Aristotle's Underlying Logic

The enlightenment thinkers of the ancient fifth century were natural heirs to the earlier thinkers who had aimed to replace mythopoeic and religious explanation of material phenomena with extensive observation and naturalistic explanation. With the advent of democracy and an increasing market economy there came a new spirit of inquiry, a new reliance on reasoned argumentation, and a new commitment to understanding nature. The aspirations of the enlightenment activists blossomed in the humanist ideals of scientific rationalism and political liberty, in profound philosophical inquiry into metaphysics, epistemology and ethics, and in artistic and cultural works of enduring value. These ancients condemned superstition in a way that recalls Hume's exhortation to employ philosophy as “the sovereign antidote … against that pestilent distemper … to restore men to their native liberty”. We might recall Heraclitus's own exhortation to listen to the logos, the aims of the Pythagorean mathêmatikoi, the nous of Anaxagoras, and the condemnation of superstition by Hippocrates in The Sacred Disease and Ancient Medicine. Sophocles especially captured this spirit in Antigone (332–375) where he has the Chorus sing that “there are many wonders, but nothing is more wonderful than a human being”. Indeed, Prometheus might well have been their patron saint, because, in providing humans with various technai, he affirmed optimism about their future. A principle underlying this optimism holds that human beings can understand themselves and nature sufficiently to govern their own destinies without the external and apparently capricious interventions of supernatural beings. These ancients embraced Kant's dictum, expressed many centuries later, “Sapere aude!” — “Dare to know!”. Their steady strides in the second half of the fifth century toward consolidating and rationally organizing the sciences helped to bring the earlier inquiries to fruition and prepared the way for the enduring accomplishments of the later philosophers, scientists, and political theorists. Many high points mark the achievements of this enlightenment, but two stand out for their progressive humanist ambition. As Protagoras in relation to the social world developed a political technê that affirmed the teachability of virtue and citizenship and thus promoted an empowering democratic activism, so Hippocrates in relation to the natural world developed a medical technê that affirmed the intelligence of human beings to intervene in the workings of nature to preserve health and to prevent disease.

Aristotle, interestingly himself the son of a physician, is an exemplary fourth century heir to enlightenment trends in science and philosophy. He affirmed the principle that nature in its diversity and human beings in their complexity are comprehensible. In Metaphysics 1.2 he openly avows a humanist ideal kindred to that of Hippocrates in Ancient Medicine.¹

The acquisition of this knowledge [

] … [has been] regarded as not suited for man. … God alone may have this prerogative, and it is fitting that a man should seek only such knowledge as becomes him [and not, as the poets say, arouse the gods’ jealousy]. But we should not believe in divine jealousy; for it is proverbial that bards tell many lies, and we ought to regard nothing more worthy of honor than such knowledge.² (982b28–983a7)

Aristotle boldly began Metaphysics by affirming that “all men naturally desire to know”. He then traced the acquisition of knowledge from sensation through memory of the same thing and finally to art and science (epistêmê), which are produced through extensive experience.

Art [

] is born when out of many bits of information derived from experience there emerges a grasp of those similarities in view of which they are a unified whole. Thus, a man is experienced who knows that when Callias was ill of this disease he was helped by this medicine, and so for Socrates and for many others, one by one; but to have art is to grasp that all members of the group of those who are ill of this disease have been helped by this medicine.

Now experience [

] seems in no respect inferior to art in a situation in which something is to be done. … The reason is that experience, like action or production, deals with things severally as concrete individuals, whereas art deals with them generally. Thus, a physician does not cure species-man (except incidentally), but he cures Callias, Socrates, or some other individual with a proper name, each of whom happens to be a man. If, then, someone lacking experience, but knowing the general principles of the art, sizes up a situation as a whole, he will often, because he is ignorant of the individuals within that whole, miss the mark and fail to cure; for it is the individual that must be cured.

