I also tell them that logic problems are more fun than sudoku. A strange line of thought by Timothy Williamson : "The larger purpose underlying my book Vagueness was to argue for realism like this: if realism is wrong about anything, it is wrong about vagueness that premise was generally agreed ; but realism is not wrong about vagueness; therefore it is not wrong about anything.
There is also Logic Programming e. Studying logic opens a whole new way of thinking about computer programming If you have a formal mathematical proof, you can apply algorithms to transform this proof. One example of such proof transformations is the system CERes for cut-elimination by resolution. You could ask him the details and his slides. I am not so familiar with verification of hardware and software yet, but I guess you could find some interesting real world examples from this application area of logic.
If I remember correctly, Luckhardt managed to prove a certain number-theoretical theorem by using techniques of Logic, and only a few years later Bombieri proved the same result by using number-theoretic techniques. In line with the "understanding your language better" reason, you could also mention the technological aspect of understanding language better: if you know it better, we can program computers to understand it better too. Imagine google accepting queries in natural language, instead of just keywords Russel's paradox within Naive set theory is a famous example of how things can go wrong if we just trust our intuition about seemingly plausible ways of reasoning such as using comprehension axioms in an unrestricted way.
This could be "sold" as an application of proof theory to theoretical computer science Automated and Interactive Theorem Proving is another motivation for the study of Logic I remember reading about a mathematical proof that had so many cases to be considered that it could only be completed with the help of automated theorem provers. Did not Russell criticize Copleston about a quantifier switch in the proof of God: every event has a cause; therefore, there must be something which is the cause of every event.
I like to point my students toJames M. Bartlett Mind 73 It gives Frege's own motivation and justification for a formal logic and it puts the enterprise in an historical context. The best reason to study logic is that it represents one of the great intellectual achievements of the 20th century: solving the classical problem, over years old, of the nature of logical truth. We can now say exactly why some truths are logical analytic, apriori, etc.
As for applications, not only does much of modern computer science depend on this breakthrough in logic, but much of the old philosophy -- more than most philosophers like to admit -- is rendered irrelevant by it.
It is simply no longer plausible to claim that there is metaphysical knowledge, of the kind historically asserted, without the examples once provided by unanalyzed logical truths. It's easy to overlook the consequences of the development of modern logic, because they are so deep that one hardly notices how much the landscape above them has shifted.
I'm going to be teaching a "Basic Logic" course as a philosophy elective for non-philosophy students; we're using The Logic Book, but they won't be tested on the metatheory chapters. So I pointed to the following ideas that logic helps one understand:-The distinction between syntax and semantics, form and content, object language and metalanguage, etc. Students really get a feel for e. Searle's "Chinese room" thought experiment when they have some hands-on experience with logic.
I don't have good examples of famous fallacies, unfortunately, but the discussion thread had some good ideas I'll try and implement over the semester I've observed during my teaching practice , that students frequently make certain mistakes.
Another example is that when they translate entailments from natural language to formal logics, often fail with inversions, mistaking antecedent with consequence. Perhaps the cause isn't the lack of capability to reason properly, but the fact, that they sometimes don't know what they're doing Anyway, apart from formalisms, logic is the theory of proper reasoning.
It is a good question: what makes a reasoning proper or improper. Albeit the answer is problematic, it seems that some forms or rules just feel right -- and the formal system is just a tool to catch and express this rightness.
In addition, the study of formal logic helps to understand the abstract concept of rule, in its purest form. It provides rules of applying rules.
At this point, it might be encouraging to show some everyday examples of the rules we live by, and to point out our natural transparent capability of apprehending rules. Sincerity requires us to present not only the advantages of formal logics, but it's limitations as well. I think it is important to explicate the assumptions that underly the propositional and predicate calculus: that it isn't necessarily natural to think of sentences as of something that is just either true or false there's much more to it!
On the other hand, it is an interesting experiment to see how far we can get with this approach. There are, indeed, many other advantages of studying formal logic -- especially for those who are interested in artificial intelligence and computation theory. It is also a good question, to what extent our mental activity can be implemented on a computer. I'm not sure if the medical database example is a good choice.
