Misplaced Pages

#206793

190-410: Chrysippus excelled in logic , the theory of knowledge , ethics , and physics . He created an original system of propositional logic in order to better understand the workings of the universe and role of humanity within it. He adhered to a fatalistic view of fate , but nevertheless sought a role for personal agency in thought and action. Ethics, he thought, depended on understanding the nature of

380-501: A r y ) ∧ Q ( J o h n ) ) {\displaystyle \exists Q(Q(Mary)\land Q(John))} " . In this case, the existential quantifier is applied to the predicate variable " Q {\displaystyle Q} " . The added expressive power is especially useful for mathematics since it allows for more succinct formulations of mathematical theories. But it has drawbacks in regard to its meta-logical properties and ontological implications, which

570-444: A countable noun , the term "a logic" refers to a specific logical formal system that articulates a proof system . Logic plays a central role in many fields, such as philosophy , mathematics , computer science , and linguistics . Logic studies arguments, which consist of a set of premises that leads to a conclusion. An example is the argument from the premises "it's Sunday" and "if it's Sunday then I don't have to work" leading to

760-445: A central role in many arguments found in everyday discourse and the sciences. Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. The support they provide for their conclusion usually comes in degrees. This means that strong ampliative arguments make their conclusion very likely while weak ones are less certain. As a consequence, the line between correct and incorrect arguments

950-480: A certain cost: the premises support the conclusion in the sense that they make its truth more likely but they do not ensure its truth. This means that the conclusion of an ampliative argument may be false even though all its premises are true. This characteristic is closely related to non-monotonicity and defeasibility : it may be necessary to retract an earlier conclusion upon receiving new information or in light of new inferences drawn. Ampliative reasoning plays

1140-535: A clear doctrine of categories may be the work of later Stoics. Chrysippus came to be renowned as one of the foremost logicians of ancient Greece. When Clement of Alexandria wanted to mention one who was master among logicians, as Homer was master among poets , it was Chrysippus, not Aristotle, he chose. Diogenes Laërtius wrote: "If the gods use dialectic , they would use none other than that of Chrysippus." The logical work by Chrysippus came to be neglected and forgotten. Aristotle's logic prevailed, partly because it

1330-573: A complex argument to be successful, each link of the chain has to be successful. Arguments and inferences are either correct or incorrect. If they are correct then their premises support their conclusion. In the incorrect case, this support is missing. It can take different forms corresponding to the different types of reasoning . The strongest form of support corresponds to deductive reasoning . But even arguments that are not deductively valid may still be good arguments because their premises offer non-deductive support to their conclusions. For such cases,

1520-401: A complex fact. All human actions – in fact, our destiny – are decided by our relation to things, or as Chrysippus put it, events are "co-fated" to occur: The non-destruction of one's coat, he says, is not fated simply, but co-fated with its being taken care of, and someone's being saved from his enemies is co-fated with his fleeing those enemies; and having children

1710-425: A conclusion. Logic is interested in whether arguments are correct, i.e. whether their premises support the conclusion. These general characterizations apply to logic in the widest sense, i.e., to both formal and informal logic since they are both concerned with assessing the correctness of arguments. Formal logic is the traditionally dominant field, and some logicians restrict logic to formal logic. Formal logic

1900-399: A context in which people are understood to be rational beings, reason is developed out of these notions. Chrysippus insisted on the organic unity of the universe , as well as the correlation and mutual interdependence of all of its parts. He said, the universe is "the soul and guide of itself." Following Zeno, Chrysippus determined fiery breath or aether to be the primitive substance of

2090-424: A debate between Diodorus and Philo. Philo regarded all conditionals as true except those which with a correct antecedent had an incorrect consequent , and this meant a proposition such as "if it is day, then I am talking," is true unless it is day and I fall silent. But Diodorus argued that a true conditional is one in which the antecedent clause could never lead to an untrue conclusion – thus, because

SECTION 10

#1732851342207

2280-545: A feast, and died soon after. In the second account, he was watching a donkey eat some figs and cried out: "Now give the donkey a drink of pure wine to wash down the figs", whereupon he died in a fit of laughter . His nephew Aristocreon erected a statue in his honour in the Kerameikos . Chrysippus was succeeded as head of the Stoic school by his pupil Zeno of Tarsus . Of his written works, none survived except as fragments quoted in

2470-510: A formal language together with a set of axioms and a proof system used to draw inferences from these axioms. In logic, axioms are statements that are accepted without proof. They are used to justify other statements. Some theorists also include a semantics that specifies how the expressions of the formal language relate to real objects. Starting in the late 19th century, many new formal systems have been proposed. A formal language consists of an alphabet and syntactic rules. The alphabet

2660-686: A formal language while informal logic investigates them in their original form. On this view, the argument "Birds fly. Tweety is a bird. Therefore, Tweety flies." belongs to natural language and is examined by informal logic. But the formal translation "(1) ∀ x ( B i r d ( x ) → F l i e s ( x ) ) {\displaystyle \forall x(Bird(x)\to Flies(x))} ; (2) B i r d ( T w e e t y ) {\displaystyle Bird(Tweety)} ; (3) F l i e s ( T w e e t y ) {\displaystyle Flies(Tweety)} "

2850-415: A given argument is valid. Because of the reliance on formal language, natural language arguments cannot be studied directly. Instead, they need to be translated into formal language before their validity can be assessed. The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, a logic is a logical formal system. Distinct logics differ from each other concerning

3040-551: A given conclusion based on a set of premises. This distinction does not just apply to logic but also to games. In chess , for example, the definitory rules dictate that bishops may only move diagonally. The strategic rules, on the other hand, describe how the allowed moves may be used to win a game, for instance, by controlling the center and by defending one's king . It has been argued that logicians should give more emphasis to strategic rules since they are highly relevant for effective reasoning. A formal system of logic consists of

3230-402: A great variety of topics. They include metaphysical theses about ontological categories and problems of scientific explanation. But in a more narrow sense, it is identical to term logic or syllogistics. A syllogism is a form of argument involving three propositions: two premises and a conclusion. Each proposition has three essential parts: a subject , a predicate, and a copula connecting

3420-480: A life according to Nature. As reasoning creatures, humans have a share in Nature's rationality. The good for a human is to be fully rational, behaving as Nature does to maintain the natural order. This means to know the logic of the good, to understand the rational explanation of the universe, and the nature and possibilities of being human. The only evil for a human is to behave irrationally—to fail to act upon reason—such

3610-614: A logical connective like "and" to form a new complex proposition. In Aristotelian logic, the subject can be universal , particular , indefinite , or singular . For example, the term "all humans" is a universal subject in the proposition "all humans are mortal". A similar proposition could be formed by replacing it with the particular term "some humans", the indefinite term "a human", or the singular term "Socrates". Aristotelian logic only includes predicates for simple properties of entities. But it lacks predicates corresponding to relations between entities. The predicate can be linked to

3800-478: A new cycle begins again. Since the world operates through reason, all things are determined . But the Stoics adopted a compatibilist view which allowed humans freedom and responsibility within the causal network of fate. Humans are part of the  logos  which permeates the cosmos. The human soul is a physical unity of reason and mind. The good for a human is thus to be fully rational, behaving as Nature does in

3990-400: A part of air is transmuted into water or earth), so the pneuma itself does not wholly transform into the elements. From the elements the one substance is transformed into the multitude of individual things in the orderly cosmos, which is itself a living thing or being, and the pneuma pervading it, and conditioning life and growth everywhere, is its soul . The process of differentiation

SECTION 20

#1732851342207

4180-440: A person is insane. To the Stoics nothing passes unexplained; there is a reason ( Logos ) for everything in nature. Because of the Stoics' commitment to the unity and cohesion of the cosmos and its all-encompassing reason, they fully embraced determinism . However instead of a single chain of causal events, there is instead a many-dimensional network of events interacting within the framework of fate. Out of this swarm of causes,

4370-664: A sentence like "yesterday was Sunday and the weather was good". It is only true if both of its input variables, p {\displaystyle p} ("yesterday was Sunday") and q {\displaystyle q} ("the weather was good"), are true. In all other cases, the expression as a whole is false. Other important logical connectives are ¬ {\displaystyle \lnot } ( not ), ∨ {\displaystyle \lor } ( or ), → {\displaystyle \to } ( if...then ), and ↑ {\displaystyle \uparrow } ( Sheffer stroke ). Given

