates is a man – S a M1
A man is an animal – M1 a M2
(Omitted conclusion and minor premise: Socrates is an animal.)
An animal is an organism – M2 a M3
(Omitted conclusion and minor premise: Socrates is an organism.)
An organism is a body – M3 a M4
(Omitted conclusion and minor premise: Socrates is a body.)
A body is a substance – M4 a P
Socrates is a substance – S a P
Goclenian sorites:

The first proposition is the major premise of its syllogism and all the rest are minor premises, except the last which is a conclusion.

The omitted conclusion in each syllogism becomes the major premise for the following syllogism.

A body is a substance – M1 a P
An organism is a body – M2 a M1 (Omitted conclusion and major premise: An organism is a substance.)
An animal is an organism – M3 a M2
(Omitted conclusion and major premise: An animal is a substance.)
A man is an animal – M4 a M3
(Omitted conclusion and major premise: A man is a substance.)
Socrates is a man – S a M4
Socrates is a substance – S a P
Aristotelian sorites expanded
Socrates is a man – S a M1
Man is an animal – M1 a M2
Socrates is an animal – S a M2

Socrates is an animal – S a M2
An animal is an organism – M2 a M3
Socrates is an organism – S a M3

Socrates is an organism – S a M3
An organism is a body – M3 a M4
Socrates is a body – S a M4

Socrates is a body – S a M4
A body is a substance – M4 a P
Socrates is a substance – S a P

Goclenian sorites expanded
A body is a substance – M1 a P
An organism is a body – M2 a M1
An organism is a substance – M2 a P

An organism is a substance – M2 a P
An animal is an organism – M3 a M2
An animal is a substance – M3 a P

An animal is a substance – M3 a P
A man is an animal – M4 a M3
A man is a substance – M4 a P

A man is a substance – M4 a P
Socrates is a man – S a M4
Socrates is a substance – S a P

Aristotelian sorites rules
1. Only the last premise may be negative.

2. Only the first (minor) premise may be partial, contingent or singular.

Goclenian sorites rules
1. Only the first premise can be negative.

2. Only the last (minor) premise may be partial, contingent or singular.

Epicheirma
An abridged polysyllogism combining any figures, one whose premises is an enthymeme.
Double epicheirema
Both premises are enthymemes
Example of single epicheirma
Beefsteak is not stored in the body because it is protein.
Food that is not stored in the body is not fattening.
Beefsteak is not fattening.
Expanded single epicheirma
Beefsteak (S) is protein (M).
Protein ( M) is food that is not stored in the body (P).
Beefsteak (S) is food that is not stored in the body (P).

Fig. 4, mood AAA, valid.

Beefsteak (S) is food thay is not stored in the body (M).
Food not stored in the body (M) is not fattening (P).
Beefsteak (S) is not fattening (P).

Fig. 4, mood AEE, valid

Multiple enthymeme v epicheirema
The multiple enthymeme has only one conclusion and many reasons supporting it.

The epicheirma has two conclusions and the double has three because they become premises which lead to a third conclusion.

Given A is true
The O and E are false, I is true
Given A is false
O is true, E and I are unknown
Exame of Analogical inference
Sparks (S1) from an electrical machine are electrical charges (P), for they (S1) have rapid motion and conductivity (M):
S1 is P because S1 is M.

Lightening (S2) resembles these sparks (S1) in rapid motion and conductivity (M):
S2 resembles S1 in M.

Lightening (S2) is probably an electrical charge (P):
S2 is probably P.

Validity of Analogical inference
Requires that the point of resemblance (M) be a property of P.
Example of Mediate opposition using mediated contraries
Mediate Contraries (AE):
X: The witness is lying.
Z’: The witness is telling the truth.

An immediate opposition uses Contradictories (AO, EI):
X: The witness is lying.
X’: The witness is not lying.

Rules of Mediated Opposition
1. The syllogism must be formally valid.

2. The third proposition, Y, must be materially true.

Illustrated Mediated Opposition
Mediated Opposition:
The witness is lying.
The witness is telling the truth.