Obviously you can try if you want, but my mind switched off on that part of your post. It seems that among all the non-technical university departments, law students are best motivated to study formal logic, because it's application is graspable to them.
I know this post is long dead, but I have recently had a thought about how I might try motivating intro logic next term: start with a batch of results from the psychology of reasoning literature showing how bad humans can be at reasoning.
The Wason selection task is of course the most famous, but I get the impressions that there are a lot of other results out there too. Within informal arguments, premises may be acceptable in a variety of ways. In many circumstances, they are acceptable if they are likely true and unacceptable if likely false.
It is worth noting that this truth criterion can be expanded to apply to visual premises. A photograph or an image may be unacceptable because it is untrustworthy or categorizes a situation in a misleading way. Other kinds of multimodal premises can be understood as likely true to the extent that they are a reliable basis for an inference.
In some kinds of dialogue — in the exchanges that characterize negotiation, bargaining, eristic, and persuasion — acceptable premises may not need to be true. This counts as an acceptable premise even if it is an idle threat that the buyer will never carry out, for threats of this sort are an acceptable element of the arguments that take place in this kind of dialogue.
In other cases, informal logics use acceptability rather than truth as a criterion for judging premises in contexts in which it is difficult to judge premises as true or false. In such cases, acceptable premises may be plausible or exploratory hypotheses, claims that can only be said to be generally accepted or assumed, or ethical or aesthetic judgments which are not easily categorized as true or false.
In still other circumstances, truth may be required for acceptability, but only one of a number of conditions that must be satisfied. Even when a premise is true, it may be unacceptable because it violates the rules of interaction that govern the dialogue in which it is embedded. In a legal proceeding or a formal hearing, premises and arguments must not entertain premises that violate rules of procedure. In situations in which arguments are attempts to convince a specific audience of a conclusion, an acceptable premise may need to be true, but also acceptable to the members of this audience.
As Aristotle suggests in the Rhetoric , successful arguments may need to have premises that are in keeping with the pathos of an audience and do so in a way that does not undermine the character — the ethos — of the arguer. As Gilbert , has emphasized, there are many real life circumstances in which the emotional acceptability of a premise is required for argument success.
In its attempt to account for a broad range of real life arguing, informal logics have expanded traditional notions of premise acceptability. Something similar has happened in the case of inference validity. The end result is an expansion of both sides of the AV criteria for good argument. Sometimes informal logic systems understand inductive arguments narrowly, as inductive generalizations.
Sometimes more broadly, as arguments which have premises that imply that a conclusion is only probable or plausible, leaving open the possibility that it is false. They are valid when they collective enough reasons to warrant their conclusions. They recognize some facts, point out that they are entailed by some hypothesis, and conclude that the hypothesis is true.
AV criteria are in many ways an extension of the notion of good argument enshrined in classical logic. In the search for ways to deal with real life arguments, some informal logicians have moved in a different direction, reviving fallacy theory as an alternative. Hamblin has become a touchstone for moves in this direction. Systems of informal logic that rely on fallacies test arguments by asking whether their proponents are guilty of fallacious reasoning.
While there is no agreed-upon taxonomy of fallacies, many canonical fallacies have been emphasized in the analysis of informal arguments.
Woods and Walton and Hansen and Pinto contain detailed discussions of the definition, analysis and assessment of fallacies. Some fallacies — e. The issues this raises include its unsystematic nature, disagreements about the definition and nature of specific fallacies, and the emphasis that fallacy theory places on faulty reasoning rather than good argument. The theoretical issues raised by fallacy theory are compounded by instances of traditional fallacies which have a reasonable role to play in real life arguing.
Appeals to pity and other appeals to emotion have, to take one example, a legitimate role to play in moral, political and aesthetic debate. The following examples highlight other circumstances in which arguments which fit the definition of a traditional fallacy cannot be so readily dismissed.
He wants to sever the Danish church from the state for his own personal sake. If there is reason to believe that an arguer favors a point of view because they have something to gain from it say, the purchase of a company in which they own shares , this does raise questions about the extent to which their arguments should be entertained.