4560-419: A significant achievement. For the Stoics, truth is distinguished from error by the sage who possesses right reason. Chrysippus's theory of knowledge was empirical . The senses transmit messages from the external world, and their reports are controlled not by referring them to innate ideas, but by comparing them to previous reports stored in the mind. Zeno had defined impressions of sense as "an impression in

4750-484: A single whole, its variety being referred to varying stages of condensation in pneuma . So, too, the human soul must possess absolute simplicity, its varying functions being conditioned by the degrees of its tension. There are no separate "parts" of the soul, as previous thinkers imagined. With this psychology is intimately connected the Stoic theory of knowledge . From the unity of soul it follows that all mental processes—sensation, assent, impulse—proceed from reason,

4940-420: Is sound when its proof system cannot derive a conclusion from a set of premises unless it is semantically entailed by them. In other words, its proof system cannot lead to false conclusions, as defined by the semantics. A system is complete when its proof system can derive every conclusion that is semantically entailed by its premises. In other words, its proof system can lead to any true conclusion, as defined by

5130-471: Is a red planet". For most types of logic, it is accepted that premises and conclusions have to be truth-bearers . This means that they have a truth value : they are either true or false. Contemporary philosophy generally sees them either as propositions or as sentences . Propositions are the denotations of sentences and are usually seen as abstract objects . For example, the English sentence "the tree

5320-441: Is a restricted version of classical logic. It uses the same symbols but excludes some rules of inference. For example, according to the law of double negation elimination, if a sentence is not not true, then it is true. This means that A {\displaystyle A} follows from ¬ ¬ A {\displaystyle \lnot \lnot A} . This is a valid rule of inference in classical logic but it

5510-416: Is also known as symbolic logic and is widely used in mathematical logic . It uses a formal approach to study reasoning: it replaces concrete expressions with abstract symbols to examine the logical form of arguments independent of their concrete content. In this sense, it is topic-neutral since it is only concerned with the abstract structure of arguments and not with their concrete content. Formal logic

5700-453: Is an example of the existential quantifier " ∃ {\displaystyle \exists } " applied to the individual variable " x {\displaystyle x} " . In higher-order logics, quantification is also allowed over predicates. This increases its expressive power. For example, to express the idea that Mary and John share some qualities, one could use the formula " ∃ Q ( Q ( M

5890-415: Is blurry in some cases, such as when the premises offer weak but non-negligible support. This contrasts with deductive arguments, which are either valid or invalid with nothing in-between. The terminology used to categorize ampliative arguments is inconsistent. Some authors, like James Hawthorne, use the term " induction " to cover all forms of non-deductive arguments. But in a more narrow sense, induction

Chrysippus - Misplaced Pages Continue

6080-442: Is co-fated with being willing to lie with a woman. ... For many things cannot occur without our being willing and indeed contributing a most strenuous eagerness and zeal for these things, since, he says, it was fated for these things to occur in conjunction with this personal effort. ... But it will be in our power, he says, with what is in our power being included in fate. Thus our actions are predetermined, and are causally related to

6270-421: Is commonly defined in terms of arguments or inferences as the study of their correctness. An argument is a set of premises together with a conclusion. An inference is the process of reasoning from these premises to the conclusion. But these terms are often used interchangeably in logic. Arguments are correct or incorrect depending on whether their premises support their conclusion. Premises and conclusions, on

6460-407: Is controversial because it belongs to the field of psychology , not logic, and because appearances may be different for different people. Fallacies are usually divided into formal and informal fallacies. For formal fallacies, the source of the error is found in the form of the argument. For example, denying the antecedent is one type of formal fallacy, as in "if Othello is a bachelor, then he

6650-453: Is deductively valid. For deductive validity, it does not matter whether the premises or the conclusion are actually true. So the argument "(1) all frogs are mammals; (2) no cats are mammals; (3) therefore no cats are frogs" is also valid because the conclusion follows necessarily from the premises. According to an influential view by Alfred Tarski , deductive arguments have three essential features: (1) they are formal, i.e. they depend only on

6840-436: Is determined by destiny, there is no individual responsibility, since what has been once foreordained must happen, come what may, Chrysippus replied that there is a distinction to be made between simple and complex predestination. Becoming ill may be fated whatever happens but, if a person's recovery is linked to consulting a doctor, then consulting the doctor is fated to occur together with that person's recovery, and this becomes

7030-498: Is divided by a plane parallel to its base, are the surfaces of the segments equal or unequal? If they are equal, then the cone becomes a cylinder ; if they are unequal, then the surface of the cone must be stepped . The reply of Chrysippus was that the surfaces are both equal and unequal. Chrysippus was, in effect, negating the law of excluded middle with respect to the equal and unequal, and thus he may have anticipated an important principle of modern infinitesimal calculus , namely,

7220-413: Is either true or false, and this must apply to future events as well: If any motion exists without a cause, then not every proposition will be either true or false. For that which has not efficient causes is neither true nor false. But every proposition is either true or false. Therefore, there is no motion without a cause. And if this is so, then all effects owe their existence to prior causes. And if this

7410-467: Is established by verification using a proof. Intuitionistic logic is especially prominent in the field of constructive mathematics , which emphasizes the need to find or construct a specific example to prove its existence. Stoic physics Stoic physics refers to the natural philosophy of the Stoic philosophers of ancient Greece and Rome which they used to explain the natural processes at work in

7600-444: Is experience and discursive thought, which manipulates the materials of sense . Our ideas are copied from stored-up sensations. Just as a relaxation in tension brings about the dissolution of the universe; so in the body, a relaxation of tension, accounts for sleep , decay, and death for the human body . After death the disembodied soul can only maintain its separate existence, even for a limited time, by mounting to that region of

7790-606: Is green" is different from the German sentence "der Baum ist grün" but both express the same proposition. Propositional theories of premises and conclusions are often criticized because they rely on abstract objects. For instance, philosophical naturalists usually reject the existence of abstract objects. Other arguments concern the challenges involved in specifying the identity criteria of propositions. These objections are avoided by seeing premises and conclusions not as propositions but as sentences, i.e. as concrete linguistic objects like

Chrysippus - Misplaced Pages Continue

7980-481: Is in itself" and gave examples of propositions such as "it is day" and "Dion is walking." He distinguished between simple and non-simple propositions, which in modern terminology are known as atomic and molecular propositions. A simple proposition is an elementary statement such as "it is day." Simple propositions are linked together to form non-simple propositions by the use of logical connectives . Chrysippus enumerated five kinds of molecular propositions according to

8170-432: Is interested in deductively valid arguments, for which the truth of their premises ensures the truth of their conclusion. This means that it is impossible for the premises to be true and the conclusion to be false. For valid arguments, the logical structure of the premises and the conclusion follows a pattern called a rule of inference . For example, modus ponens is a rule of inference according to which all arguments of

8360-415: Is invalid in intuitionistic logic. Another classical principle not part of intuitionistic logic is the law of excluded middle . It states that for every sentence, either it or its negation is true. This means that every proposition of the form A ∨ ¬ A {\displaystyle A\lor \lnot A} is true. These deviations from classical logic are based on the idea that truth

8550-432: Is light"). Though the earlier Megarian dialecticians –  Diodorus Cronus and Philo  – had worked in this field and the pupils of Aristotle –  Theophrastus and Eudemus  – had investigated hypothetical syllogisms , it was Chrysippus who developed these principles into a coherent system of propositional logic. Chrysippus defined a proposition as "that which is capable of being denied or affirmed as it

8740-400: Is likely they saw them as merely two aspects of the single material cosmos. Pneuma , from this perspective, is not a special substance intermingled with passive matter, but rather it could be said that the material world has pneumatic qualities. The diversity of the world is explained through the transformations and products of this eternal principle. Like Aristotle, the Stoics conceived of

8930-447: Is male; Othello is not a bachelor; therefore Othello is not male". But most fallacies fall into the category of informal fallacies, of which a great variety is discussed in the academic literature. The source of their error is usually found in the content or the context of the argument. Informal fallacies are sometimes categorized as fallacies of ambiguity, fallacies of presumption, or fallacies of relevance. For fallacies of ambiguity,

