Post by JoeC on Jun 21, 2014 20:58:37 GMT -5
Modes and syllogisms are, I believe, a common part of mathematical curricula in modern schools. However, as these are an important part of Stoic Logic, particularly in the are of dialectics. What follows are my notes on these subjects and they are by no means complete. If anyone has anything to add to them or an argument for or against, please feel free to do so. This is simply an example of my "writing in order to better understand".
Zeno is said to have studied with the Megarian philosopher Stilpo. This is likely an important contributing factor in the development of Stoic logic. Chrysippus is the most celebrated of the early Stoic logicians and it is suggested that he rivaled Aristotle as such, and it is he who is credited with the development of Stoic propositional logic. Stoic Logic is often contrasted with Aristotelian Logic and so understanding the differences are an important factor in understanding Stoicism itself.
Aristotelian Logic
Aristotelian theory is based on the fundamental assumption that propositions are composed of two terms and that the reasoning process is in turn built from propositions.
The term is a part of speech representing something, but which is not inherently true or false in its own right, such as "human" or "mortal".
The proposition consists of two terms, in which one term (the "predicate") is "affirmed" or "denied" of the other (the "subject"), and which is capable of truth or falsity.
Ex.
1. All humans (subject) are animals (predicate)
Propositions may be universal or particular, and they may be affirmative or negative. Traditionally, the four kinds of propositions are:
This is called fourfold scheme of propositions.
As is demonstrated above, this logic makes use of four basic logical terms:
"All", "some", "is" & "is not" or "no"
Traditionally "is" and "is not" are used as the copula (that is, the word joining the two terms). In modern logic we sometimes us "are" or "are not" for grammatical correctness.
The syllogism is an inference in which one proposition (the "conclusion") follows of necessity from two others (the "premises").
Ex.
1. All humans are animals, (Major Premise)
2. All animals are mortal, (Minor Premise)
3. Therefore, all humans are mortal (Conclusion)
If it is accepted that both premises are true then it must be accepted that the conclusion also is true.
The formal structure can be more clearly displayed as:
All A are B;
All B are C;
Therefore, all A are C.
The formal structure of this argument is known as valid.
The weakness in this form of logic becomes apparent when dealing with singular terms. A singular term in Aristotelian substance theory is primary substance, which can only be predicated of itself. "Socrates", for example, is not predicable of any other thing, thus one does not say "every Socrates" one would say "every human". The subject of one premise, must be the predicate of the other, and so it is necessary to eliminate from the logic any terms which cannot function both as subject and predicate - i.e. singular terms. Thus, Aristotelian logic is concerned solely with universal terms.
For more on Aristotelian Logic refer to Organon; De Interpretatione(On Interpretation) & Analytica Priora(Prior Analytics)
Stoic Propositional Logic
Now, concerning ourselves with Stoic Logic. In contrast to Aristotelian logic which concerned itself with the interrelated nature of terms ("human" or "mortal"), Stoic logic was concerned with the interrelation of propositions (i.e. "Dion is walking" or "it is night"). These propositions are called "assertibles"(axioma) & an assertible is a complete sayable. That is to say assertibles are not statements, as it takes a person to make a statement. Rather an assertible may be a statement, if it is said; however, it subsists even when not said.
To the Stoic the Aristotelian term "writes" is considered a defective expression, for it leaves us to inquire "Who?" Whereas, "Socrates writes" is considered a complete expression as it is a finished, complete thought.
Of defective expressions the Stoics tell us are ranged all predicates.
Of complete expressions are judgments, syllogisms, questions & inquiries.
According to Apollodorus, a predicate is what is said of something; i.e., a thing associated with one or more subjects.
Chrysippus also distinguished assertibles as either "simple" (atomic) which cannot be broken down into simpler sentences and "not simple" or complex (molecular) which contained within them more than one simple assertible.
Ex.
Simple Assertible: Dion is walking
Not Simple Assertible: If Dion is walking, Dion is moving
So we see this differs significantly from Aristotelian logic in that even the simple assertible is a declarative sentence which may inherently be true or false and can be denied or affirmed. Also when you say a complex assertible, you are not actually stating the simple assertibles within it, you are state one assertible; namely the complex one.
The Stoics also catalog different types of both simple and not simple assertibles.
Of Simple Assertibles they classify:
Affirmation or Assertory: One that consists of a noun in the nominative case and a predicate
"Dion is walking"
Negation: "Not"
"It is not day."
Denial: Negative part or particle and a predication
"No one is walking."
Privation: One that contains a privative particle reversing the effect of a judgement
"This man is unkind."
Definitive: One that consists of a demonstrative in the nominative case and a predicate
"This man is walking."