Let X = minor premise
Y = major premise
Z = conclusion

Disputant 1:
X = The witness is lying.
Y = Whoever lies doesn’t tell the truth.
Z = The witness is not telling the truth.

Let X’ = contradictory of X
Let Z’ = contradictory of Z

Disputant 2:
X’ = The witness is not lying.
Y = Whoever lies doesn’t tell the truth.
Z’ = The witness is telling the truth.

Y is the omitted but shared proposition held by disputants 1 and 2 in the mediated opposition.

Relations of Mediated Opposition
If Y is materially true:

X (the witness is lying) and Z’ (the witness is telling the truth) are mediated contraries (A and E) and both cannot be true, but both can be false.

Z (the witness is not telling the truth) and X’ (the witness is not lying) are subcontraries (I and O) and both cannot be false but both may be true.

Syllogistic rules of inference
1. If both premises are true, the conclusion is.

2. If the conclusion is false, at least one of the premises must be false.

3. If one or both premises are false, the conclusion is unknown.

4. If the conclusion is true, the value of the premises are unknown.

5. If one or both of the premises are probable, the conclusion can only be probable.

6. If the conclusion is probable, the value of the premises is unknown

Figure 2 special rules
Figure 2’s middle term, M, is predicate in both premises:
S__M
P__M
S__P

Hence,
1. One premise must be negative to distribute M.

Because a negative premise requires a negative conclusion (rule 6), the major term P will be distributed in the conclusion.

Yet it cannot be distributed in the conclusion without having already been distributed in its own premise. There, however, the major term P is the subject, and only a total or necessary proposition distributes its subject. So

2. The major premise must be total or necessary and so take the mood of either A or E.

So the valid moods of figure 2 are:
AEE, EAE, IEO, OAO.

Figure 1 special rules
Figure 1’s middle term (M) is the predicate of the minor term (S) and the subject of the major term (P):
S__M
M__P
S__P

Hence,
1. The minor premise (S) must be affirmative.

This is because if the minor premise were negative, the conclusion would be negative (rule 6). But then the major term P would be distributed in the conclusion, and it could not be distributed in the conclusion without also being distributed in its own premise (rule 4). For that to happen, however, would require the major premise to also be negative, and yet two negative premises lead to no conclusion (rule 5).

2. The major premise must be total or necessary to avoid an undistributed middle term, because that term was not distributed in the (affirmative) minor premise.

So the valid moods of figure 1 are:
AAA, AEE, IAI, IEO

Figure 3 special rules
Figure 3’s middle term (M) is the subject of the minor (S) and major (P) premises:
M__S
M__P
S__P

Hence,
1. The minor premise (S) must be affirmative.

2. Because S is formally undistributed it will be undisturbed in the conclusion, and only partial or contingent propositions have undisturbed subjects. Thus the conclusion is partial and contingent.

So the valid moods of figure 3 are:
AAI, AII, IAI, AEO, AOO, IEO

Figure 4 special rules
The middle term is the subject of the minor term (S) and predicate of the major term (P):
M__S
P__M
S__P

1. If the major premise is affirmative, M is undistributed and so must be distributed in the minor (rule 3) where it is subject. So M__S must be total or necessary.

2. If the minor is affirmative, the conclusion must be partial or contingent (rule 2 of fig. 3).

3. If the conclusion is negative, the major premise must be total or necessary (rule 2 of fig. 2).

Moods:
AAI, EAE, AII, AEO, IEO

Figure 1 alone can yield
A total or necessary general affirmative proposition as conclusion
Figure 2 can only yield
Negative conclusions, unless one premise is a definition.

It’s adapted to disproof.

Figure 3 can only yield
Partial, singular or contingent conclusions, unless one premise is the definition.

It’s adapted to proving exceptions.

Syllogistic reduction
A process where a syllogism in an imperfect figure (2, 3 or 4) is expressed as a syllogism of the first figure.

Assumes;
1. Imperfect figures’ premises are true.
2. That the first figure is formally valid.

