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This article is about arguments in Logic.
In Logic, an argument is a set of declarative sentences (statements) known as the premises, and another declarative sentence (statement) known as the conclusion in which it is asserted that the truth of the conclusion follows from (is entailed by) the premisses. Such an argument may or may not be valid. Note: in Logic declarative sentences (statements) are either true or false (not valid or invalid); arguments are valid or invalid (not true or false). Many authors in Logic now use the term 'sentence' to mean a declarative sentence rather than 'statement' or 'proposition' to avoid certain philosophical implications of these last two terms.
A valid argument is one in which a specific structure is followed. An invalid argument is one in which a specfic structure is NOT followed.
The validity of an argument does not guarantee the truth of its conclusion, since a valid argument may have false premises. 'Only a valid argument with true premises must have a true conclusion.'
The validity of an argument depends on its form, not on the truth or falsity of its premises and conclusions. Logic seeks to discover the forms of valid arguments. Since a ''valid'' argument is one such that if the premises are true then the conclusion must be true it follows that a 'valid argument cannot have true premises and a false conclusion'. Since the validity of an argument depends on its form, an argument can be shown to be invalid by showing that its form is invalid because other arguments of the same form have true premises and false conclusions. In informal logic this is called a counter argument.
A proof is a demonstration that an argument is valid (see Proof procedure).
Some authors define a sound argument is a valid argument with true premises (see also Validity, Soundness, Truth.)
Arguments can be invalid for a variety of reasons. There are well-established patterns of reasoning that arguments may follow which render them invalid; these patterns are known as logical fallacies.
Even if an argument is sound (and hence also valid), an argument may still fail in its primary task of persuading us of the truth of its conclusion. Such an argument is then sound, but ineffective. An argument may fail to be effective because it is not ''scrutinizable'', in the sense that it is not open to public examination. This may be because the argument is too long or too complex, because the terms occurring in it are obscure, or because the reasoning it employs is not well understood. The validity and soundness of an argument are logical properties of it, known as semantic properties. Effectiveness, on the other hand, is not a logical notion but a practical concern.
In mathematics, an argument can often be formalized by writing each of its statements in a formal language such as first-order Peano Arithmetic. A formalized argument should have the following properties:
★ its premises are clearly identified as such
★ each of the inferences is justified by appeal to a specific rule of reasoning of the formal language in which the argument is written
★ the conclusion of the argument appears as the final inference
Checking the validity of a formal argument is thus a straightforward matter, since the presence of these three properties is easily verified.
Most arguments used in mathematics are not formal in quite so strict a sense. Strictly formal proofs of all but the most trivial assertions are extremely tedious to construct and often so long as to be hard to follow without assistance from a computer. Automated theorem proving is sometimes used to overcome these problems.
In general mathematical practice arguments are formal insofar as they are formalizable in theory; this is sometimes expressed by saying that mathematical arguments are ''rigorous''. Mathematicians are happy to make a single inference that would, if formalized, amount to a long chain of inferences, because they are confident that the formal chain could be constructed if required.
Nevertheless, one advantage of formalizing arguments is the possibility of constructing a theory of valid mathematical arguments such as proof theory. Proof theory investigates the class of valid arguments in mathematics as a whole, and hence elucidates what kinds of statements can occur as conclusions to sound mathematical arguments. Gödel's incompleteness theorems are proof-theoretic results which show the surprising fact that not all true mathematical statements can occur as the conclusion of formalized, sound mathematical arguments. In effect, not all true statements of mathematics are provable.
In ordinary, philosophical and scientific argumentation abductive arguments and arguments by analogy are also commonly used. Arguments can be ''valid'' or ''invalid'', although how arguments are determined to be in either of these two categories can often itself be an object of much discussion. Informally one should expect that a valid argument should be ''compelling'' in the sense that it is capable of convincing someone about the truth of the conclusion. However, such a criterion for validity is inadequate or even misleading since it depends more on the skill of the person constructing the argument to manipulate the person who is being convinced and less on the objective truth or undeniability of the argument itself.
Less subjective criteria for validity of arguments are often clearly desirable, and in some cases we should even expect an argument to be rigorous, that is, to adhere to precise rules of validity. This is the case for arguments used in mathematical proofs. Note that a rigorous proof does not have to be a formal proof.