Nevertheless, we believe that knowing and understanding

characterize art rather than experience. And so we take experts

in an art to be wiser than men of mere experience; because wisdom presumably comes only with knowledge, and we believe that the experts can analyze and explain, whereas others cannot. Men of experience discern the fact “that”, but not the reason “why”. Hence we also hold master workmen [

] in each craft to be more valuable and discerning and wise than manual laborers [

], because the former can discriminate the various factors relevant to the various effects produced; whereas the latter, like inanimate objects, produce effects, as fire burns, without knowing what they are doing. Inanimate objects produce their effects somehow by nature; and manual workers, by habit. Master workers are presumably wiser, then, not because they are practical, but because they have their reasons and can explain what they are doing [

] (981a5–981b6)

Until recently the difference between traditional or ‘Aristotelian’ logic and Aristotle's own ancient logic had been blurred. This is similar to the blurring of a similar distinction between Christian religion and the teachings of Jesus, or the difference between various ‘Marxian’ philosophies and the teachings of Karl Marx. It is remarkable, for example, that for Aristotle in every syllogism the conclusion follows logically from the premisses. This contrasts with the usage of traditional logicians, who continue to speak of invalid syllogisms. For Aristotle this is a contradiction in terms, an oxymoron. In addition, Aristotle would never have tested the validity or invalidity of a syllogism according to rules of quality, quantity, and distribution. He had his own methods for establishing validity and invalidity. However, it was really not possible meaningfully to distinguish the historical logic of Aristotle from its later accretions and compare the two until modern logicians examined Aristotle's syllogistic through the lens of mathematical logic — that is, until modern logicians turned their attention specifically to the formal aspects of deductive discourses apart from their subject matters. As a result, studies of Aristotle's logic since the early 20^th century have established his genius as a logician of considerable originality and insight. Indeed, we can now recognize many aspects of his logical investigations that are themselves modern, in the sense that modern logicians are making discoveries that Aristotle had already made or had anticipated. Perhaps the longevity of this oversight about the nature and accomplishments of his logical investigations is attributable to scholars not having recognized that Aristotle expressly treated the deduction process itself.

Jan Łukasiewicz initiated the reassessment of Aristotle's syllogistic in the 1920s. He was followed by James W. Miller, I. M. Bochenski, and Günther Patzig among others. This reassessment culminated in the 1970s and 1980s with the works of John Corcoran, Timothy Smiley, and Robin Smith. These modern logicians used mathematical logic to model Aristotle's logic and discovered a logical sophistication long overlooked by traditionalist logicians such as R. Whately, H. W. B. Joseph, J. N. Keynes, W. D. Ross, and R. M. Eaton. These traditionalists, whose modern origin can be traced to the Port Royal Logic, believe that Aristotle composed Prior Analytics as a logic manual for studying categorical arguments or syllogisms. They take a syllogism to be a fully interpreted premiss-conclusion argument whose validity or invalidity is determined by applying rules of quality, quantity, and distribution, all of which really only help to define a syllogism. However, traditionalists tend to conflate this sense of a syllogism with another sense when they take a syllogism also to be a relatively uninterpreted argument pattern whose instances are valid or invalid arguments.

Now, in spite of their equally criticizing traditionalist interpreters, mathematical logicians themselves tend to fall into two camps concerning Aristotle's project in Prior Analytics. In fact, when modern logicians mathematically modeled Aristotle's logic, they tacitly distinguished two tendencies in the traditionalist interpretation, the one treating what it believed were Aristotle's axiomatic interests, the other treating Aristotle's argumental interests. The axiomaticist interpretation by Łukasiewicz, Bochenski, Miller, and Patzig takes a syllogism to be a single, logically true conditional proposition, some of which are taken to be axioms. On this interpretation Prior Analytics contains an axiomatized deductive system with an implicit underlying propositional logic. Euclid's Elements is an ancient analogue. The axiomaticists examine Aristotle's syllogistic mathematically from a Frege-Russell view of logic as formal ontology. On the other hand, deductionists examine Aristotle's logic mathematically from a Quinian view of logic as formal epistemology.³ The deductionist interpretation of Corcoran, Smiley, and Smith takes a syllogism to be a deduction, that is, to be a fully interpreted argumentation having a cogent chain of reasoning in addition to premisses and a conclusion. On this interpretation the number of premisses is not restricted to two. This interpretation sees Prior Analytics as having proof-theoretic interests relating to a natural deduction system. Interpretive lines, then, are drawn along what each view considers a syllogism to be and what each takes to be Aristotle's accomplishment in Prior Analytics.