Figure reduction-Valid Moods
First Figure:
Barbara (AAA), Celarent (EAE), Darii (AII), Ferio (EIO), que prioris.

Second Figure:
Cesare (EAE), Camestres (AEE), Festino (EIO), Baroco (AOO), secundae.

Third Figure:
Tertia Darapti (AAI), Disamis (IAI), Datisi (AII), Felapton (EAO)
Bocardo (OAO), Ferison (EIO) habet.

Fourth Figure:
Quarta in super addit
Bramantip (AAI), Camenes (AEE), Dimaris (IAI), Fesapo (EAO), Fresison (EIO).

Vowels = mood in traditional order:
Major premise, minor premise, conclusion.

B, C, D, and F names in second to fourth syllogisms can have their moods reduced to the first figure moods whose names begin with the same letter. E.g.,
Camestres (AEE) of the second figure can be reduced to Celarent (EAE) of the first figure.

Mode of reduction to first figure form
Barbara (AAA), Celarent (EAE), Darii (AII), Ferio (EIO), que prioris.

Cesare (EAE), Camestres (AEE), Festino (EIO), Baroco (AOO), secundae.

Tertia Darapti (AAI), Disamis (IAI), Datisi (AII), Felapton (EAO)
Bocardo (OAO), Ferison (EIO) habet.

Quarta in super addit
Bramantip (AAI), Camenes (AEE), Dimaris (IAI), Fesapo (EAO), Fresison (EIO).

A name with the letter “s” in it signifies that the vowel preceding it (Disamis-IAI-of the 3rd syllogism) is reduced to the first figure mood (Darii-AII) by simple conversion (reverse subject and predicate).

“P” signifies that the proposition indicated by the proceeding vowel must be converted by limitation (A to I in one case or I to A) to the first figure mood.

“M” signifies premises are to be transposed; “c” by showing that the conclusion in the first figure contradicts a premise given as true in another figure; r, b, l, n, t, d mean nothing.

Reducing Camestres to Celarent (A to B)
CaMeStreS (AEE) or A:

All circles (P) are (a) curvilinear (M).
M-Transpose

No square (S) is (e) curvilinear (M).
S-Convert

No square (S) is (e) a circle (P).
S-Convert

Celarent or B:

No curvilinear figure (M) is (E) a square (P).
All circles (S) are (A) curvilinear (M).
No circle (S) is (E) a square (P).

Reduction by “c”, per contradictorian propositionem: BoCardo (OAO) to Barbara (AAA)
From Barbara, using as premises the A of Bocardo and the contradictory of its conclusion draw the conclusion implicit in these premises:

Bocardo or A:

Some lions (M) are not (o) tame (P).
All lions (M) are (a) animals (S).
Some animals (S) are not (o) tame (P).

Barbara or B:

All animals (M) are (a) tame (P).
All lions (S) are (a) animals (M).
All lions (S) are (a) tame (P).

Because premise O of Bocardo is true and the conclusion of Barbara is its contradictory, that conclusion, though of a valid syllogism, is false. So the matter rather than the form is to blame, meaning one of the premises must be false. And because the minor premise of Barbara was given by Bocardo as true, then its major premise, All animals are tame, must be false. That premise is the contradictory of Bocardo’s conclusion, Some animals are not tame, which means that Bocardo’s conclusion is true.

Hypothetical syllogism
Expresses a conditional relation of propositions
Simple syllogism
Expresses a relation of terms
3 term hypothetical proposition’s formula
If S is M, it is P
4 term hypothetical proposition’s formula
If B is C, D is E
Reduction of three term hypothetical proposition
If S is M, it is P>SM is P
Reduction of four term hypothetical proposition
If B is C, D is E> BC is DE
Hypothetical proposition and its contradictory
If a man drinks water, he will die.

If a man drinks water, he will not die.

Taken in relation to the first proposition, which is false, the second, its contradictory, is true, though not by itself.

First of disjunctive propositions
1. S is P or Q or R.