In ordinary language, people refer to the ''logic of an argument'' or use terminology that suggests that an argument is based on inference rules of formal logic. Though arguments do use inferences that are indisputably purely logical (such as syllogisms), other kinds of inferences are almost always used in practical arguments. For example, arguments commonly deal with causality, probability and statistics or even specialized areas such as economics. In these cases, rather than to the well-defined principles of pure logic as explicitly set out and agreed upon in an academic, professional or other strictly understood context, ''logic'' in everyday usage almost always refers to something the reader or audience of the argument believe they perceive in the argument, and which drives them inexorably towards some conclusion, something perhaps ill-defined in their own minds (as opposed to putting the emphasis on examining by what criteria they actually accept this apparently compelling force as correct, which is how the formal rules of pure logic are constructed). And yet this feeling of inexorable conviction is also the foundation of those begrudgingly somewhat unsatisfying definitions we give of "logic"; in that we who are driven to construct these most conscientious, circumspect and clear definitions were initially drawn to do so by a similar belief that we recognized some intrinsic logic or compelling rational force in the world- even in the most everyday arguments, although such a view may have been naive, and is in any case incapable of being tested in any objective and/or universally satisfying fashion.
Theories of arguments are closely related to theories of informal logic. Ideally, a theory of argument should provide some mechanism for explaining validity of arguments.
One natural approach would follow the mathematical paradigm and attempt to define validity in terms of semantics of the assertions in the argument. Though such an approach is appealing in its simplicity, the obstacles to proceeding this way are very difficult for anything other than purely logical arguments. Among other problems, we need to interpret not only entire sentences, but also components of sentences, for example noun phrases such as ''The present value of government revenue for the next twelve years''.
One major difficulty of pursuing this approach is that determining an appropriate semantic domain is not an easy task, raising numerous thorny ontological issues. It also raises the discouraging prospect of having to work out acceptable semantic theories before being able to say anything useful about understanding and evaluating arguments. For this reason the purely semantic approach is usually replaced with other approaches that are more easily applicable to practical discourse.
For arguments regarding topics such as probability, economics or physics, some of the semantic problems can be conveniently shoved under the rug if we can avail ourselves of a model of the phenomenon under discussion. In this case, we can establish a limited semantic interpretation using the terms of the model and the validity of the argument is reduced to that of the abstract model. This kind of reduction is used in the natural sciences generally, and would be particularly helpful in arguing about social issues if the parties can agree on a model. Unfortunately, this prior reduction seldom occurs, with the result that arguments about social policy rarely have a satisfactory resolution.
Another approach is to develop a theory of argument pragmatics, at least in certain cases where argument and social interaction are closely related. This is most useful when the goal of logical argument is to establish a mutually satisfactory resolution of a difference of opinion between individuals.
Arguments as discussed in the preceding paragraphs are static, such as one might find in a textbook or research article. They serve as a published record of justification for an assertion. Arguments can also be interactive, in which the proposer and the interlocutor have a more symmetrical relationship. The premises are discussed, as well the validity of the intermediate inferences. For example, consider the following exchange, illustrated by the No true Scotsman fallacy:
: Argument: "No Scotsman puts sugar on his porridge."
: Reply: "But my friend Angus likes sugar with his porridge."
: Rebuttal: "Ah yes, but no true Scotsman puts sugar on his porridge."
In this dialogue, the proposer first offers a premise, the premise is challenged by the interlocutor, and finally the proposer offers a modification of the premise. This exchange could be part of a larger discussion, for example a murder trial, in which the defendant is a Scotsman, and it had been established earlier that the murderer was eating sugared porridge when he or she committed the murder.
In argumentative dialogue, the rules of interaction may be negotiated by the parties to the dialogue, although in many cases the rules are already determined by social mores. In the most symmetrical case, argumentative dialogue can be regarded as a process of discovery more than one of justification of a conclusion. Ideally, the goal of argumentative dialogue is for participants to arrive jointly at a conclusion by mutually accepted inferences. In some cases however, the validity of the conclusion is secondary. For example; emotional outlet, scoring points with an audience, wearing down an opponent or lowering the sale price of an item may instead be the actual goals of the dialogue. Walton distinguishes several types of argumentative dialogue which illustrate these various goals:
★ Personal quarrel.
★ Forensic debate.
★ Persuasion dialogue.
★ Bargaining dialogue.
★ Action seeking dialogue.
★ Educational dialogue.