Such arguments play a central role in the civil rights movement. If it is true that some action will precipitate a chain of consequences that lead down an alleged slippery slope, this is a good reason to question it. Examples of this sort have forced careful accounts of fallacies to make room for reasonable arguments which share the form of traditional fallacies. In doing so, it is helpful to distinguish between fallacies which do and do not have non-fallacious instances.
Equivocation, post hoc ergo propter hoc , non sequitor and hasty generalization are commonly classified as forms of argument that are inherently mistaken. In contrast, traditional fallacies like ad hominem , two wrongs reasoning, guilt by association, and appeal to pity are patterns of reasoning which can, when they are constructed in the right way, play a legitimate role within real life reasoning and are, in view of this, sometimes treated as argument schemes rather than fallacies.
It suggests that we should interpret informal arguments as attempts to create deductively valid inferences which can be analyzed and assessed accordingly. In a deductivist system of informal logic, the V in the AV criteria for good arguments is this classical notion of validity. It is true that ordinary arguing rarely satisfies the strict proof procedures they imply, but deductive validity is not restricted to this compass and there are many instances of ordinary argument which are clear examples of deductively valid argument.
In cases of deductive reasoning, the conclusion of an argument need not be certain, but only as certain the premises, creating ample room for conclusions which are merely likely, plausible, or probable.
In this case, the premise of the argument is not certain, but reasonably thought to be true — because it was in the commentary backed by an extrapolation from well established population trends. The deductively valid inference based upon it makes it reasonable to judge the conclusion of the argument true as well, though it is not certain, as all predictions about population growth are, at best, plausible conjectures.
A blog by a professional dietician Dr. Oz says [that gracinia cambogia is a miracle weight loss pill]. Conclusion : This must be true. Oz says Oz could say gracinia cambogia is a miracle weight loss pill and be wrong. That said, anyone using this argument must assume an associated conditional which can be understood as an implicit premise, allowing us to standardize the argument as:. Premise : Dr. Implicit Premise : If Dr. Oz says this, it must be true. So understood, the Dr.
Oz argument is deductively valid, but not sound, as it is a valid argument with a problematic implicit premise. NLD deals with inductive generalizations in a similar way. Consider the following example from a conversation about French men. I have worked with many and this was what I found. In this example, the move from the premise to the conclusion of the argument assumes that the sample of French men the arguer is familiar with are a representative sample of French men.
If this is not likely, then the sample does not provide good reasons for concluding that French men are, in general, as fastidious if they are. If we recognize this assumption as an implicit premise when we standardize the argument, then the argument is deductively valid, for a representative sample is a subset of a population that accurately reflects the characteristics of the larger group. This does not eliminate such uncertainty, but maintains it as a key consideration in argument evaluation, for it makes the status of this warrant an essential element of premise acceptability.
In favor of NLD, it has been argued that reconstruction of many arguments it proposes is a dialectically useful way to make explicit the key assumptions that arguments depend on, and frees us from the need to distinguish between different kinds of validity in ways that can be problematic when they are applied to real life arguing. The pragma-dialectical account of indirect speech acts Eemeren and Grootendorst , Groarke provides a way to reconstruct arguments as deductive arguments when NLD requires it, though Johnson and Godden argue that NLD is an artificial theory which forces informal arguments to adhere to an overly restrictive model of inference.
Once identified they can be used to evaluate an argument which is instance of a scheme, or as a template or recipe arguers can use when they construct an argument which is an instance of a scheme.
Walton, Reed and Macagno provide a compendium of 96 schemes. Wageman has developed a Periodic Table of Arguments which provides a systematized account of basic schemes. Rules of inference like modus ponens and modus tollens can be understood as deductive schemes.
Dove and Groarke have shown how visual arguments with non-verbal visual elements may be instances of common schemes, and have identified some schemes which are inherently visual. Taking this approach, the scheme Argument from Authority can be defined as follows. A is a credible authority in the domain D. A asserted X, which implies T.
A can be trusted and T is within domain D. T is consistent with what other experts in domain D assert. Therefore T is true. In both cases, the result is an account of premise acceptability and validity which is tailored to specifically apply to arguments from authority. Different argument schemes are a refinement of general AV criteria, creating specific criteria which can be applied to different kinds of argument.