However, notwithstanding significant differences among modern interpretations, there are two striking similarities. (1) All three interpretations consider the process of reduction

In great measure, interpretive problems are attributable to scholars not having sufficiently recognized Aristotle's acumen in distinguishing logical and metalogical discourses. Deductions are equally performed in different languages: (1) in an object language about a given subject matter and (2) in a metalanguage, which is used to model formal aspects of object language discourses relating to sentences, arguments, argumentations and deductions. Discourses in these categorially different languages may or may not use the same deduction system or the same logic terms with the same or different denotations. We distinguish an object language deduction from a metalogical deduction. Aristotle understood his syllogistic deduction system to function at both levels. In Prior Analytics he both studied his syllogistic logic and its applications and he used this logic in that study. Traditionalists, however, altogether missed Aristotle's making this distinction by their conflating two senses of a syllogism and, consequently, they overlooked a syllogistic deduction process. Axiomaticists mistook a conditional sentence corresponding to a syllogism for the syllogism itself to confuse the two levels of discourse and thereby they lost sight of Aristotle's principal concern with deduction. Still, they were correct to focus on his metalogical treatment of ‘syllogistic forms’, even if in their enthusiasm to apply mathematical logic to Aristotle's work they mistakenly saw an axiomatized deductive system in Prior Analytics. Deductionists correctly focused attention on Aristotle's concern with the process of deduction and a natural deduction system. However, in reacting to the axiomaticists, they did not take Aristotle as himself modeling object language discourses by means of a metalogical discourse. Nor, then, did they consider his metalogical discourse to be sufficiently formal for his having distinguished logical syntax from semantics. Deductionists modeled Aristotle's logic but did not recognize Aristotle as himself providing an ancient model of an underlying logic with a formal language.

Aristotle would have agreed with Alonzo Church that “(formal) logic is concerned with the analysis of sentences or of propositions and of proof with attention to the form in abstraction from the matter” (1956: 1; author's emphasis). Thus, for Church the science of logic is a metalogical study of underlying logics (1956: 57–58). The difference between logic and metalogic is drawn between using a logic to process information about a given subject matter with a given object language and studying a logic or an underlying logic, which involves a language, a semantics, and a deduction system. Logicians use a metalanguage to study the formal aspects of an object language apart from its subject matter, often to study an underlying logic's deduction system. Aristotle undertook just such a study in Prior Analytics. Indeed, part of Aristotle's philosophical genius is to have established a formal logic, while at the same time making the study of logic scientific. He recognized that deductions about a given subject matter are topic specific and pertain to a given domain, say to geometry or to arithmetic or to biology, but that such deductions employ a topic neutral deduction system to establish knowledge of logical consequence.

In having a keen interest in epistemics, Aristotle shares with modern logicians the notion that central to the study of logic is examining the formal conditions for establishing knowledge of logical consequence — that logic, then, is a part of epistemology. He composed Prior Analytics and Posterior Analytics to establish a firm theoretical and methodological foundation for

Aristotle's promethean contribution to science and philosophy, then, concerns his study of the deduction process itself. He knew that a given sentence is either true or false; and he recognized this to be the case independent of a participant. He also knew from his familiarity with mathematical argumentation and dialectical reasoning that a given sentence either follows necessarily or does not follow necessarily from other given sentences; likewise, he recognized this to be the case independent of a participant. These are ontic matters having to do with being. In addition, Aristotle knew that the truth or falsity of a given sentence or the validity or invalidity of a given argument might not be known to one or another participant. Now, a given axiomatic science aims to establish knowledge about its proper subject matter (Po. An. A1: 71a1–11 & A3: 72b19–22). Since Aristotle took such a science to consist principally in the collection of sentences — definitions, axioms, theorems — of its extended discourse, the project of such a science is to decide which sentences pertaining to its subject matter are true, or theorems, and which sentences, for that matter, are false and not theorems. Procedures for deciding a sentence's truth or falsity are epistemic matters having to do with knowing.