Van Eemeren and Grootendorst identify various stages of argumentative dialogue. These stages can be regarded as an argument protocol. In a somewhat loose interpretation, the stages are as follows:
★ Confrontation: Presentation of the problem, such as a debate question or a political disagreement
★ Opening: Agreement on rules, such as for example, how evidence is to be presented, which sources of facts are to be used, how to handle divergent interpretations, determination of closing conditions.
★ Argumentation: Application of logical principles according to the agreed-upon rules
★ Closing: This occurs when the termination conditions are met. Among these could be for example, a time limitation or the determination of an arbiter.
Van Eemeren and Grootendorst provide a detailed list of rules that must be applied at each stage of the protocol. Moreover, in the account of argumentation given by these authors, there are specified roles of protagonist and antagonist in the protocol which are determined by the conditions which set up the need for argument.
Many cases of argument are highly unsymmetrical, although in some sense they are dialogues. A particularly important case of this is political argument.
Much of the recent work on argument theory has considered argumentation as an integral part of language and perhaps the most important function of language (Grice, Searle, Austin, Popper). This tendency has removed argumentation theory away from the realm of pure formal logic.
One of the original contributors to this trend is the philosopher Chaim Perelman, who together with Lucie Olbrechts-Tyteca, introduced the French term ''La nouvelle rhetorique'' in 1958 to describe an approach to argument which is not reduced to application of formal rules of inference. Perelman's view of argumentation is much closer to a juridical one, in which rules for presenting evidence and rebuttals play an important role. Though this would apparently invalidate semantic concepts of truth, this approach seems useful in situations in which the possibility of reasoning within some commonly accepted model does not exist or this possibility has broken down because of ideological conflict. Retaining the notion enunciated in the introduction to this article that ''logic'' usually refers to the structure of argument, we can regard the logic of rhetoric as a set of protocols for argumentation.
In recent decades one of the more influential discussions of philosophical arguments is that by Nicholas Rescher in his book ''The Strife of Systems''. Rescher models philosophical problems on what he calls aporia or an aporetic cluster: a set of statements, each of which has initial plausibility but which are jointly inconsistent. The only way to solve the problem, then, is to reject one of the statements. If this is correct, it constrains how philosophical arguments are formulated.
★ Robert Audi, ''Epistemology'', Routledge, 1998. Particularly relevant is Chapter 6, which explores the relationship between knowledge, inference and argument.
★ J. L. Austin ''How to Do Things With Words'', Oxford University Press, 1976.
★ H. P. Grice, ''Logic and Conversation'' in ''The Logic of Grammar'', Dickenson, 1975.
★ Vincent F. Hendricks, ''Thought 2 Talk: A Crash Course in Reflection and Expression'', New York: Automatic Press / VIP, 2005, ISBN 87-991013-7-8
★ R. A. DeMillo, R. J. Lipton and A. J. Perlis, ''Social Processes and Proofs of Theorems and Programs'', Communications of the ACM, Vol. 22, No. 5, 1979. A classic article on the social process of acceptance of proofs in mathematics.
★ Yu. Manin, ''A Course in Mathematical Logic'', Springer Verlag, 1977. A mathematical view of logic. This book is different from most books on mathematical logic in that it emphasizes the mathematics of logic, as opposed to the formal structure of logic.
★ Ch. Perelman and L. Olbrechts-Tyteca, ''The New Rhetoric'', Notre Dame, 1970. This classic was originally published in French in 1958.
★ Henri Poincaré, ''Science and Hypothesis'', Dover Publications, 1952
★ Frans van Eemeren and Rob Grootendorst, ''Speech Acts in Argumentative Discussions'', Foris Publications, 1984.
★ K. R. Popper ''Objective Knowledge; An Evolutionary Approach'', Oxford: Clarendon Press, 1972.
★ L. S. Stebbing, ''A Modern Introduction to Logic'', Methuen and Co., 1948. An account of logic that covers the classic topics of logic and argument while carefully considering modern developments in logic.
★ Douglas Walton, ''Informal Logic: A Handbook for Critical Argumentation'', Cambridge, 1998
★ Carlos Chesñevar, Ana Maguitman and Ronald Loui, ''Logical Models of Argument'', ACM Computing Surveys, vol. 32, num. 4, pp.337-383, 2000.