9120-688: Is necessary, then it is also possible. This means that ◊ A {\displaystyle \Diamond A} follows from ◻ A {\displaystyle \Box A} . Another principle states that if a proposition is necessary then its negation is impossible and vice versa. This means that ◻ A {\displaystyle \Box A} is equivalent to ¬ ◊ ¬ A {\displaystyle \lnot \Diamond \lnot A} . Other forms of modal logic introduce similar symbols but associate different meanings with them to apply modal logic to other fields. For example, deontic logic concerns

9310-518: Is necessary. For example, if the formula B ( s ) {\displaystyle B(s)} stands for the sentence "Socrates is a banker" then the formula ◊ B ( s ) {\displaystyle \Diamond B(s)} articulates the sentence "It is possible that Socrates is a banker". To include these symbols in the logical formalism, modal logic introduces new rules of inference that govern what role they play in inferences. One rule of inference states that, if something

9500-417: Is no longer, as Plato maintained, hot or hard or bright by partaking in abstract heat or hardness or brightness, but by containing within its own substance the material of these pneuma currents in various degrees of tension. As to the relation between the active and the passive principles there was no clear difference. Although the Stoics talked about the active and passive as two separate types of body, it

9690-476: Is not eternal; it continues only until the time of the restoration of all things. For the cosmos will in turn decay, and the tension which has been relaxed will again be tightened. Things will gradually resolve into elements, and the elements into the primary substance, to be consumed in a general conflagration when once more the world will be absorbed in God. This ekpyrôsis is not so much a catastrophic event, but rather

SECTION 50

#1732851342207

9880-407: Is not the best or most likely explanation. Not all arguments live up to the standards of correct reasoning. When they do not, they are usually referred to as fallacies . Their central aspect is not that their conclusion is false but that there is some flaw with the reasoning leading to this conclusion. So the argument "it is sunny today; therefore spiders have eight legs" is fallacious even though

10070-541: Is only one type of ampliative argument alongside abductive arguments . Some philosophers, like Leo Groarke, also allow conductive arguments as another type. In this narrow sense, induction is often defined as a form of statistical generalization. In this case, the premises of an inductive argument are many individual observations that all show a certain pattern. The conclusion then is a general law that this pattern always obtains. In this sense, one may infer that "all elephants are gray" based on one's past observations of

10260-443: Is possible for us to act differently or not. This rather subtle position which attempts to reconcile determinism with human responsibility is known as soft-determinism, or compatibilism . Chrysippus also argued for the existence of fate based on divination , which he thought there was good evidence for. It would not be possible for diviners to predict the future if the future itself was accidental. Omens and portents, he believed, are

10450-463: Is possible; here the Stoics agreed with the Epicureans. It is necessary, therefore, that assent should not be given indiscriminately; we must determine a criterion of truth, a special formal test whereby reason may recognize the merely plausible and hold fast the true. The earlier Stoics made right reason the standard of truth. Zeno compared sensation to the outstretched hand , flat and open; bending

10640-440: Is so, all things happen by fate. It follows therefore that whatever happens, happens by fate. The Stoic view of fate is entirely based on a view of the universe as a whole. Individual things and persons only come into consideration as dependent parts of this whole. Everything is, in every respect, determined by this relation, and is consequently subject to the general order of the world. If his opponents objected that, if everything

10830-403: Is something acting within them, "a spirit deeply interfused," germinating and developing from within. In one sense the Stoics believed that this is the best of all possible worlds . Only God or Nature is good, and Nature is perfectly rational. It is an organic unity and completely ordered. The goodness of Nature manifests in the way it works to arrange things in the most rational way. For

11020-430: Is studied by formal logic. The study of natural language arguments comes with various difficulties. For example, natural language expressions are often ambiguous, vague, and context-dependent. Another approach defines informal logic in a wide sense as the normative study of the standards, criteria, and procedures of argumentation. In this sense, it includes questions about the role of rationality , critical thinking , and

11210-450: Is the body which confines and shelters the atoms of soul. This corporeal soul is reason , mind , and ruling principle; in virtue of its divine origin Cleanthes can say to Zeus, "We too are thy offspring," and Seneca can calmly insist that, if man and God are not on perfect equality, the superiority rests rather on our side. What God is for the world, the soul is for humans. The cosmos is

11400-464: Is the creative force (God) which develops and shapes the universal order ( cosmos ). God is everything that exists. In the original state, the pneuma-God and the cosmos are absolutely identical; but even then tension, the essential attribute of matter, is at work. In the primitive pneuma there resides the utmost heat and tension, within which there is a pressure , an expansive and dispersive tendency. Motion backwards and forwards once set up cools

11590-404: Is the origin of the elements , life, and human rationality. The cosmos proceeds from an original state in utmost heat, and, in the cooling and separation that occurs, all things appear which are only different and stages in the change of primitive being. Eventually though, the world will be reabsorbed into the primary substance, to be consumed in a general conflagration ( ekpyrôsis ), out of which

SECTION 60

#1732851342207

11780-410: Is the set of basic symbols used in expressions . The syntactic rules determine how these symbols may be arranged to result in well-formed formulas. For instance, the syntactic rules of propositional logic determine that " P ∧ Q {\displaystyle P\land Q} " is a well-formed formula but " ∧ Q {\displaystyle \land Q} " is not since

11970-430: Is the study of deductively valid inferences or logical truths . It examines how conclusions follow from premises based on the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies , critical thinking , and argumentation theory . Informal logic examines arguments expressed in natural language whereas formal logic uses formal language . When used as

12160-668: Is to live in accordance with one's experience of the actual course of nature. A person's individual nature is part of the nature of the whole universe, and thus life should be lived in accordance with one's own human nature as well as that of the universe. Human nature is ethical, and humanity is akin to the Divine, emanating from the primal fire or aether, which, though material, is the embodiment of reason; and people should conduct themselves accordingly. People have freedom, and this freedom consists in emancipation from irrational desires (lust, riches, position in life, domination, etc.) and in subjecting

12350-432: Is to study the criteria according to which an argument is correct or incorrect. A fallacy is committed if these criteria are violated. In the case of formal logic, they are known as rules of inference . They are definitory rules, which determine whether an inference is correct or which inferences are allowed. Definitory rules contrast with strategic rules. Strategic rules specify which inferential moves are necessary to reach

12540-540: Is unable to address. Both provide criteria for assessing the correctness of arguments and distinguishing them from fallacies. Many characterizations of informal logic have been suggested but there is no general agreement on its precise definition. The most literal approach sees the terms "formal" and "informal" as applying to the language used to express arguments. On this view, informal logic studies arguments that are in informal or natural language. Formal logic can only examine them indirectly by translating them first into

12730-599: Is used to represent the ideas of knowing something in contrast to merely believing it to be the case. Higher-order logics extend classical logic not by using modal operators but by introducing new forms of quantification. Quantifiers correspond to terms like "all" or "some". In classical first-order logic, quantifiers are only applied to individuals. The formula " ∃ x ( A p p l e ( x ) ∧ S w e e t ( x ) ) {\displaystyle \exists x(Apple(x)\land Sweet(x))} " ( some apples are sweet)

12920-567: Is vulnerable to blows. Thirdly, evils are distributed according to the rational will of Zeus, either to punish the wicked or because they are important to the world-order as a whole. Thus evil is good under disguise, and is ultimately conducive to the best. Chrysippus compared evil to the coarse jest in the comedy ; for, just as the jest, though offensive in itself, improves the piece as a whole, "so too you may criticize evil regarded by itself, yet allow that, taken with all else, it has its use." Chrysippus regarded bodies , surfaces, lines , places ,

13110-431: Is why first-order logic is still more commonly used. Deviant logics are logical systems that reject some of the basic intuitions of classical logic. Because of this, they are usually seen not as its supplements but as its rivals. Deviant logical systems differ from each other either because they reject different classical intuitions or because they propose different alternatives to the same issue. Intuitionistic logic

13300-475: The Herculaneum papyri . Presumably of Phoenician descent, Chrysippus was the son of Apollonius of Tarsus , and he was born at Soli, Cilicia . He was slight in stature, and is reputed to have trained as a long-distance runner. While still young, he lost his substantial inherited property when it was confiscated to the king's treasury. Chrysippus moved to Athens , where he became the disciple of Cleanthes , who