In respect of epistemics Aristotle recognized two ways to establish the truth of a given sentence: (1) by induction

By a demonstration

I mean a scientific deduction

; and by scientific I mean a deduction by possessing which we understand something … demonstrative understanding

in particular must proceed from items that are true and primitive and immediate and more familiar than and prior to and explanatory of the conclusions. There can be a deduction [

(sullogisrnos)]even if these conditions are not met, but there cannot be a demonstration [

] — for it will not bring about understanding [

]; in respect of a given subject matter]. (71b17–25; cf. Top. A1: 100a27–29)

A deduction per se, then, establishes knowledge, not that the sentence that is the conclusion of a given argument is true, but only that it follows necessarily, or logically, from the sentences in a premiss-set. Aristotle made an important distinction in his logical investigations between epistemic concerns and ontic concerns. This is especially evident in Prior Analytics B1–4 where he treated the deducibility of true and false sentences from various combinations of true and false sentences taken as premisses. This distinction indicates an understanding of logical consequence that modern logicians will recognize. The confidence one acquires from a demonstration derives from knowing, as Aristotle often pointed out, that it is impossible for true sentences to imply a false sentence (Pr. An. B2–4). Given true sentences as premisses, established (initially) by means independent of deduction, one can be certain that the conclusion sentence of a demonstration also is true⁵ precisely because it is shown to be a logical consequence of other true sentences. In Prior Analytics Aristotle was especially concerned to determine which formal patterns of argumentation might be used to establish knowledge that a given sentence necessarily follows from other given sentences. In particular, he saw his project as determining “how every syllogism is generated” (25b26–31) by identifying which elementary argument patterns could serve as rules analogous to such patterns as modus ponens, modus tollens, and disjunctive syllogism for modern propositional logic.

Looking back, we see that mathematicians of the fourth century had been assiduously attending to axiomatizing geometry. This activity principally concerned condensing the entire wealth of geometric knowledge into small sets of definitions and axioms from which the theorems of geometry could be derived and set out as a long, extended discourse. Euclid's Elements is an extant fruit of this activity. Except for identifying a small set of common notions

Undoubtedly Aristotle had participated in discussions, in the Academy and elsewhere, about axiomatizing geometry. He may have asked about deduction rules used to establish geometric theorems. Indications that he did include his attention to various proofs such as that of the incommensurability of the diagonal with the side of a square and those related to properties of triangles, and his frequent attention to the common notions of the mathematical sciences; there is also his curious mention of the middle term and syllogistic reasoning in connection with geometric demonstration (Po. An. A9: 76a4–10; A12: 77b27–28; cf. Pr. An. A35 & A24). Aristotle surely wondered how one could be assured in geometric demonstrations that a conclusion necessarily follows from premisses. This matter is all the more interesting in the case of longer, more involved demonstrations. Still, we cannot say that he undertook a metalogical study of geometric proof.

Perhaps it was Aristotle's own insight or an implicit part of the philosophical discussion of the time that the axiomatization of geometry could serve in some way as a model for formalizing the non-mathematical sciences such as botany and zoology. Some such notion seems to have animated his scientific and logical investigations.⁶ Now, the actual project of establishing a given science's definitions and axioms and then its theorems, which would conform to the criteria set out in Posterior Analytics A2, did not concern Aristotle in the Organon. This project lies outside the scope of logic. Rather, his preeminent concern there was to study deduction and demonstration per se: not with, that is, one or another distinct subject matter, but with the formal deduction process that has no similar subject matter. Aristotle in his logical investigations subordinated a concern with the what or the why and wherefore to focus on the that and the how. Thus, presupposing various axiomatic sciences with distinct domains, he took up the narrower and more poignant questions about the epistemic process of deriving theorems from axioms. Aristotle especially examined the deductive foundations of demonstration, that is, of demonstrative knowledge or axiomatic science.

The first chapters of Metaphysics 1 reveal Aristotle's intellectual disposition toward scientific knowledge and signal the importance he attributed to metalogical study of deduction. In fact, Aristotle identified this task as a province proper only to philosophy. In Metaphysics 2.1 he writes that “philosophy is the science of truth

But it is clear that the axioms extend to all things as being (since they all have being in common); hence the theory of axioms [

] also belongs to him who knows being as being. (1005a27–29)

After indicating the limitations of the special sciences for examining these axioms, Aristotle writes that

the philosopher, who examines the most general features of primary being, must investigate also the principles of deductive reasoning [

]. … So that he who gets the best grasp of beings as beings must be able to discuss the basic principles of all being

, and he is the philosopher. (1005b5-6-11; cf. Meta. 11.4)

Hence, if contraries cannot at the same time belong to the same thing … and if an opinion [

] stated in opposition to another opinion is directly contrary to it, then it is evidently absurd for the same man at the same time to believe the same thing to be and not to be; for whoever denies this would at the same time hold contrary opinions

. It is for this reason that all who carry out a demonstration rest it on this as on an ultimate belief