★ T. Edward Damer. Attacking Faulty Reasoning, 5th Edition, Wadsworth, 2005. ISBN 0-534-60516-8
In Logic, an argument is a set of declarative sentences (statements) known as the premises, and another declarative sentence (statement) known as the conclusion in which it is asserted that the truth of the conclusion follows from (is entailed by) the premisses. Such an argument may or may not be valid. Note: in Logic declarative sentences (statements) are either true or false (not valid or invalid); arguments are valid or invalid (not true or false). Many authors in Logic now use the term 'sentence' to mean a declarative sentence rather than 'statement' or 'proposition' to avoid certain philosophical implications of these last two terms.
Validity
A valid argument is one in which a specific structure is followed. An invalid argument is one in which a specfic structure is NOT followed.
The validity of an argument does not guarantee the truth of its conclusion, since a valid argument may have false premises. 'Only a valid argument with true premises must have a true conclusion.'
The validity of an argument depends on its form, not on the truth or falsity of its premises and conclusions. Logic seeks to discover the forms of valid arguments. Since a ''valid'' argument is one such that if the premises are true then the conclusion must be true it follows that a 'valid argument cannot have true premises and a false conclusion'. Since the validity of an argument depends on its form, an argument can be shown to be invalid by showing that its form is invalid because other arguments of the same form have true premises and false conclusions. In informal logic this is called a counter argument.
Proof
A proof is a demonstration that an argument is valid (see Proof procedure).
Validity, soundness and effectiveness
Some authors define a sound argument is a valid argument with true premises (see also Validity, Soundness, Truth.)
Arguments can be invalid for a variety of reasons. There are well-established patterns of reasoning that arguments may follow which render them invalid; these patterns are known as logical fallacies.
Even if an argument is sound (and hence also valid), an argument may still fail in its primary task of persuading us of the truth of its conclusion. Such an argument is then sound, but ineffective. An argument may fail to be effective because it is not ''scrutinizable'', in the sense that it is not open to public examination. This may be because the argument is too long or too complex, because the terms occurring in it are obscure, or because the reasoning it employs is not well understood. The validity and soundness of an argument are logical properties of it, known as semantic properties. Effectiveness, on the other hand, is not a logical notion but a practical concern.
Formal arguments and mathematical arguments
In mathematics, an argument can often be formalized by writing each of its statements in a formal language such as first-order Peano Arithmetic. A formalized argument should have the following properties:
★ its premises are clearly identified as such
★ each of the inferences is justified by appeal to a specific rule of reasoning of the formal language in which the argument is written
★ the conclusion of the argument appears as the final inference
Checking the validity of a formal argument is thus a straightforward matter, since the presence of these three properties is easily verified.
Most arguments used in mathematics are not formal in quite so strict a sense. Strictly formal proofs of all but the most trivial assertions are extremely tedious to construct and often so long as to be hard to follow without assistance from a computer. Automated theorem proving is sometimes used to overcome these problems.
In general mathematical practice arguments are formal insofar as they are formalizable in theory; this is sometimes expressed by saying that mathematical arguments are ''rigorous''. Mathematicians are happy to make a single inference that would, if formalized, amount to a long chain of inferences, because they are confident that the formal chain could be constructed if required.
Nevertheless, one advantage of formalizing arguments is the possibility of constructing a theory of valid mathematical arguments such as proof theory. Proof theory investigates the class of valid arguments in mathematics as a whole, and hence elucidates what kinds of statements can occur as conclusions to sound mathematical arguments. Gödel's incompleteness theorems are proof-theoretic results which show the surprising fact that not all true mathematical statements can occur as the conclusion of formalized, sound mathematical arguments. In effect, not all true statements of mathematics are provable.
Logical arguments in science
In ordinary, philosophical and scientific argumentation abductive arguments and arguments by analogy are also commonly used. Arguments can be ''valid'' or ''invalid'', although how arguments are determined to be in either of these two categories can often itself be an object of much discussion. Informally one should expect that a valid argument should be ''compelling'' in the sense that it is capable of convincing someone about the truth of the conclusion. However, such a criterion for validity is inadequate or even misleading since it depends more on the skill of the person constructing the argument to manipulate the person who is being convinced and less on the objective truth or undeniability of the argument itself.
Less subjective criteria for validity of arguments are often clearly desirable, and in some cases we should even expect an argument to be rigorous, that is, to adhere to precise rules of validity. This is the case for arguments used in mathematical proofs. Note that a rigorous proof does not have to be a formal proof.