13490-510: The cosmos as being finite with the Earth at the centre and the moon, sun, planets, and fixed stars surrounding it. Similarly, they rejected the possibility of any void (i.e. vacuum) within the cosmos since that would destroy the coherence of the universe and the sympathy of its parts. However, unlike Aristotle, the Stoics saw the cosmos as an island embedded in an infinite void. The cosmos has its own hexis which holds it together and protects it and

13680-406: The earth , hot springs , sparks from the flint , were claimed as the last remnant of pneuma not yet utterly slackened and cold. They appealed also to the speed and expansion of gaseous bodies, to whirlwinds and inflated balloons . In the rational creatures pneuma is manifested in the highest degree of purity and intensity as an emanation from the world-soul . Humans have souls because

13870-488: The existence of God , making use of a teleological argument : If there is anything that humanity cannot produce, the being who produces it is better than humanity. But humanity cannot produce the things that are in the universe – the heavenly bodies, etc. The being, therefore, who produces them is superior to humanity. But who is there that is superior to humanity, except God? Therefore, God exists. Chrysippus spoke of God and gods interchangeably. He interpreted

14060-645: The fingers was assent; the clenched fist was "simple apprehension," the mental grasp of an object; knowledge was the clenched fist tightly held in the other hand. But this criterion was open to the persistent attacks of Epicureans and Academics , who made clear (1) that reason is dependent upon, if not derived from, sense, and (2) that the utterances of reason lack consistency. Chrysippus, therefore, did much to develop Stoic logic , and more clearly defined and safeguarded his predecessors' position. a. Some historians prefer to describe Stoic doctrine as "corporealism" rather than "materialism". One objection to

14250-407: The gods of traditional Greek religion by viewing them as different aspects of the one reality. Cicero tells us that "he further maintained that aether is that which people call Zeus , and that the air which permeates the seas is Poseidon , and that the earth is what is known by the name of Demeter , and he treated in similar style the names of the other gods." In addition, the universe exists for

14440-522: The limit and the process of convergence towards a limit . Chrysippus was notable for claiming that " one " is a number . One was not always considered a number by the ancient Greeks since they viewed one as that by which things are measured. Aristotle in his Metaphysics wrote, "... a measure is not the things measured, but the measure or the One is the beginning of number." Chrysippus asserted that one had "magnitude one" ( Greek : πλῆθος ἕν ), although this

14630-448: The periodic conflagration , when the universe is renewed . There were no universals or abstract objects for Chrysippus, making him a kind of nominalist. For Chrysippus, all things happen according to fate : what seems to be accidental has always some hidden cause. The unity of the world consists in the chain-like dependence of cause upon cause. Nothing can take place without a sufficient cause. According to Chrysippus, every proposition

14820-406: The pneuma of the world-soul pervades the whole universe, this allows human souls to be influenced by divine souls. Omens and portents, Chrysippus explained, are the natural symptoms of certain occurrences. There must be countless indications of the course of providence , for the most part unobserved, the meaning of only a few having become known to humanity. To those who argued that divination

15010-455: The universe . To the Stoics, the cosmos is a single pantheistic god, one which is rational and creative, and which is the basis of everything which exists. Nothing incorporeal exists. The nature of the world is one of unceasing change, driven by the active part or reason ( logos ) of God which pervades all things. The active substance of the world is characterized as a 'breath', or pneuma , which provides form and motion to matter, and

15200-406: The void and time as all being infinitely divisible . He determined one of the principal features of the infinite set : since a man and a finger have an infinite number of parts as do the universe and a man, it cannot be said that a man has more parts than his finger, nor that the universe has more parts than a man. Chrysippus also responded to a problem first posed by Democritus . If a cone

15390-465: The Greek word "logos", which has a variety of translations, such as reason , discourse , or language . Logic is traditionally defined as the study of the laws of thought or correct reasoning , and is usually understood in terms of inferences or arguments . Reasoning is the activity of drawing inferences. Arguments are the outward expression of inferences. An argument is a set of premises together with

15580-403: The Stoic system are the two principles, the active and the passive : everything which exists is capable of acting and being acted upon. The active principle is God acting as the rational principle ( logos ), and which has a higher status than the passive matter ( ousia ). In their earlier writings the Stoics characterised the rational principle as a creative fire, but later accounts stress

15770-501: The Stoics this is therefore the most reasonable, the most rational , of all possible worlds. None of the events which occur by Nature are inherently bad; but nor are they intrinsically 'good' even though they have been caused by a good agent. The natural patterning of the world—life, death, sickness, health, etc.—is made up of morally indifferent events which in themselves are neither good nor bad. Such events are not unimportant, but they only have value in as far as they contribute to

15960-418: The ambiguity and vagueness of natural language are responsible for their flaw, as in "feathers are light; what is light cannot be dark; therefore feathers cannot be dark". Fallacies of presumption have a wrong or unjustified premise but may be valid otherwise. In the case of fallacies of relevance, the premises do not support the conclusion because they are not relevant to it. The main focus of most logicians

16150-402: The antecedent. This corresponds to the modern-day strict conditional . Chrysippus developed a syllogistic or system of deduction in which he made use of five types of basic arguments or argument forms called indemonstrable syllogisms, which played the role of axioms, and four inference rules , called themata by means of which complex syllogisms could be reduced to these axioms. The forms of

16340-399: The assessment of arguments. Premises and conclusions are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises, or in other words, the premises support the conclusion. For instance, the premises "Mars is red" and "Mars is a planet" support the conclusion "Mars

16530-495: The basic principles of classical logic. They introduce additional symbols and principles to apply it to fields like metaphysics , ethics , and epistemology . Modal logic is an extension of classical logic. In its original form, sometimes called "alethic modal logic", it introduces two new symbols: ◊ {\displaystyle \Diamond } expresses that something is possible while ◻ {\displaystyle \Box } expresses that something

16720-450: The benefit of the universal god: We should infer in the case of a beautiful dwelling-place that it was built for its owners and not for mice ; we ought, therefore, in the same way to regard the universe as the dwelling-place of the gods. In response to the question of how evil could exist in a good universe, Chrysippus replied "evil cannot be removed, nor is it well that it should be removed." Firstly, he argued, following Plato , that it

16910-487: The best explanation, for example, when a doctor concludes that a patient has a certain disease which explains the symptoms they suffer. Arguments that fall short of the standards of correct reasoning often embody fallacies . Systems of logic are theoretical frameworks for assessing the correctness of arguments. Logic has been studied since antiquity . Early approaches include Aristotelian logic , Stoic logic , Nyaya , and Mohism . Aristotelian logic focuses on reasoning in

17100-432: The chance movements of primitive atoms . In the Stoic system material substance has a continuous structure, held together by tension ( tonos ) as the essential attribute of body. This tension is a property of the pneuma , and physical bodies are held together by the pneuma which is in a continual state of motion. The various pneuma currents combining give objects their stable, physical properties ( hexis ). A thing

17290-505: The change of primitive being which Heraclitus had called "a progress up and down". Out of it is separated elemental fire , the fire which we know, which burns and destroys; and this condenses into air ; a further step in the downward path produces water and earth from the solidification of air. At every stage the degree of tension is slackened, and the resulting element approaches more and more to "inert" matter. But, just as one element does not wholly transform into another (e.g. only

17480-408: The character of the agent. Our actions would only be involuntary if they were produced by external causes alone, without any co-operation, on the part of our wills, with external causes. Virtue and vice are set down as things in our power, for which, consequently, we are responsible. Moral responsibility depends only on freedom of the will, and what emanates from our will is our own, no matter whether it

17670-645: The claim "either it is raining, or it is not". These two definitions of formal logic are not identical, but they are closely related. For example, if the inference from p to q is deductively valid then the claim "if p then q " is a logical truth. Formal logic uses formal languages to express and analyze arguments. They normally have a very limited vocabulary and exact syntactic rules . These rules specify how their symbols can be combined to construct sentences, so-called well-formed formulas . This simplicity and exactness of formal logic make it capable of formulating precise rules of inference. They determine whether

17860-495: The color of elephants. A closely related form of inductive inference has as its conclusion not a general law but one more specific instance, as when it is inferred that an elephant one has not seen yet is also gray. Some theorists, like Igor Douven, stipulate that inductive inferences rest only on statistical considerations. This way, they can be distinguished from abductive inference. Abductive inference may or may not take statistical observations into consideration. In either case,