Indefinitive: One that consists of an indefinite word or words and a predicate
"Some one is walking", "There's some one walking", "He is in motion"
Of Not Simple Assertibles they class:
the Hypothetical (Conditional: "If")
"If it is day, it is light"
the Inferential (Conjunction: "Since")
"Since it is day-time, it is light"
the Coupled ("And")
"It is day-time and it is light"
the Disjunctive ("Either" and "Or")
"Either it is day or it is night"
the Casual ("Because")
"Because it is day, it is light"
that which indicates more or less ("Rather" and "than")
"It is rather day-time than night"
Assertibles can also be distinguished by their modality:
Consider the following:
Propositions can also be time-dependent for their truth as in the example "It is night", which, of course, would only be true at night.
The formal structure of this argument is written as:
If p, the q;
p;
Therefore, q.
Note how the letters express not (defective) things, but whole propositions.
Or, to the Stoics:
If the first, then the second;
The first;
Therefore, the second.
Once formalized in this way the argument structure is known as a "mode". A mode is not an argument itself, but rather the structural form of a certain type of argument that particular arguments can have.
Stoic logic holds that there were 5 most basic, "indemonstrable" modes that were valid, to which all others could be reduced. These were held to be obviously valid and not in need of further proof.
They are:
However, according to Sextus:
The five basic modes "were defined by five standardized meta-linguistic descriptions of the forms of the arguments:
1. A first indemonstrable is an argument composed of a conditional and its antecedent as premises, having the consequent of the conditional as conclusion.
2. A second indemonstrable is an argument composed of a conditional and the contradictory of its consequent as premises, having the contradictory of its antecedent as conclusion.
3. A third indemonstrable is an argument composed of a negated conjunction and one of its conjuncts as premises, having the contradictory of the other conjunct as conclusion.
4. A fourth indemonstrable is an argument composed of a disjunctive assertible and one of its disjuncts as premises, having the contradictory of the remaining disjunct as conclusion.
5. A fifth indemonstrable, finally, is an argument composed of a disjunctive assertible and the contradictory of one of its disjuncts as premises, having the remaining disjunct as conclusion.
If we disregard complex arguments, there are thirty-two modes corresponding to the five meta-linguistic descriptions; the latter thus prove noticeably more economical. The almost universal assumption among historians of logic that the Stoics represented their five (types of) indemonstrables by five modes is false and not supported by textual evidence."
...
Zeno is said to have studied with the Megarian philosopher Stilpo. This is likely an important contributing factor in the development of Stoic logic. Chrysippus is the most celebrated of the early Stoic logicians and it is suggested that he rivaled Aristotle as such, and it is he who is credited with the development of Stoic propositional logic. Stoic Logic is often contrasted with Aristotelian Logic and so understanding the differences are an important factor in understanding Stoicism itself.
Aristotelian Logic
Aristotelian theory is based on the fundamental assumption that propositions are composed of two terms and that the reasoning process is in turn built from propositions.
The term is a part of speech representing something, but which is not inherently true or false in its own right, such as "human" or "mortal".
The proposition consists of two terms, in which one term (the "predicate") is "affirmed" or "denied" of the other (the "subject"), and which is capable of truth or falsity.
Ex.
1. All humans (subject) are animals (predicate)
Propositions may be universal or particular, and they may be affirmative or negative. Traditionally, the four kinds of propositions are:
- A-type: Universal and affirmatives ("All humans are mortal")
- E-type: Universal and negatives ("No humans are perfect")
- I-type: Particular and affirmatives ("Some humans are healthy")
- O-type: Particular and negatives ("Some humans are not clever")
This is called fourfold scheme of propositions.
As is demonstrated above, this logic makes use of four basic logical terms:
"All", "some", "is" & "is not" or "no"
Traditionally "is" and "is not" are used as the copula (that is, the word joining the two terms). In modern logic we sometimes us "are" or "are not" for grammatical correctness.
The syllogism is an inference in which one proposition (the "conclusion") follows of necessity from two others (the "premises").
Ex.
1. All humans are animals, (Major Premise)
2. All animals are mortal, (Minor Premise)
3. Therefore, all humans are mortal (Conclusion)
If it is accepted that both premises are true then it must be accepted that the conclusion also is true.
The formal structure can be more clearly displayed as:
All A are B;
All B are C;
Therefore, all A are C.
The formal structure of this argument is known as valid.
The weakness in this form of logic becomes apparent when dealing with singular terms. A singular term in Aristotelian substance theory is primary substance, which can only be predicated of itself. "Socrates", for example, is not predicable of any other thing, thus one does not say "every Socrates" one would say "every human". The subject of one premise, must be the predicate of the other, and so it is necessary to eliminate from the logic any terms which cannot function both as subject and predicate - i.e. singular terms. Thus, Aristotelian logic is concerned solely with universal terms.