; for this is naturally a foundation also of all other axioms [

]. (1005b26–34)

Now, Aristotle writes in Metaphysics 11.4 that “taking equals from equals leaves equal remainders” is an example of a notion common to all quantitative being (1061b20–21). He had recognized that the common notions belonging to the mathematical sciences belonged equally to all and specially to none.⁸ And, although he never articulated a complete list of common notions for the non-mathematical sciences, nor for that matter in any systematic way for the mathematical sciences, he stated in Metaphysics 4 that the principles of contradiction and of the excluded middle are among the common notions applicable to all rational discourse. Aristotle noticed that the common notions relating to different branches of mathematics, while stipulative of magnitudes in general, do not stipulate any one domain, such as arithmetic or geometry, in particular. They generally do not specify any content however much they anticipate establishing relationships among magnitudes special to a mathematical science. They are topic neutral in this circumscribed sense, and, thus, they have a relative independence unlike lines, angles, or numbers. They are neither embedded in nor mentally inextricable from the objects of a particular science. This is not the case with a science's principles. The common notions, then, may be taken in abstractum and treated on their own account irrespective of the subject matter of a given quantitative science.

These common notions, moreover, generally express formal relationships among magnitudes within a quantitative science and accordingly apply equally to a variety of different, quantitative domains. Because of their relative formality and their special universality, the common notions were applied as inference rules across the mathematical sciences. Their use in this epistemic manner is evident in Euclid's Elements.⁹ Aristotle's having recognized the common notions as principles of reasoning had profound consequences for the development of ancient logic. Understanding this and that Aristotle took a syllogism to fit an elementary argument pattern with only valid instances help to confirm the rule-nature of his statements in Prior Analytics A4–6 relating to when a syllogism comes about. In the case of Euclid's common notions, two magnitudes remain incommensurable without there being a third, or middle, that unites them as extremes in a particular way — using a common notion makes this evident. An exactly analogous relationship applies in the case of the patterns of perfect or complete syllogisms, the

Finally in this connection, besides stating syllogistic deduction rules and his actually using the patterns of the teleioi sullogismoi as rules in Prior Analytics, Aristotle virtually stated his taking them formally as rules in Prior Analytics A30 (46a10–12/15). There he used the expression ‘the principles of deduction

Our concern here is to present Aristotle's system of logic while also revealing the mathematical sophistication of his logical investigations. Modern logicians believe that the possibility of mathematical logic, an important part of which involves generating models, consists in making a clear distinction between syntax and semantics. They also believe that the clear distinction between syntax and semantics resulted from borrowing symbolic notations from mathematical practice and then applying them to studies of deductive logics, but that earlier thinkers, lacking such notations, could not have made such distinctions. However, mathematical logic, considered as a discipline in general, has a formal and a material aspect. Its formal aspect has principally to do with the symbolic notations that have helped to illuminate underlying structural, or logical, features of deductive discourses. Yet, the substance of mathematical logic does not consist in its sophisticated notations, but in the problems logicians consider when studying underlying logics — that is, in particular, when they distinguish a logic's syntax and its semantics and then ask questions about their relationships. Principal in this respect have been questions about a logic's consistency, soundness, and completeness, which involve determining relationships between deducibility and logical consequence. A distinguishing feature of mathematical logic, then, consists precisely in these substantive matters.¹¹ Remarkably, with only a rudimentary notation Aristotle considered just such mathematical matters in his concern to establish the practical, epistemic power of his logic for establishing scientific knowledge. In this connection we can grasp the revolution in the history and philosophy of logic — the “hypostatization of proof” — consolidated by Aristotle's works on logic. “Prior Analytics is the earliest known work which treats proofs as timeless abstractions amenable to investigation similar to the investigations already directed toward numbers and geometrical figures” (Corcoran: personal communication). Thus, Prior Analytics is a proof-theoretic treatise on the deduction system of an underlying logic. Aristotle recognized the epistemic efficacy of certain elementary argument patterns, to wit, those of the syllogisms, and he formulated them as rules of natural deduction. Having raised important metalogical questions about the properties of his syllogistic deduction system, he successfully established a set of formal, epistemic conditions for recognizing logical necessity, and in this way he became the founder of formal logic.