In order for it to be a convincing argument from authority, it would need to fully satisfy the conditions outlined in our definition of the scheme Argument from Authority. The attempt to satisfy the requirements this implies produces a version of the argument which can be summarized as follows. Einstein A is a credible authority on nuclear weapons D.
Einstein A can be trusted and questions about stockpiling nuclear weapons T are questions about nuclear weapons D. The claim that we should not stockpile nuclear weapons T is consistent with what other experts on nuclear weapons D assert.
Therefore We should not stockpile nuclear weapons T. This attempt to satisfy the conditions for a good instance of argument from authority fails because it produces a number of problematic premises. More fundamentally, the proposed argument is founded on too loose an account of nuclear weapons as a domain of expertise. Einstein is a renowned expert on nuclear physics but this it does not make him an expert on the social and political issues raised by nuclear weapons.
This is something that a convincing version of the argument would have to establish. In some ways, the scheme approach to argument assessment rectifies problems that arise in systems of informal logic that adopt the fallacy approach to argument evaluation.
For traditional fallacies which have non-fallacious instances can be understood, not as fallacies, but as argumentation schemes which are legitimate forms of reasoning when they are properly employed. Fallacy definitions can be turned into scheme definitions by identifying a list of critical questions or required premises that specify what is required to make the arguments in question valid.
Ad hominem is a case in point, for there are many instances in which criticisms of an arguer rather than their position are a reasonable way to cast doubt on their views. We can specify when this is so by listing critical questions that determine whether this is so in a particular case of argument. Treated in this way, ad hominem is a legitimate scheme of argument, but there are as there are in the case of all schemes many situations in which ad hominem arguments are poor instances of the scheme.
In this and many other cases, traditional fallacies can be regarded as deviations from an inherently correct scheme of reasoning. Most informal logics combine different ways of evaluating arguments.
In this way, T in a particular system tends to be some combined set of tools that can be used in this endeavor. But systems of informal logic can accommodate other, less common criteria for deciding whether arguments are good or bad, strong or weak. A system might, for example, incorporate criteria which are founded on a virtue-based approach to argument see Virtues and Arguments , on feminist principles, or on notions derived from rhetoric, theories of communication, or other cognate fields.
One cognate field of note is Artificial Intelligence AI. It relies on step by step accounts of informal reasoning in a wide array of contexts. Informal logics provide this in a general way that has influenced the attempt to model argumentation between agents in multi-agent systems which mimic or assist human reasoning.
Automated argument assistance functions as a computational aid that can assist in the construction of an argument. Verheij provides an overview of the issues that this raises.
In an early study of this sort, Jorgenson, Kock and Rorbech analyzed a series of 37 one-hour televised debates from Danish public TV. The debates featured well-known public figures arguing for and against policy proposals. A representative audience of voters voted before and after the debate.
Other studies consider corpora made up of large databases of selected written texts see, e. Argument mining is a subfield of data mining, or text mining and computational linguistics. It uses software and algorithms that automatically process texts looking for argument structures — for premises, conclusions, argumentation schemes, and extended webs of argument. The texts studied include legal documents , on-line debates, product reviews, academic literature, user comments on proposed regulations, newspaper articles and court cases, as well as dialogical domains.
Other research relevant to informal logic has highlighted many ways in which the success of real life arguments depends on aspects of argumentation which are not well integrated into standard systems of informal logic. The study of these and other pragmatic, social, dialectical, semiotic and rhetorical features of arguing will probably play a role in the continued development of informal logic.
The expansion of informal logic to account for an ever broader range of argument is evident in discussions of the use of narratives within argument. In this and many other situations, stories of various kinds accounts of some historical event, biographies, fables, parables, morality plays, etc are designed to provide support for some conclusion.
It is often been said that a novel or some other work of fiction is an argument for socialism, freedom of expression, or some other value. One can understand the argumentative use of narratives in a variety of ways: as rhetorical embellishment, as a form of argument by analogy, as implicit generalization the characters in a story functioning as variables within such generalizations , or reasoning that requires the development of unique standards of argument assessment.