For more on Aristotelian Logic refer to Organon; De Interpretatione(On Interpretation) & Analytica Priora(Prior Analytics)
Stoic Propositional Logic
Now, concerning ourselves with Stoic Logic. In contrast to Aristotelian logic which concerned itself with the interrelated nature of terms ("human" or "mortal"), Stoic logic was concerned with the interrelation of propositions (i.e. "Dion is walking" or "it is night"). These propositions are called "assertibles"(axioma) & an assertible is a complete sayable. That is to say assertibles are not statements, as it takes a person to make a statement. Rather an assertible may be a statement, if it is said; however, it subsists even when not said.
To the Stoic the Aristotelian term "writes" is considered a defective expression, for it leaves us to inquire "Who?" Whereas, "Socrates writes" is considered a complete expression as it is a finished, complete thought.
Of defective expressions the Stoics tell us are ranged all predicates.
Of complete expressions are judgments, syllogisms, questions & inquiries.
According to Apollodorus, a predicate is what is said of something; i.e., a thing associated with one or more subjects.
Chrysippus also distinguished assertibles as either "simple" (atomic) which cannot be broken down into simpler sentences and "not simple" or complex (molecular) which contained within them more than one simple assertible.
Ex.
Simple Assertible: Dion is walking
Not Simple Assertible: If Dion is walking, Dion is moving
So we see this differs significantly from Aristotelian logic in that even the simple assertible is a declarative sentence which may inherently be true or false and can be denied or affirmed. Also when you say a complex assertible, you are not actually stating the simple assertibles within it, you are state one assertible; namely the complex one.
The Stoics also catalog different types of both simple and not simple assertibles.
Of Simple Assertibles they classify:
Affirmation or Assertory: One that consists of a noun in the nominative case and a predicate
"Dion is walking"
Negation: "Not"
"It is not day."
Denial: Negative part or particle and a predication
"No one is walking."
Privation: One that contains a privative particle reversing the effect of a judgement
"This man is unkind."
Definitive: One that consists of a demonstrative in the nominative case and a predicate
"This man is walking."
Indefinitive: One that consists of an indefinite word or words and a predicate
"Some one is walking", "There's some one walking", "He is in motion"
Of Not Simple Assertibles they class:
the Hypothetical (Conditional: "If")
"If it is day, it is light"
the Inferential (Conjunction: "Since")
"Since it is day-time, it is light"
the Coupled ("And")
"It is day-time and it is light"
the Disjunctive ("Either" and "Or")
"Either it is day or it is night"
the Casual ("Because")
"Because it is day, it is light"
that which indicates more or less ("Rather" and "than")
"It is rather day-time than night"
Assertibles can also be distinguished by their modality:
- Possible/Impossible
- Necessary/Non-necessary
Consider the following:
- If it is raining this afternoon, then I shall not go out for a walk;
- It is raining this afternoon;
- Therefore, I shall not go out for a walk.
Propositions can also be time-dependent for their truth as in the example "It is night", which, of course, would only be true at night.
The formal structure of this argument is written as:
If p, the q;
p;
Therefore, q.
Note how the letters express not (defective) things, but whole propositions.
Or, to the Stoics:
If the first, then the second;
The first;
Therefore, the second.
Once formalized in this way the argument structure is known as a "mode". A mode is not an argument itself, but rather the structural form of a certain type of argument that particular arguments can have.
Stoic logic holds that there were 5 most basic, "indemonstrable" modes that were valid, to which all others could be reduced. These were held to be obviously valid and not in need of further proof.
They are:
- If the first, then the second; the first; therefore, the second. modus ponendo ponens
- If the first, then the second; not the second; therefore not the first. modus tollendo tollens
- Not both the first and the second; the first; therefore, not the second. modus ponendo tollens (1)
- Either the first or the second; the first; therefore, not the second. modus ponendo tollens (2)
- Either the first or the second; not the second; therefore, the first. modus tollendo ponens
However, according to Sextus:
The five basic modes "were defined by five standardized meta-linguistic descriptions of the forms of the arguments:
1. A first indemonstrable is an argument composed of a conditional and its antecedent as premises, having the consequent of the conditional as conclusion.
2. A second indemonstrable is an argument composed of a conditional and the contradictory of its consequent as premises, having the contradictory of its antecedent as conclusion.
3. A third indemonstrable is an argument composed of a negated conjunction and one of its conjuncts as premises, having the contradictory of the other conjunct as conclusion.
4. A fourth indemonstrable is an argument composed of a disjunctive assertible and one of its disjuncts as premises, having the contradictory of the remaining disjunct as conclusion.
5. A fifth indemonstrable, finally, is an argument composed of a disjunctive assertible and the contradictory of one of its disjuncts as premises, having the remaining disjunct as conclusion.
If we disregard complex arguments, there are thirty-two modes corresponding to the five meta-linguistic descriptions; the latter thus prove noticeably more economical. The almost universal assumption among historians of logic that the Stoics represented their five (types of) indemonstrables by five modes is false and not supported by textual evidence."
...