18050-502: The conclusion "I don't have to work". Premises and conclusions express propositions or claims that can be true or false. An important feature of propositions is their internal structure. For example, complex propositions are made up of simpler propositions linked by logical vocabulary like ∧ {\displaystyle \land } ( and ) or → {\displaystyle \to } ( if...then ). Simple propositions also have parts, like "Sunday" or "work" in

18240-511: The conclusion "all ravens are black". A further approach is to define informal logic as the study of informal fallacies . Informal fallacies are incorrect arguments in which errors are present in the content and the context of the argument. A false dilemma , for example, involves an error of content by excluding viable options. This is the case in the fallacy "you are either with us or against us; you are not with us; therefore, you are against us". Some theorists state that formal logic studies

18430-458: The conclusion is true. Some theorists, like John Stuart Mill , give a more restrictive definition of fallacies by additionally requiring that they appear to be correct. This way, genuine fallacies can be distinguished from mere mistakes of reasoning due to carelessness. This explains why people tend to commit fallacies: because they have an alluring element that seduces people into committing and accepting them. However, this reference to appearances

18620-591: The conditional proposition p → q {\displaystyle p\to q} , one can form truth tables of its converse q → p {\displaystyle q\to p} , its inverse ( ¬ p → ¬ q {\displaystyle \lnot p\to \lnot q} ) , and its contrapositive ( ¬ q → ¬ p {\displaystyle \lnot q\to \lnot p} ) . Truth tables can also be defined for more complex expressions that use several propositional connectives. Logic

18810-414: The connective used: Thus several types of molecular propositions, familiar to modern logic, were listed by Chrysippus, including the conjunction , the disjunction , and the conditional , and Chrysippus studied their criteria of truth closely. The first logicians to debate conditional statements were Diodorus Cronus and his pupil Philo . Writing five-hundred years later, Sextus Empiricus refers to

19000-438: The contrast between necessity and possibility and the problem of ethical obligation and permission. Similarly, it does not address the relations between past, present, and future. Such issues are addressed by extended logics. They build on the basic intuitions of classical logic and expand it by introducing new logical vocabulary. This way, the exact logical approach is applied to fields like ethics or epistemology that lie beyond

19190-596: The cosmos always unfolds according to the best possible reason , any succeeding world is likely to be identical to the previous one. Thus in the same way that the cosmos occupies a finite space in an infinite void, so it can be understood to occupy a finite period in an infinite span of time. Ekpyrosis itself however, was not a universally accepted theory by all Stoics. Other prominent stoics such as Panaetius , Zeno of Tarsus , Boethus of Sidon , and others either rejected Ekpyrosis or had differing opinions regarding its degree. A strong acceptance of Aristotle 's theories of

19380-476: The course of events is fully realised. Humans appear to have free will because personal actions participate in the determined chain of events independently of external conditions. This " soft-determinism " allows humans to be responsible for their own actions, alleviating the apparent arbitrariness of fate. Divination was an essential element of Greek religion , and the Stoics attempted to reconcile it with their own rational doctrine of strict causation. Since

19570-451: The depth level. But they can be highly informative on the surface level by making implicit information explicit. This happens, for example, in mathematical proofs. Ampliative arguments are arguments whose conclusions contain additional information not found in their premises. In this regard, they are more interesting since they contain information on the depth level and the thinker may learn something genuinely new. But this feature comes with

19760-436: The disturbing element in right judgment. Chrysippus wrote a whole book, On Passions ( Greek : Περὶ παθῶν ), concerning the therapy of the emotions. The passions are like diseases which depress and crush the soul, thus he sought to eradicate them ( apatheia ). Wrong judgements turn into passions when they gather an impetus of their own, just as, when one has started running, it is difficult to stop. One cannot hope to eradicate

19950-498: The divine reason ( logos ) which governs the universe , of which human beings are a part. The goal was to find valid rules of inference and forms of proof to help people find their way in life. According to Sextus Empiricus, Chrysippus held that dogs use disjunctive syllogism, such as when using scent to pick which path to run down. This was in contrast to a tradition since Aristotle, who saw reasoning (and reasoning deductively) as man's defining aspect. Chrysippus analyzed speech and

20140-409: The example. The truth of a proposition usually depends on the meanings of all of its parts. However, this is not the case for logically true propositions. They are true only because of their logical structure independent of the specific meanings of the individual parts. Arguments can be either correct or incorrect. An argument is correct if its premises support its conclusion. Deductive arguments have

20330-434: The field of ethics and introduces symbols to express the ideas of obligation and permission , i.e. to describe whether an agent has to perform a certain action or is allowed to perform it. The modal operators in temporal modal logic articulate temporal relations. They can be used to express, for example, that something happened at one time or that something is happening all the time. In epistemology, epistemic modal logic

20520-409: The five indemonstrables were: Of the four inference rules (themata, θέματα), only two survived. One, the so-called first thema , was a rule of antilogism. The other, the third thema , was a cut rule by which chain syllogisms could be reduced to simple syllogisms. The purpose of Stoic syllogistic was not merely to create a formal system. It was also understood as the study of the operations of reason,

20710-485: The form "(1) p , (2) if p then q , (3) therefore q " are valid, independent of what the terms p and q stand for. In this sense, formal logic can be defined as the science of valid inferences. An alternative definition sees logic as the study of logical truths . A proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true in all possible worlds and under all interpretations of its non-logical terms, like

20900-508: The form of syllogisms . It was considered the main system of logic in the Western world until it was replaced by modern formal logic, which has its roots in the work of late 19th-century mathematicians such as Gottlob Frege . Today, the most commonly used system is classical logic . It consists of propositional logic and first-order logic . Propositional logic only considers logical relations between full propositions. First-order logic also takes

21090-523: The form of the premises and the conclusion: how they have to be structured for the inference to be valid. Arguments that do not follow any rule of inference are deductively invalid. The modus ponens is a prominent rule of inference. It has the form " p ; if p , then q ; therefore q ". Knowing that it has just rained ( p {\displaystyle p} ) and that after rain the streets are wet ( p → q {\displaystyle p\to q} ), one can use modus ponens to deduce that

21280-419: The form of the premises and the conclusion; (2) they are a priori, i.e. no sense experience is needed to determine whether they obtain; (3) they are modal, i.e. that they hold by logical necessity for the given propositions, independent of any other circumstances. Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference. Rules of inference specify

21470-421: The general form of arguments while informal logic studies particular instances of arguments. Another approach is to hold that formal logic only considers the role of logical constants for correct inferences while informal logic also takes the meaning of substantive concepts into account. Further approaches focus on the discussion of logical topics with or without formal devices and on the role of epistemology for

21660-481: The glowing mass of fiery vapour and weakens the tension. Thus follows the first differentiation of primitive substance—the separation of force from matter, the emanation of the world from God. The seminal Logos which, in virtue of its tension, slumbered in pneuma , now proceeds upon its creative task. The cycle of its transformations and successive condensations constitutes the life of the cosmos. The cosmos and all its parts are only different embodiments and stages in

21850-445: The handling of names and terms. He also devoted much effort in refuting fallacies and paradoxes. According to Diogenes Laërtius, Chrysippus wrote twelve works in 23 books on the Liar paradox ; seven works in 17 books on amphiboly ; and another nine works in 26 books on other conundrums. In all, 28 works or 66 books were given over to puzzles or paradoxes. Chrysippus is the first Stoic for whom

22040-412: The idea of breath, or pneuma , as the active substance. The cosmos is thus filled with an all-pervading pneuma which allows for the cohesion of matter and permits contact between all parts of the cosmos. The pneuma is everywhere coextensive with matter, pervading and permeating it, and, together with it, occupying and filling space. The Epicureans had placed the form and movement of matter in

22230-407: The individual components maintain their own properties, and they can be separated again. The second type was a fusion, whereby a new substance is created leading to the loss of the properties of the individual components, this roughly corresponds to the modern concept of a chemical change. The third type was a commingling, or total blending: there is complete interpenetration of the components down to

22420-473: The infinitesimal, but each component maintains its own properties. In this third type of mixture a new substance is created, but since it still has the qualities of the two original substances, it is possible to extract them again. In the words of Chrysippus: "there is nothing to prevent one drop of wine from mixing with the whole ocean". Ancient critics often regarded this type of mixing as paradoxical since it apparently implied that each constituent substance be