In ordinary language, people refer to the ''logic of an argument'' or use terminology that suggests that an argument is based on inference rules of formal logic. Though arguments do use inferences that are indisputably purely logical (such as syllogisms), other kinds of inferences are almost always used in practical arguments. For example, arguments commonly deal with causality, probability and statistics or even specialized areas such as economics. In these cases, rather than to the well-defined principles of pure logic as explicitly set out and agreed upon in an academic, professional or other strictly understood context, ''logic'' in everyday usage almost always refers to something the reader or audience of the argument believe they perceive in the argument, and which drives them inexorably towards some conclusion, something perhaps ill-defined in their own minds (as opposed to putting the emphasis on examining by what criteria they actually accept this apparently compelling force as correct, which is how the formal rules of pure logic are constructed). And yet this feeling of inexorable conviction is also the foundation of those begrudgingly somewhat unsatisfying definitions we give of "logic"; in that we who are driven to construct these most conscientious, circumspect and clear definitions were initially drawn to do so by a similar belief that we recognized some intrinsic logic or compelling rational force in the world- even in the most everyday arguments, although such a view may have been naive, and is in any case incapable of being tested in any objective and/or universally satisfying fashion.
Theories of arguments
Theories of arguments are closely related to theories of informal logic. Ideally, a theory of argument should provide some mechanism for explaining validity of arguments.
One natural approach would follow the mathematical paradigm and attempt to define validity in terms of semantics of the assertions in the argument. Though such an approach is appealing in its simplicity, the obstacles to proceeding this way are very difficult for anything other than purely logical arguments. Among other problems, we need to interpret not only entire sentences, but also components of sentences, for example noun phrases such as ''The present value of government revenue for the next twelve years''.
One major difficulty of pursuing this approach is that determining an appropriate semantic domain is not an easy task, raising numerous thorny ontological issues. It also raises the discouraging prospect of having to work out acceptable semantic theories before being able to say anything useful about understanding and evaluating arguments. For this reason the purely semantic approach is usually replaced with other approaches that are more easily applicable to practical discourse.
For arguments regarding topics such as probability, economics or physics, some of the semantic problems can be conveniently shoved under the rug if we can avail ourselves of a model of the phenomenon under discussion. In this case, we can establish a limited semantic interpretation using the terms of the model and the validity of the argument is reduced to that of the abstract model. This kind of reduction is used in the natural sciences generally, and would be particularly helpful in arguing about social issues if the parties can agree on a model. Unfortunately, this prior reduction seldom occurs, with the result that arguments about social policy rarely have a satisfactory resolution.
Another approach is to develop a theory of argument pragmatics, at least in certain cases where argument and social interaction are closely related. This is most useful when the goal of logical argument is to establish a mutually satisfactory resolution of a difference of opinion between individuals.
Argumentative dialogue
Arguments as discussed in the preceding paragraphs are static, such as one might find in a textbook or research article. They serve as a published record of justification for an assertion. Arguments can also be interactive, in which the proposer and the interlocutor have a more symmetrical relationship. The premises are discussed, as well the validity of the intermediate inferences. For example, consider the following exchange, illustrated by the No true Scotsman fallacy:
: Argument: "No Scotsman puts sugar on his porridge."
: Reply: "But my friend Angus likes sugar with his porridge."
: Rebuttal: "Ah yes, but no true Scotsman puts sugar on his porridge."
In this dialogue, the proposer first offers a premise, the premise is challenged by the interlocutor, and finally the proposer offers a modification of the premise. This exchange could be part of a larger discussion, for example a murder trial, in which the defendant is a Scotsman, and it had been established earlier that the murderer was eating sugared porridge when he or she committed the murder.
In argumentative dialogue, the rules of interaction may be negotiated by the parties to the dialogue, although in many cases the rules are already determined by social mores. In the most symmetrical case, argumentative dialogue can be regarded as a process of discovery more than one of justification of a conclusion. Ideally, the goal of argumentative dialogue is for participants to arrive jointly at a conclusion by mutually accepted inferences. In some cases however, the validity of the conclusion is secondary. For example; emotional outlet, scoring points with an audience, wearing down an opponent or lowering the sale price of an item may instead be the actual goals of the dialogue. Walton distinguishes several types of argumentative dialogue which illustrate these various goals:
★ Personal quarrel.
★ Forensic debate.
★ Persuasion dialogue.
★ Bargaining dialogue.
★ Action seeking dialogue.
★ Educational dialogue.