According to Fisher , argument itself is best understood as narrative. According to Nussbaum , literature is a way to better understand, and argue about, complex moral situations. Within informal logic, Walton, Reed, and Macagno identify narrative-based schemes of argument while others continue to debate the role that narratives play in ordinary argument see Govier and Ayer , Olmos , Plumer Some of the educational issues raised by informal logic are manifest in the development of critical thinking tests which attempt to measure argumentation skills.
They are used to test the abilities of students or others and, in a self-reflective way, as an empirical way to test the success of attempts to teach informal reasoning.
Critical thinking or, even more so, creative thinking skills are not easily assessed using standardized tests which are designed for large scale use, and typically rely on multiple choice question and answers see Sobocan In real life contexts, what counts as good arguing and thinking is open ended and unpredictable, dialectical, and influenced by pragmatic and contextual considerations which are difficult to incorporate within standard tests.
Ennis provides a comprehensive proposal for dealing with the issues raised by critical thinking tests, and with other challenges raised by attempts to teach critical thinking. The field of informal logic is a recent invention, but one that continues historical attempts to understand and teach others how to argue.
In Aristotle it is manifest in his systematic account of reasoning, which is expressly designed to teach others how to argue well. Within the history of philosophy, one finds numerous other attempts to formulate accounts of argument that can be used to explain, evaluate, and teach real life reasoning.
The practice of philosophy itself assumes and frequently develops an account of argument as it assembles evidence for different philosophical perspectives. Systems of informal logic assume, and often depend upon, the resulting views of reason, rationality and what counts as evidence and knowledge. The philosophical issues in play are tied to complex, unsettled epistemological questions about evidence and knowledge.
Mercier and Sperber argue that reasoning is a practice which has evolved from, and needs to be understood in terms of, the social practice of argumentation. Johnson pushes in the opposite direction, arguing that a comprehensive account of argument must be built upon a philosophical account of rationality.
Goldman situates knowledge in the social interactions that take place within interpersonal exchange and knowledge institutions, emphasizing informal argument and the constraints which make it a valuable practice. Some aspects of informal logic raise deep questions that have implications for logic and philosophy. One notable feature of informal logic as it is now practiced is a proliferation of different systems of informal logic which approach the analysis and evaluation of informal reasoning in different ways — employing fallacies, AV criteria, argumentation schemes, methods of formal analysis, and other models of good argument.
One implication is a broadening of the conditions for argument felicity. However one understands visual and multimodal arguments, there is no easy way to reduce them to sets of sentences, for there is no precise way to translate what we see, hear, experience, etc. In North America and elsewhere, informal logic is a field in which philosophers apply theories of argument rationality, knowledge, etc.
In keeping with this, philosophers continue to be the core contributors to informal logic; philosophy departments in colleges and universities continue to be the core departments that teach the courses that are its pedagogical focus. Though informal logic addresses many issues relevant to core philosophical disciplines most notably, epistemology and philosophy of mind; evident in the work of Goldman, Crosswhite, Thagard, and others , it has had limited impact on mainstream approaches to their subject matter.
Woods has speculated on the reasons why. The scale and complexity of the enterprise is such that if one seeks in contemporary American philosophy for a consensus on the problem agenda, let alone for agreement on the substantive issues, then one is predestined to look in vain.
The goal of applied philosophy is philosophically informed and nuanced reasoning that addresses complex real life situations. Informal logic is one field which has made a valuable contribution to this goal. History 2. Systems of Informal Logic 3. Standardizing Arguments 4. Testing Arguments 5. History Puppo provides a recent collection of articles on the history of informal logic and the issues it addresses. Systems of Informal Logic As Hansen emphasizes, there are many different methods that informal logicians use to analyze instances of argument.
The fact is, fetuses are being aborted whether conservatives like it or not. Post-abortion, the embryos are literally being thrown away when they could be used in lifesaving medical research Lives could be saved and vastly improved if only scientists were allowed to use embryos that are otherwise being tossed in the garbage. We can summarize the elements of this argument as follows. Premise : Fetuses are being aborted anyway whether conservatives like it or not.