Below we set out Aristotle's underlying logic much as he himself did in the works of the Organon. We include Metaphysics among the treatises of his logical investigations. It is natural and not surprising that modern logicians and commentators, when treating Aristotle's logic, focus principally on Prior Analytics: Prior Analytics is the most ‘logical’ of the treatises. In truth, the attraction of Prior Analytics has consisted in a scholar's implicit recognition that Aristotle there treated the deduction system of an ancient underlying logic. We say ‘implicit’ because it was not until the studies of J. Corcoran and T. Smiley, and later those of R. Smith, that there is a growing explicit recognition that this is so. In any case, a deduction system is only one part of an underlying logic, which also contains a grammar and a semantics. Our contribution takes this recognition a little farther to hold that Aristotle intentionally aimed to develop an underlying logic along the lines of modernist thinking. This means that Aristotle invented a formal language to model his logic. However, since Aristotle did not set out his underlying logic in as systematic a manner as a modern logician, while, nevertheless, accomplishing much the same result, we employ the theoretical apparatus of modern mathematical logic to structure his account. With the aid of this template we show in Aristotle's own words that he was concerned with exactly similar matters as a modern logician. We begin by presenting Aristotle's treatment of the syntax and semantics of natural language in Categories, On Interpretation, and Metaphysics. These studies laid a foundation for his developing the formal language found in Prior Analytics for modeling axiomatic discourse. We then proceed to extract the syntax of sentence transformations leading to his establishing a set of deduction rules in Prior Analytics. Next we treat the logical methodology by which Aristotle established his deduction rules. We conclude with a statement of his understandings of “formal deducibility” and “logical consequence” and with a final section that summarizes four proof-theoretic accomplishments of his logical investigations.

The following terminology assists in our study of Aristotle's logic. We use Aristotle's own terminology wherever it exists, which, interestingly, often corresponds exactly to ours. An argument is a two part system consisting in a set of sentences in the role of premisses and a single sentence in the role of conclusion; an argument is either valid or invalid. A sentence is either true or false. We sometimes use ‘conclusion’ elliptically for ‘sentence in the role of conclusion’ or ‘conclusion sentence’, and similarly for ‘premiss’. An argumentation is a three part system consisting in a chain of reasoning in addition to premisses and conclusion; an argumentation is either cogent, in which case it is a deduction, or fallacious, a fallacy. A sentence, an argument, and an argumentation are object language phenomena and domain specific. An argument pattern is a two part system consisting in a set of sentence patterns in the role of a premiss-set and a single sentence pattern in the role of a conclusion. A pattern is a metalinguistic object distinguishable from a form and is commonly represented schematically. An argument is said to fit, or to be an instance of, one or more argument patterns.¹² A given argument pattern may have all valid instances, all invalid instances, or some valid and some invalid instances. An argument pattern is not properly valid or invalid, although logicians have used ‘valid’ in this connection, but we distinguish these category differences. An argument pattern with all valid instances is panvalid, that with all invalid instances is paninvalid, and that having instances of both is neutrovalid. We add that an elementary panvalid argument pattern is one having a simple premiss-set pattern whose epistemic value consists, in many cases, in its being quickly evident, or ‘evident through itself’, that its conclusion follows necessarily. An elementary argument pattern may be formulated in a corresponding sentence to express a rule of deduction. In addition, we follow G. Patzig (1968; cf. Rose 1968) to distinguish in Aristotle's logic a concludent pattern of two sentences in the role of premisses, or a premiss-pair pattern, from an inconcludent premiss-pair pattern. A concludent pattern has a necessary result, that is, it results in a panvalid pattern all of whose instances are syllogisms, while an inconcludent pattern has no necessary result, that is, it cannot result in a panvalid pattern but only in a paninvalid pattern with only invalid argument instances.

In addition, we understand Aristotle to have considered a

Finally, we take treating patterns of sentences, patterns of arguments, and patterns of argumentations to constitute a large part of modeling a logic. Thus, for example, while there are numerous simple sentences in a given object language, each of them, nevertheless, consists in a subject and a predicate. Extracting this elementary pattern and representing it abstractly, or metalinguistically, is modeling a simple sentence — either by means of another sentence, using the language of the given object language (but, nevertheless, in the metalanguage), or by means of mathematical notation. In either case, a sentence is modeled and becomes an object of logical investigation. Thus, we take a formal language to be a model of one or another object language, with one or another degree of precision. In this way a logician can model arguments, deductions, and deduction systems better to study their respective properties and logical relationships.