22610-406: The internal parts of propositions into account, like predicates and quantifiers . Extended logics accept the basic intuitions behind classical logic and apply it to other fields, such as metaphysics , ethics , and epistemology . Deviant logics, on the other hand, reject certain classical intuitions and provide alternative explanations of the basic laws of logic. The word "logic" originates from

22800-407: The internal structure of propositions. This happens through devices such as singular terms, which refer to particular objects, predicates , which refer to properties and relations, and quantifiers, which treat notions like "some" and "all". For example, to express the proposition "this raven is black", one may use the predicate B {\displaystyle B} for the property "black" and

22990-522: The logical conjunction ∧ {\displaystyle \land } requires terms on both sides. A proof system is a collection of rules to construct formal proofs. It is a tool to arrive at conclusions from a set of axioms. Rules in a proof system are defined in terms of the syntactic form of formulas independent of their specific content. For instance, the classical rule of conjunction introduction states that P ∧ Q {\displaystyle P\land Q} follows from

23180-482: The materialism label relates to a narrow 17th/18th-century conception of materialism whereby things must be "explained by the movements and combination of passive matter" ( Gourinat 2009 , p. 48). Since Stoicism is vitalistic it is "not materialism in the strict sense" ( Gourinat 2009 , p. 68). A second objection refers to a Stoic distinction between mere bodies (which extend in three dimensions and offer resistance), and material bodies which are "constituted by

23370-754: The natural order. In pursuing their physics the Stoics wanted to create a picture of the world which would be completely coherent. Stoic physics can be described in terms of (a) monism , (b) materialism , and (c) dynamism. Stoicism is a pantheistic philosophy. The cosmos is active, life-giving, rational and creative. It is a single cohesive unit, a self-supporting entity containing within it all that it needs, and all parts depending on mutual exchange with each other. Different parts of this unified structure are able to interact and have an affinity with each other ( sympatheia ). The Stoics explained everything from natural events to human conduct as manifestations of an all-pervading reason ( logos ). Thus they identified

23560-435: The natural symptoms of certain occurrences. There must be countless indications of the course of providence, for the most part unobserved, the meaning of only a few having become known to humanity. To those who argued that divination was superfluous as all events are foreordained, he replied that both divination and our behaviour under the warnings which it affords are included in the chain of causation. The Stoics believed that

23750-424: The object. True presentations are distinguished from those that are false by the use of memory, classification and comparison. If the sense organ and the mind are healthy – and provided that an external object can be really seen or heard – the presentation, due to its clearness and distinctness, has the power to extort the assent that always lies in our power, to give or to withhold. In

23940-409: The other hand, are true or false depending on whether they are in accord with reality. In formal logic, a sound argument is an argument that is both correct and has only true premises. Sometimes a distinction is made between simple and complex arguments. A complex argument is made up of a chain of simple arguments. This means that the conclusion of one argument acts as a premise of later arguments. For

24130-444: The other hand, do not have propositional parts. But they can also be conceived as having an internal structure: they are made up of subpropositional parts, like singular terms and predicates . For example, the simple proposition "Mars is red" can be formed by applying the predicate "red" to the singular term "Mars". In contrast, the complex proposition "Mars is red and Venus is white" is made up of two simple propositions connected by

24320-401: The overarching network of fate, but nevertheless the moral responsibility of how we respond to impressions remains our own. The one all-determining power is active everywhere, working in each particular being according to its nature, whether in rational or irrational creatures or in inorganic objects. Every action is brought about by the co-operation of causes depending on the nature of things and

24510-439: The passions when one is in the heat of love or anger: this can only be done when one is calm. Therefore, one should prepare in advance, and deal with the passions in the mind as if they were present. By applying reason to passions such as greed, pride, or lust, one can understand the harm which they cause. Logic Logic is the study of correct reasoning . It includes both formal and informal logic . Formal logic

24700-462: The past, but also against all possible attack in the future. He took the doctrines of Zeno and Cleanthes and crystallized them into what became the definitive system of Stoicism . He elaborated the physical doctrines of the Stoics and their theory of knowledge and he created much of their formal logic. In short, Chrysippus made the Stoic system what it was. It was said that "without Chrysippus, there would have been no Stoa". Chrysippus wrote much on

24890-440: The period of the cosmic cycle when the preponderance of the fiery element once again reaches its maximum. All matter is consumed becoming completely fiery and wholly soul-like. God, at this point, can be regarded as completely existing in itself. In due order a new cycle of the cosmos begins ( palingenesis ), reproducing the previous world, and so on forever. Therefore, the same events play out again repeated endlessly. Since

25080-485: The premises P {\displaystyle P} and Q {\displaystyle Q} . Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are different types of proof systems including natural deduction and sequent calculi . A semantics is a system for mapping expressions of a formal language to their denotations. In many systems of logic, denotations are truth values. For instance,

25270-413: The premises offer support for the conclusion because the conclusion is the best explanation of why the premises are true. In this sense, abduction is also called the inference to the best explanation . For example, given the premise that there is a plate with breadcrumbs in the kitchen in the early morning, one may infer the conclusion that one's house-mate had a midnight snack and was too tired to clean

25460-470: The premises. But this point is not always accepted since it would mean, for example, that most of mathematics is uninformative. A different characterization distinguishes between surface and depth information. The surface information of a sentence is the information it presents explicitly. Depth information is the totality of the information contained in the sentence, both explicitly and implicitly. According to this view, deductive inferences are uninformative on

25650-415: The presence of incorporeal things such as void, place and time, but although real they could not exist and were said to "subsist". Stoicism was thus fully materialistic; the answers to metaphysics are to be sought in physics; particularly the problem of the causes of things for which Plato's theory of forms and Aristotle 's " substantial form " had been put forth as solutions. A dualistic feature of

25840-481: The presence with one another of both [active and passive] principles, and by the effects of one principle on the other". The active and passive principles are bodies but not material bodies under this definition ( Cooper 2009 , p. 100). b. The concept of pneuma (as a "vital breath") was prominent in the Hellenistic medical schools. Its precise relationship to the "creative fire" ( pyr technikon ) of

26030-414: The proposition "if it is day, then I am talking" can be false, it is invalid. However, paradoxical propositions were still possible such as "if atomic elements of things do not exist, atomic elements exists." Chrysippus adopted a much stricter view regarding conditional propositions, which made such paradoxes impossible: to him, a conditional is true if denial of the consequent is logically incompatible with

26220-485: The propositional connective "and". Whether a proposition is true depends, at least in part, on its constituents. For complex propositions formed using truth-functional propositional connectives, their truth only depends on the truth values of their parts. But this relation is more complicated in the case of simple propositions and their subpropositional parts. These subpropositional parts have meanings of their own, like referring to objects or classes of objects. Whether

26410-406: The propositions are formed. For example, the syllogism "all men are mortal; Socrates is a man; therefore Socrates is mortal" is valid. The syllogism "all cats are mortal; Socrates is mortal; therefore Socrates is a cat", on the other hand, is invalid. Classical logic is distinct from traditional or Aristotelian logic. It encompasses propositional logic and first-order logic. It is "classical" in

26600-417: The psychology of argumentation. Another characterization identifies informal logic with the study of non-deductive arguments. In this way, it contrasts with deductive reasoning examined by formal logic. Non-deductive arguments make their conclusion probable but do not ensure that it is true. An example is the inductive argument from the empirical observation that "all ravens I have seen so far are black" to

26790-437: The quotations of others. He was considered diffuse and obscure in his utterances and careless in his style, but his abilities were highly regarded, and he came to be seen as a preeminent authority for the school. He died during the 143rd Olympiad (208–204 BC) at the age of 73. Diogenes Laërtius gives two different accounts of his death. In the first account, Chrysippus was seized with dizziness having drunk undiluted wine at

26980-476: The receptacle of each other. However to the Stoics, the pneuma is like a force, a continuous field interpenetrating matter and spreading through all of space. Every character and property of a particular thing is determined solely by the tension in it of pneuma , and pneuma , though present in all things, varies indefinitely in quantity and intensity. A certain warmth, akin to the vital heat of organic being, seems to be found in inorganic nature: vapours from