Premise : Lives could be saved and vastly improved if scientists were allowed to use embryos that are otherwise being tossed in the garbage. Conclusion : The conservative opposition to embryonic research is shortsighted and stubborn.
Bibliography Ajdukiewicz, K. Pragmatic Logic , O. Wojtasiewicz trans. English translation of Logika pragmatyczna , originally published Arnauld, Antoine, and Pierre Nicole, Bynum and James H. Moor eds. Battersby, Mark and Sharon Bailin, Battersby, Mark, Is That a Fact? See for example, Montague [], Davidson [], Lycan [] and the entry on logical form.
Another view, held at least in part by Gottlob Frege and Wilhelm Leibniz, is that because natural languages are fraught with vagueness and ambiguity, they should be replaced by formal languages. A similar view, held by W.
Quine e. One desideratum of the enterprise is that the logical structures in the regimented language should be transparent. A regimented language is similar to a formal language regarding, for example, the explicitly presented rigor of its syntax and its truth conditions.
On a view like this, deducibility and validity represent idealizations of correct reasoning in natural language. A chunk of reasoning is correct to the extent that it corresponds to, or can be regimented by, a valid or deducible argument in a formal language.
When mathematicians and many philosophers engage in deductive reasoning, they occasionally invoke formulas in a formal language to help disambiguate, or otherwise clarify what they mean. In other words, sometimes formulas in a formal language are used in ordinary reasoning.
This suggests that one might think of a formal language as an addendum to a natural language. Then our present question concerns the relationship between this addendum and the original language.
What do deducibility and validity, as sharply defined on the addendum, tell us about correct deductive reasoning in general? Another view is that a formal language is a mathematical model of a natural language in roughly the same sense as, say, a collection of point masses is a model of a system of physical objects, and the Bohr construction is a model of an atom.
In other words, a formal language displays certain features of natural languages, or idealizations thereof, while ignoring or simplifying other features. The purpose of mathematical models is to shed light on what they are models of, without claiming that the model is accurate in all respects or that the model should replace what it is a model of. On a view like this, deducibility and validity represent mathematical models of perhaps different aspects of correct reasoning in natural languages.
Correct chunks of deductive reasoning correspond, more or less, to valid or deducible arguments; incorrect chunks of reasoning roughly correspond to invalid or non-deducible arguments. See, for example, Corcoran [], Shapiro [], and Cook [].
There is no need to adjudicate this matter here. Perhaps the truth lies in a combination of the above options, or maybe some other option is the correct, or most illuminating one. We raise the matter only to lend some philosophical perspective to the formal treatment that follows. Here we develop the basics of a formal language, or to be precise, a class of formal languages. Again, a formal language is a recursively defined set of strings on a fixed alphabet.
Some aspects of the formal languages correspond to, or have counterparts in, natural languages like English. We begin with analogues of singular terms , linguistic items whose function is to denote a person or object.
We call these terms. We assume a stock of individual constants. These are lower-case letters, near the beginning of the Roman alphabet, with or without numerical subscripts:.
We envisage a potential infinity of individual constants. In the present system each constant is a single character, and so individual constants do not have an internal syntax. Thus we have an infinite alphabet. We also assume a stock of individual variables. These are lower-case letters, near the end of the alphabet, with or without numerical subscripts:.
In ordinary mathematical reasoning, there are two functions terms need to fulfill. We need to be able to denote specific, but unspecified or arbitrary objects, and sometimes we need to express generality. In our system, we use some constants in the role of unspecified reference and variables to express generality.
Both uses are recapitulated in the formal treatment below. Constants and variables are the only terms in our formal language, so all of our terms are simple, corresponding to proper names and some uses of pronouns. We call a term closed if it is not a variable. Logic books aimed at mathematicians are likely to contain function letters, probably due to the centrality of functions in mathematical discourse.
Books aimed at a more general audience or at philosophy students , may leave out function letters, since it simplifies the syntax and theory. We follow the latter route here.
This is an instance of a general tradeoff between presenting a system with greater expressive resources, at the cost of making its formal treatment more complex. These are upper-case letters at the beginning or middle of the alphabet.