Van Eemeren and Grootendorst identify various stages of argumentative dialogue. These stages can be regarded as an argument protocol. In a somewhat loose interpretation, the stages are as follows:
★ Confrontation: Presentation of the problem, such as a debate question or a political disagreement
★ Opening: Agreement on rules, such as for example, how evidence is to be presented, which sources of facts are to be used, how to handle divergent interpretations, determination of closing conditions.
★ Argumentation: Application of logical principles according to the agreed-upon rules
★ Closing: This occurs when the termination conditions are met. Among these could be for example, a time limitation or the determination of an arbiter.
Van Eemeren and Grootendorst provide a detailed list of rules that must be applied at each stage of the protocol. Moreover, in the account of argumentation given by these authors, there are specified roles of protagonist and antagonist in the protocol which are determined by the conditions which set up the need for argument.
Many cases of argument are highly unsymmetrical, although in some sense they are dialogues. A particularly important case of this is political argument.
Much of the recent work on argument theory has considered argumentation as an integral part of language and perhaps the most important function of language (Grice, Searle, Austin, Popper). This tendency has removed argumentation theory away from the realm of pure formal logic.
One of the original contributors to this trend is the philosopher Chaim Perelman, who together with Lucie Olbrechts-Tyteca, introduced the French term ''La nouvelle rhetorique'' in 1958 to describe an approach to argument which is not reduced to application of formal rules of inference. Perelman's view of argumentation is much closer to a juridical one, in which rules for presenting evidence and rebuttals play an important role. Though this would apparently invalidate semantic concepts of truth, this approach seems useful in situations in which the possibility of reasoning within some commonly accepted model does not exist or this possibility has broken down because of ideological conflict. Retaining the notion enunciated in the introduction to this article that ''logic'' usually refers to the structure of argument, we can regard the logic of rhetoric as a set of protocols for argumentation.
Other theories
In recent decades one of the more influential discussions of philosophical arguments is that by Nicholas Rescher in his book ''The Strife of Systems''. Rescher models philosophical problems on what he calls aporia or an aporetic cluster: a set of statements, each of which has initial plausibility but which are jointly inconsistent. The only way to solve the problem, then, is to reject one of the statements. If this is correct, it constrains how philosophical arguments are formulated.
References
★ Robert Audi, ''Epistemology'', Routledge, 1998. Particularly relevant is Chapter 6, which explores the relationship between knowledge, inference and argument.
★ J. L. Austin ''How to Do Things With Words'', Oxford University Press, 1976.
★ H. P. Grice, ''Logic and Conversation'' in ''The Logic of Grammar'', Dickenson, 1975.
★ Vincent F. Hendricks, ''Thought 2 Talk: A Crash Course in Reflection and Expression'', New York: Automatic Press / VIP, 2005, ISBN 87-991013-7-8
★ R. A. DeMillo, R. J. Lipton and A. J. Perlis, ''Social Processes and Proofs of Theorems and Programs'', Communications of the ACM, Vol. 22, No. 5, 1979. A classic article on the social process of acceptance of proofs in mathematics.
★ Yu. Manin, ''A Course in Mathematical Logic'', Springer Verlag, 1977. A mathematical view of logic. This book is different from most books on mathematical logic in that it emphasizes the mathematics of logic, as opposed to the formal structure of logic.
★ Ch. Perelman and L. Olbrechts-Tyteca, ''The New Rhetoric'', Notre Dame, 1970. This classic was originally published in French in 1958.
★ Henri Poincaré, ''Science and Hypothesis'', Dover Publications, 1952
★ Frans van Eemeren and Rob Grootendorst, ''Speech Acts in Argumentative Discussions'', Foris Publications, 1984.
★ K. R. Popper ''Objective Knowledge; An Evolutionary Approach'', Oxford: Clarendon Press, 1972.
★ L. S. Stebbing, ''A Modern Introduction to Logic'', Methuen and Co., 1948. An account of logic that covers the classic topics of logic and argument while carefully considering modern developments in logic.
★ Douglas Walton, ''Informal Logic: A Handbook for Critical Argumentation'', Cambridge, 1998
★ Carlos Chesñevar, Ana Maguitman and Ronald Loui, ''Logical Models of Argument'', ACM Computing Surveys, vol. 32, num. 4, pp.337-383, 2000.
★ T. Edward Damer. Attacking Faulty Reasoning, 5th Edition, Wadsworth, 2005. ISBN 0-534-60516-8
See also
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