27170-485: The reverse order: earth being rarefied into water, water into air and air into fire. The human soul was divided by Chrysippus into eight faculties: the five senses, the power of reproduction , the power of speech , and the "ruling part" that is located in the chest rather than the head. Individual souls are perishable; but, according to the view originated by Chrysippus, the souls of wise people survive longer after their death. No individual soul can, however, survive beyond

27360-436: The rules of inference they accept as valid and the formal languages used to express them. Starting in the late 19th century, many new formal systems have been proposed. There are disagreements about what makes a formal system a logic. For example, it has been suggested that only logically complete systems, like first-order logic , qualify as logics. For such reasons, some theorists deny that higher-order logics are logics in

27550-452: The ruling part; the one rational soul alone has sensations, assents to judgments, is impelled towards objects of desire just as much as it thinks or reasons. Not that all these powers at once reach full maturity. The soul at first is empty of content; in the embryo it has not developed beyond the nutritive principle of a plant; at birth the "ruling part" is a blank tablet , although ready prepared to receive writing. The source of knowledge

27740-492: The scope of mathematics. Propositional logic comprises formal systems in which formulae are built from atomic propositions using logical connectives . For instance, propositional logic represents the conjunction of two atomic propositions P {\displaystyle P} and Q {\displaystyle Q} as the complex formula P ∧ Q {\displaystyle P\land Q} . Unlike predicate logic where terms and predicates are

27930-418: The semantics for classical propositional logic assigns the formula P ∧ Q {\displaystyle P\land Q} the denotation "true" whenever P {\displaystyle P} and Q {\displaystyle Q} are true. From the semantic point of view, a premise entails a conclusion if the conclusion is true whenever the premise is true. A system of logic

28120-604: The semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly. Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments. For over two thousand years, Aristotelian logic was treated as the canon of logic in the Western world, but modern developments in this field have led to a vast proliferation of logical systems. One prominent categorization divides modern formal logical systems into classical logic , extended logics, and deviant logics . Aristotelian logic encompasses

28310-404: The sense organ, into the percipient's mind. The quality transmitted appears as a disturbance or impression upon the corporeal surface of that "thinking thing," the soul. In the example of sight , a conical pencil of rays diverges from the pupil of the eye , so that its base covers the object seen. A presentation is conveyed, by an air-current, from the sense organ, here the eye, to the mind, i.e.

28500-518: The sense that it is based on basic logical intuitions shared by most logicians. These intuitions include the law of excluded middle , the double negation elimination , the principle of explosion , and the bivalence of truth. It was originally developed to analyze mathematical arguments and was only later applied to other fields as well. Because of this focus on mathematics, it does not include logical vocabulary relevant to many other topics of philosophical importance. Examples of concepts it overlooks are

28690-404: The simple proposition "Mars is red", are true or false. In such cases, the truth is called a logical truth: a proposition is logically true if its truth depends only on the logical vocabulary used in it. This means that it is true under all interpretations of its non-logical terms. In some modal logics , this means that the proposition is true in all possible worlds. Some theorists define logic as

28880-415: The simple proposition they form is true depends on their relation to reality, i.e. what the objects they refer to are like. This topic is studied by theories of reference . Some complex propositions are true independently of the substantive meanings of their parts. In classical logic, for example, the complex proposition "either Mars is red or Mars is not red" is true independent of whether its parts, like

29070-870: The singular term r {\displaystyle r} referring to the raven to form the expression B ( r ) {\displaystyle B(r)} . To express that some objects are black, the existential quantifier ∃ {\displaystyle \exists } is combined with the variable x {\displaystyle x} to form the proposition ∃ x B ( x ) {\displaystyle \exists xB(x)} . First-order logic contains various rules of inference that determine how expressions articulated this way can form valid arguments, for example, that one may infer ∃ x B ( x ) {\displaystyle \exists xB(x)} from B ( r ) {\displaystyle B(r)} . Extended logics are logical systems that accept

29260-474: The smallest units, propositional logic takes full propositions with truth values as its most basic component. Thus, propositional logics can only represent logical relationships that arise from the way complex propositions are built from simpler ones. But it cannot represent inferences that result from the inner structure of a proposition. First-order logic includes the same propositional connectives as propositional logic but differs from it because it articulates

29450-416: The soul is purely passive and the impression reveals not only its own existence, but that also of its cause – just as light displays itself and the elements that are in it. The power to name the object resides in the understanding. First must come the impression, and the understanding – having the power of utterance – expresses in speech the affection it receives from

29640-420: The soul" and this was interpreted literally by Cleanthes, who compared the impression on the soul to the impression made by a seal on wax. Chrysippus preferred to regard it as an alteration or change in the soul; that is, the soul receives a modification from every external object that acts upon it, just as the air receives countless strokes when many people are speaking at once. In the receipt of an impression,

29830-487: The soul's "ruling part." The presentation, besides attesting its own existence, gives further information of its object—such as colour or size. Zeno and Cleanthes compared this presentation to the impression which a seal bears upon wax , while Chrysippus determined it more vaguely as a hidden modification or mode of mind. But the mind is no mere passive recipient of impressions: the mind assents or dissents. The contents of experience are not all true or valid: hallucination

30020-418: The streets are wet ( q {\displaystyle q} ). The third feature can be expressed by stating that deductively valid inferences are truth-preserving: it is impossible for the premises to be true and the conclusion to be false. Because of this feature, it is often asserted that deductive inferences are uninformative since the conclusion cannot arrive at new information not already present in

30210-437: The strict sense. When understood in a wide sense, logic encompasses both formal and informal logic. Informal logic uses non-formal criteria and standards to analyze and assess the correctness of arguments. Its main focus is on everyday discourse. Its development was prompted by difficulties in applying the insights of formal logic to natural language arguments. In this regard, it considers problems that formal logic on its own

30400-550: The strongest form of support: if their premises are true then their conclusion must also be true. This is not the case for ampliative arguments, which arrive at genuinely new information not found in the premises. Many arguments in everyday discourse and the sciences are ampliative arguments. They are divided into inductive and abductive arguments. Inductive arguments are statistical generalizations, such as inferring that all ravens are black based on many individual observations of black ravens. Abductive arguments are inferences to

30590-438: The study of logical truths. Truth tables can be used to show how logical connectives work or how the truth values of complex propositions depends on their parts. They have a column for each input variable. Each row corresponds to one possible combination of the truth values these variables can take; for truth tables presented in the English literature, the symbols "T" and "F" or "1" and "0" are commonly used as abbreviations for

30780-700: The subject in two ways: either by affirming it or by denying it. For example, the proposition "Socrates is not a cat" involves the denial of the predicate "cat" to the subject "Socrates". Using combinations of subjects and predicates, a great variety of propositions and syllogisms can be formed. Syllogisms are characterized by the fact that the premises are linked to each other and to the conclusion by sharing one predicate in each case. Thus, these three propositions contain three predicates, referred to as major term , minor term , and middle term . The central aspect of Aristotelian logic involves classifying all possible syllogisms into valid and invalid arguments according to how

30970-400: The subject of logic and created a system of propositional logic . Aristotle 's term logic had been concerned with the interrelations of terms such as "Socrates" or "man" ("all men are mortal, Socrates is a man, so Socrates is mortal"). Stoic logic, on the other hand, was concerned with the interrelations of propositions such as "it is day" ("if it is day, it is light: but it is day: so it

31160-417: The subject to the predicate. For example, the proposition "Socrates is wise" is made up of the subject "Socrates", the predicate "wise", and the copula "is". The subject and the predicate are the terms of the proposition. Aristotelian logic does not contain complex propositions made up of simple propositions. It differs in this aspect from propositional logic, in which any two propositions can be linked using

31350-403: The surrounding void cannot affect it. The cosmos can, however, vary in volume, allowing it to expand and contract in volume through its cycles. The pneuma of the Stoics is the primitive substance which existed before the cosmos. It is the everlasting presupposition of particular things; the totality of all existence; out of it the whole of nature proceeds, eventually to be consumed by it. It

31540-423: The symbols displayed on a page of a book. But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous, meaning an argument's validity would not only depend on its parts but also on its context and on how it is interpreted. Another approach is to understand premises and conclusions in psychological terms as thoughts or judgments. This position is known as psychologism . It