A superscript indicates the number of places, and there may or may not be a subscript. For example,. We often omit the superscript, when no confusion will result. They correspond to free-standing sentences whose internal structure does not matter. And so on. The non-logical terminology of the language consists of its individual constants and predicate letters. In taking identity to be logical, we provide explicit treatment for it in the deductive system and in the model-theoretic semantics.
Most authors do the same, but there is some controversy over the issue Quine [, Chapter 5]. Examples of atomic formulas include:. If an atomic formula has no variables, then it is called an atomic sentence. If it does have variables, it is called open.
In the above list of examples, the first and second are open; the rest are sentences. Clause 8 allows us to do inductions on the complexity of formulas. If a certain property holds of the atomic formulas and is closed under the operations presented in clauses 2 — 7 , then the property holds of all formulas.
Here is a simple example:. Theorem 1. Moreover, each left parenthesis corresponds to a unique right parenthesis, which occurs to the right of the left parenthesis. Similarly, each right parenthesis corresponds to a unique left parenthesis, which occurs to the left of the given right parenthesis. If a parenthesis occurs between a matched pair of parentheses, then its mate also occurs within that matched pair.
In other words, parentheses that occur within a matched pair are themselves matched. Proof : By clause 8 , every formula is built up from the atomic formulas using clauses 2 — 7. The atomic formulas have no parentheses. Parentheses are introduced only in clauses 3 — 5 , and each time they are introduced as a matched set. So at any stage in the construction of a formula, the parentheses are paired off.
We next define the notion of an occurrence of a variable being free or bound in a formula. We do not even think of those as occurrences of the variable. All variables that occur in an atomic formula are free. That is, the unary and binary connectives do not change the status of variables that occur in them.
Although it does not create any ambiguities see below , we will avoid such formulas, as a matter of taste and clarity. These, too, will be avoided in what follows. Some treatments of logic rule out vacuous binding and double binding as a matter of syntax. That simplifies some of the treatments below, and complicates others. Free variables correspond to place-holders, while bound variables are used to express generality. If a formula has no free variables, then it is called a sentence.
If a formula has free variables, it is called open. Before turning to the deductive system and semantics, we mention a few features of the language, as developed so far. This helps draw the contrast between formal languages and natural languages like English.
We assume at the outset that all of the categories are disjoint. For example, no connective is also a quantifier or a variable, and the non-logical terms are not also parentheses or connectives. Also, the items within each category are distinct. For example, the sign for disjunction does not do double-duty as the negation symbol, and perhaps more significantly, no two-place predicate is also a one-place predicate. Consider the English sentence:. It can mean that John is married and either Mary is single or Joe is crazy, or else it can mean that either both John is married and Mary is single, or else Joe is crazy.
If our formal language did not have the parentheses in it, it would have amphibolies. The parentheses resolve what would be an amphiboly. Can we be sure that there are no other amphibolies in our language?
Our next task is to answer this question. Lemma 2. Each formula consists of a string of zero or more unary markers followed by either an atomic formula or a formula produced using a binary connective, via one of clauses 3 — 5. Proof : We proceed by induction on the complexity of the formula or, in other words, on the number of formation rules that are applied. The Lemma clearly holds for atomic formulas.
Lemma 3. Proof : Here we also proceed by induction on the number of instances of 2 — 7 used to construct the formula. Clearly, the Lemma holds for atomic formulas, since they have no parentheses. The proof proceeds by induction on the number of instances of 2 — 7 used to construct the formula, and we leave it as an exercise. Theorem 5. If the latter formula was produced via one of clauses 3 — 5 , then it begins with a left parenthesis. Theorem 6. Moreover, no formula produced by clauses 2 — 7 is atomic.
In this case, it must have been produced by one of 3 — 5 , and not by any other clause. Similar reasoning takes care of the other combinations. It shows that each formula is produced from the atomic formulas via the various clauses in exactly one way.
We apologize for the tedious details. We included them to indicate the level of precision and rigor for the syntax. As above, we define an argument to be a non-empty collection of sentences in the formal language, one of which is designated to be the conclusion.
If there are any other sentences in the argument, they are its premises.
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