The alternatives are:

A) collectively exhaustive
B) mutually exclusive
C) species resulting from division acording to single basis

Second of disjunctive propositions
2. S or T or U is P
Third type of disjunctive proposition
3. B is C or D is E
Eduction of hypothetical propositions
Original: if a tree is a pine, it is necessarily an evergreen.

Obverse: if a tree is a pine, it is necessarily not a nonevergreen.

Partial contrapositive: if a tree is a nonevergreen, it is necessarily not a pine.

Full contrapositive: if a tree is a nonevergreen, it is necessarily a nonpine.

Full inverse: if a tree is a nonpine, it may be a nonevergreen.

Partial inverse: if a tree is a nonpine, it may not be an evergreen.

Converse: if a tree is an evergreen, it may be a pine.

Obverted converse: if a tree is an evergreen, it may not be a nonpine.

Eduction of disjunctive proposition
Original: a material substance must be either a gas, liquid or solid.

Converse: a substance that is either a gas, liquid or solid must be a material substance.

Obverted converse: a substance that is either a gas, liquid, or solid cannot be a nonmaterial substance.

Partial inversion: a nonmaterial substance cannot be either a gas, liquid or solid

Full inverse: a nonmaterial substance must be neither a gas, liquid nor solid.

Full contrapositive: a substance that is either neither a gas, liquid nor a solid must be a nonmaterial substance.

Partial contrapositive: a substance that is neither a gas, liquid nor a solid cannot be a material substance.

Obverse: a material substance cannot be neither a gas, liquid nor solid.

Original.

Mixed hypothetical syllogism
The major premise is a hypothetical proposition, and the minor premise is a simple proposition
Rules for the mixed hypothetical syllogism
The minor premise must either:

1. (Modus ponens – the way of affirmation:) posit the antecedent or

2. (Modus tollens – the way of negation:) sublate the consequent of the major premise (by restating as fact its contradictory).

Example of positing the antecedent: modus ponens
Major premise:
(Antecedent) If a man is not honest, (consequent) he is not a fit public officer.

Minor Premise:
(Positing of antecedent) This man is not honest.

Conclusion;
(Positing of consequent) This man is not a fit public officer.

Example of sublating the consequent: modus tollens
Major premise:
(Antecedent) If all students were equally competent, (consequent) each would have the same grade in a given course.

Minor premise:
(Sublating of consequent) But each doesn’t acquire the same amount of knowledge from a given course.

Conclusion:
(Sublating the antecedent) All students are not equally competent.

The conclusion is the contradictory of the antecedent.

Fallacy of sublating the antecedent
If a man drinks poison, he will die.
This man has not drunk poison.
He will not die.

Syllogistically:

Whoever drinks poison (M) will (a) die (P).
This man (S) has not (e) drunk poison (M).
He (S) will not (e) die (P).

P is distributed in the conclusion but not in the major premise, hence the syllogism is invalid.

Fallacy of positing the consequent
If a man drinks poison he will die.
This man died.
He must have drunk poison.

Syllogistically:

Whoever drinks poison (P) will (a) die (M).
This man (S) died (a and M).
He (S) must have (a) drunk poison (P).

The middle term is undistributed.

Modus pollens
Whenever the minor premise posits the antecedent, the conclusion posits the consequent:

(Major premise;) If A (antecedent) then B (consequent).
(Minor:) A .
(Conclusion;) B.

Modus tollens
Whenever the minor premise sublates the consequent, the conclusion sublates the antecedent.

(Major premise:) If A (antecedent) then B (consequent).
(Minor premise:) -B.
(Conclusion:) -A.

Ponendo tollens of a disjunctive syllogism
The minor premise posits one alternative, and the conclusion sublates the other.

This woman’s missing husband (S) is either living (P) or dead (Q).
He is living (P).
He is not dead (-Q).

Tollendo ponens of disjunctive syllogism
The minor premise sublates one alternative, and the conclusion posits the other.

Th soul (S) is either spiritual (P) or material (Q)
The soul is not material (S=-Q)
The soul is spiritual (S=P).

Disjunctive fallacies
Formal: Where the minor premise and conclusion both posit and sublate each alternative.