31730-409: The table. This conclusion is justified because it is the best explanation of the current state of the kitchen. For abduction, it is not sufficient that the conclusion explains the premises. For example, the conclusion that a burglar broke into the house last night, got hungry on the job, and had a midnight snack, would also explain the state of the kitchen. But this conclusion is not justified because it

31920-399: The term ampliative or inductive reasoning is used. Deductive arguments are associated with formal logic in contrast to the relation between ampliative arguments and informal logic. A deductively valid argument is one whose premises guarantee the truth of its conclusion. For instance, the argument "(1) all frogs are amphibians; (2) no cats are amphibians; (3) therefore no cats are frogs"

32110-411: The third of the four Stoic categories , i.e. the category somehow disposed is attested. In the surviving evidence, Chrysippus frequently makes use of the categories of substance and quality , but makes little use of the other two Stoic categories ( somehow disposed and somehow disposed in relation to something ). It is not clear whether the categories had any special significance for Chrysippus, and

32300-479: The truth values "true" and "false". The first columns present all the possible truth-value combinations for the input variables. Entries in the other columns present the truth values of the corresponding expressions as determined by the input values. For example, the expression " p ∧ q {\displaystyle p\land q} " uses the logical connective ∧ {\displaystyle \land } ( and ). It could be used to express

32490-515: The universe and God with Zeus , as the ruler and upholder, and at the same time the law, of the universe. The Stoic God is not a transcendent omniscient being standing outside nature, but rather it is immanent —the divine element is immersed in nature itself. God orders the world for the good, and every element of the world contains a portion of the divine element that accounts for its behaviour. The reason of things—that which accounts for them—is not some external end to which they are tending; it

32680-454: The universe has a soul, and human rationality is the same as God's rationality. The pneuma that is soul pervades the entire human body. The soul is corporeal, else it would have no real existence, would be incapable of extension in three dimensions (i.e. to diffuse all over the body), incapable of holding the body together, herein presenting a sharp contrast to the Epicurean tenet that it

32870-431: The universe is God , and Chrysippus affirmed that "the universe itself is God and the universal outpouring of its soul." It is the guiding principle of the universe, "operating in mind and reason, together with the common nature of things and the totality which embraces all existence." Based on these beliefs, physicist and philosopher Max Bernhard Weinstein identified Chrysippus as a Pandeist . Chrysippus sought to prove

33060-411: The universe which is akin to its nature. It was a moot point whether all souls so survive, as Cleanthes thought, or the souls of the wise and good alone, which was the opinion of Chrysippus; in any case, sooner or later individual souls are merged in the soul of the universe, from which they originated. The Stoics explained perception as a transmission of the perceived quality of an object, by means of

33250-518: The universe with God, and the diversity of the world is explained through the transformations and products of God as the rational principle of the cosmos. Philosophers since the time of Plato had asked whether abstract qualities such as justice and wisdom , have an independent existence. Plato in his Sophist dialogue (245e–249d) had argued that since qualities such as virtue and vice cannot be 'touched', they must be something very different from ordinary bodies. The Stoics' answer to this dilemma

33440-543: The universe, and he taught a therapy of extirpating the unruly passions which depress and crush the soul. He initiated the success of Stoicism as one of the most influential philosophical movements for centuries in the Greek and Roman world. The linguistic orientation of Chrysippus' work made it difficult for its students even within the Stoic school. Of his several written works, none have survived except as fragments. Recently, segments of some of his works were discovered among

33630-706: The universe, combined with a more practical lifestyle practiced by the Roman people, caused the later Stoics to focus their main effort on their own social well-being on earth, not on the cosmos. A prime example are the Stoic-influenced writings of the Roman Emperor Marcus Aurelius (121–180). In his Meditations , he chooses to discuss how one should act and live their life, rather than speculate on cosmological theories. The Stoics attempted to incorporate traditional polytheism into their philosophy. Not only

33820-511: The universe. Objects are made up of inert formless matter and an informing soul, " pneuma ", provides form to the undifferentiated matter. The pneuma pervades all of substance and maintains the unity of the universe and constitutes the soul of the human being. The classical elements change into one another by a process of condensation and rarefaction . Fire first becomes solidified into air ; then air into water ; and lastly, water into earth . The process of dissolution takes place in

34010-404: The will to reason. Chrysippus laid the greatest stress on the worth and dignity of the individual, and on the power of will. The Stoics admitted between the good and the bad a third class of things – the indifferent ( adiaphora ). Of things morally indifferent, the best includes health, and riches, and honour, and the worst includes sickness and poverty. Chrysippus accepted that it

34200-448: The works of later authors like Cicero , Seneca , Galen , Plutarch , and others. Recently, segments from Logical Questions and On Providence were discovered among the Herculaneum papyri . A third work by Chrysippus may also be among them. Chrysippus had a long and successful career of resisting the attacks of the Academy and hoped not simply to defend Stoicism against the assaults of

34390-405: Was discussed at length around the turn of the 20th century but it is not widely accepted today. Premises and conclusions have an internal structure. As propositions or sentences, they can be either simple or complex. A complex proposition has other propositions as its constituents, which are linked to each other through propositional connectives like "and" or "if...then". Simple propositions, on

34580-405: Was impossible for good to exist without evil, for justice could not be known without injustice, courage without cowardice, temperance without intemperance or wisdom without foolishness. Secondly, apparent evils exist as a consequent of nature's goodness, thus it was necessary for the human skull to be made from small and thin bones for reasons of utility, but this superior utility meant that the skull

34770-492: Was normal in ordinary usage to refer to the preferred indifferent things as "good", but the wise person, said Chrysippus, uses such things without requiring them. Practice and habit are necessary to render virtue perfect in the individual – in other words, there is such a thing as moral progress, and character has to be built up. The Stoics sought to be free of the unruly emotions , which they regarded as being contrary to nature. The passions or emotions ( pathe ) are

34960-486: Was not generally accepted by the Greeks, and Iamblichus wrote that "magnitude one" was a contradiction in terms. Chrysippus taught that ethics depended on physics. In his Physical Theses , he stated: "for there is no other or more appropriate way of approaching the subject of good and evil on the virtues or happiness than from the nature of all things and the administration of the universe." The goal of life, said Chrysippus,

35150-465: Was seen as more practical, and partly because it was taken up by the Neoplatonists . As recently as the 19th century, Stoic logic was treated with contempt, a barren formulaic system, which was merely clothing the logic of Aristotle with new terminology. It was not until the 20th century, with the advances in logic, and the modern propositional calculus , that it became clear that Stoic logic constituted

35340-512: Was shown, among other things, in the request he is supposed to have made to Cleanthes: "Give me the principles, and I will find the proofs myself." He succeeded Cleanthes as head of the Stoic school when Cleanthes died, in around 230 BC. Chrysippus was a prolific writer. He is said to rarely have gone without writing 500 lines a day and he composed more than 705 works. His desire to be comprehensive meant that he would take both sides of an argument and his opponents accused him of filling his books with

35530-404: Was superfluous as all events are foreordained, he replied that both divination and our behaviour under the warnings which it affords are included in the chain of causation. To fully characterize the physical world, the Stoics developed a theory of mixing in which they recognised three types of mixture. The first type was a purely mechanical mixture such as mixing barley and wheat grains together:

35720-454: Was the primitive substance God, the one supreme being, but divinity could be ascribed to the manifestations—to the heavenly bodies, to the forces of nature, even to deified persons; and thus the world was peopled with divine agencies. Prayer is of apparently little help in a rationally ordered cosmos, and surviving examples of Stoic prayers appear similar to self-meditation rather than appeals for divine intervention. The Stoics often identified

35910-465: Was then the head ( scholarch ) of the Stoic school. He is believed to have attended the courses of Arcesilaus and his successor Lacydes , in the Platonic Academy . Chrysippus threw himself eagerly into the study of the Stoic system. His reputation for learning among his contemporaries was considerable. He was noted for intellectual audacity and self-confidence and his reliance on his own ability

36100-402: Was to assert that everything, including wisdom, justice, etc., are bodies. Plato had defined being as "that which has the power to act or be acted upon," and for the Stoics this meant that all action proceeds by bodily contact; every form of causation is reduced to the efficient cause, which implies the communication of motion from one body to another. Only Body exists. The Stoics did recognise

#206793