Material: Where the alternatives are not mutually exclusive or not collectively exhaustive.

Dilemma
A syllogism with:

1. A disjunctive minor premise

2. A compound hypothetical proposition for its major premise

3. A simple or disjunctive proposition for its conclusion.

Dilemma’s moods
1. Simple constructive

2. Complex constructive

3. Simple destructive

4. Complex destructive

Fallacies of the dilemma
1. False major premise

2. Imperfect disjunction in the minor premise

3. The dilemmatic fallacy, from a shifting point of view

Exposing the dilemma with a false major premise
Taking the dilemma by the horns
Exposing the dilemma of the imperfect disjunction in the minor premise
Escaping between the horns
Exposing the dilemmatic fallacy
Rebutting the dilemma:

Used when the dilemma open to rebutal and the rebutting dilemma contain a formal and material fallacy.

Accept the alternatives in the minor premise of the original dilemma but transpose the consequents of the major premise and change them into contraries.

Constructive dilemma
The minor premise posits the two antecedents of the major
Destructive dilemma
The minor premise sublates the two consequents of the major premise
Fallacies in dictione
1. Equivocation
2. Amphiboly
3. Composition
4. Division
5. Accent
6. Verbal form
Fallacy of equivocation
Feathers are light.
Light is the opposite of darkness.
Feathers are the opposite of darkness.
Fallacy of amphiboly
Produced by ambiguity of syntax or grammatical structure.

He told his brother that he had won the prize. (Who?)

Fallacy of composition
Properties of parts are illicitly predicated of the whole.
Fallacy of division
When properties of the whole are illicitly predicated of the parts
Fallacy of verbal form
Inspiration and inexplicable: if “in” means “not” in one, it means “not” in the other.
Fallacy of accident
A shifting of usage in one term. For example, to shift from first imposition to second:

1st Imposition – Feathers are light.
2nd Imposition – Light is an adjective.
Feathers are adjectives.

Feathers is first used as an adjective and then as a noun (when, ironically, it is called an adjective).

Second intention
Of logic
Second imposition
Of grammar
First imposition and first intention
Of reality
Zero imposition
Of phonetics and spelling
Fallacy with grammar used in two impositions
Carry is a verb.
Verb is a noun.
Carry is a noun.

Verb shifts from first to second imposition

Valid syllogism with grammar
Sing is verb.
A verb has tense.
Sing has tense.
Secundum quid
Assumes that a proposition true in certain respects is true absolutely
Fallacy of consequent
There are two types of material fallacy of consequent:

1. Positing the consequent – Falsely assumes that because a consequent follows upon the antecedent, the antecedent must follow upon the consequent.

If it rains, the ground is wet.
The ground is wet.
It rained.

2. Sublating the antecedent – Falsely assumes that from the contrary of the antecedent the contrary of the consequent must follow.

If it rains, the ground is wet.
It did not rain.
The ground is not wet.

Begging the question
1. Tautological argument:

Shakespeare is famous because his plays are known all over.

2. The shuttle argument:

The boy is insane. Why? Because he murdered his mom. Why? Because he’s insane.

3. Arguing in a circle:

Shuttle argument plus additional propositions.

4. Question-begging epithet:

Using a phrase that assumes the point to be proved.

Induction
The derivation of general propositions from individual instances.

The apprehension of the cause common to a number of observed facts.

4 causes
1. Efficient cause – the agent and instruments.

2. Final cause – the end or purpose that moved the agent. It is first in intention, last in execution.

3. Material cause – that out of which a thing is made.

4. Formal cause – the kind of thing which is made; its essence.

KRegressive syllogism
Links induction to deduction.

Seeks the cause of natural phenomena, a middle term, which is the formal cause of the relation of the terms in the conclusion of the syllogism.

So we see S is P, and we posit that S is P because of M. We want M to be the only antecedent of P so that M is P and P is M. So we want to say
Pif S is M
it is P
and if S is P
it is M.

If so we get this regressive syllogism:

S is P.
P is M.
Therefore S is M.

So

If S is M, it is P.
But S is M.
Therefore S is P.

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