Complete the system of natural deduction by adding a new category of justification-a justified assumption. Then see how this concept is used in conditional and indirect proofs. With these additions, you are now fully equipped to evaluate the validity of arguments from everyday life. So far, you have learned two approaches to logic: Aristotle's categorical method and truth-functional logic. Now add a third, hybrid approach, first-order predicate logic, which allows you to get inside sentences to map the logical structure within them.
For all of their power, truth tables won't work to demonstrate validity in first-order predicate arguments. For that, you need natural deduction proofs-plus four additional rules of inference and one new equivalence. Review these procedures and then try several examples. Study two techniques for demonstrating that an argument in first-order predicate logic is invalid.
The method of counter-example involves scrupulous attention to the full meaning of the words in a sentence, which is an unusual requirement, given the symbolic nature of logic. The method of expansion has no such requirement. Hone your skill with first-order predicate logic by expanding into relations. An example: If I am taller than my son and my son is taller than my wife, then I am taller than my wife.
Still missing from our logical toolkit is the ability to validate identity. Known as equivalence relations, these proofs have three important criteria: equivalence is reflexive, symmetric, and transitive. Test the techniques by validating the identity of an unknown party in an office romance.
See how all that you have learned in the course relates to mathematics-and vice versa. Trace the origin of deductive logic to the ancient geometrician Euclid. Then consider the development of non-Euclidean geometries in the 19th century and the puzzle this posed for mathematicians. Delve deeper into the effort to prove that the logical consistency of mathematics can be reduced to basic arithmetic.
Add two new operators to your first-order predicate vocabulary: a symbol for possibility and another for necessity. These allow you to deal with modal concepts, which are contingent or necessary truths.
See how philosophers have used modal logic to investigate ethical obligations. See what happens if we deny the central claim of classical logic, that a proposition is either true or false. This step leads to new and useful types of reasoning called multi-valued logic and fuzzy logic. Wind up the course by considering where you've been and what logic is ultimately about.
Read reviews for An Introduction to Formal Logic 4. Reviews Write a review. This action will open a modal dialog. Rating Snapshot. Select to filter reviews with 5 stars. Select to filter reviews with 4 stars. Select to filter reviews with 3 stars. Select to filter reviews with 2 stars. Select to filter reviews with 1 star. Average Customer Ratings. Most Helpful Favorable Review. Review by PacificNW. Written 6 years ago.
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Adjust Line Height. Align Left. Adjust Letter Spacing. Align Right. Color Adjustments. Dark Contrast. Light Contrast. Suppose that I accept A as well. This requires a hypothetical proof: ask me to suppose for the time being that I accept A, and show me that it would be logically necessary, in that supposed situation, to accept B. That economy has some famous consequences, some of which are disturbing.
The implication arrow is directional: it leads from A to B. Reading the implication arrow backwards, seeing causes where there may be none, is called abduction.
Beware of it in logical arguments. Abduction seems to be essential if you are to guess truths from insufficient evidence. A necessary step of reasoning which is outside logic? No wonder we can be so easily led astray — see the case of the drowned Major in chapter 5.
Figure 3. I have accepted for almost all my life that my name is Richard. But here is a hypothetical proof which seems to show logically that it does! Irrelevant implications, like the one in the price-of-tomatoes argument, are a consequence of the monotonicity principle. An uncle who keeps his promises is surely a kind uncle similar remarks apply, of course, to aunts and to large donations in any currency. An unkind uncle might casually make the same promise, and then break it: your birthday comes around, but no money.
Unkind uncles are surely the worst. Or are they? Is he kind or unkind: that is, has he broken or kept his promise? Logic takes a very literal legalistic view of these things, and it says that a promise is kept until it is broken. The cunning uncle gets away with it, in logical terms. Logicians call it vacuous implication. If you never see A, then when you see A you see B. The point of this definition is its uncertainty.
So you might accept just A, or you might accept just B, or you might accept both. Either 1. I accept A, or 2. I accept B, or 3. I accept both. There is no hope of tricking me into revealing which of the three possibilities is the case, but there is something you can do: you can argue by cases, picking off the alternatives one by one, showing that each leads to the same conclusion so that the uncertainty is resolved.
You persuade me first that C is a logical consequence of A, picking whatever C suits your purposes. You persuade me next that the same C is also a consequence of B. Then you have persuaded me that I must accept C in any case: in case I accept A, then I must accept C; in case I accept B then I must accept C; in case I accept both, then by the monotonicity principle either the A-argument or the B-argument already shows that I accept C.
The elim rule, which captures the use argument, looks fearsome but check it! The introduction rules, on the other, use certainty and then hide it. This is formal logic, not Boolean-algebra arithmetic. To persuade me that I already accept a contradiction, show me that I accept some claim and at the same time it would be wrong to accept it the elim rule in table 3. If negation depends on contradiction, what does contradiction mean? It turns out that there are two ways to proceed from here.
Technically it can all be boiled down to a single choice of rule; philosophically the choice is complicated and important. That makes particular sense in the world of computer programming and computer science. These two positions sound as if they are universes apart and incomparable. The battle is an old one, going back at least a century. Science contains lots of undecided questions: must the universe continue expanding? The margin is too small to contain it. If Golbach2 is ever proved right, tbere are plenty more unproved conjectures in the rich field of prime numbers: this chapter will only need minor changes.
Who knows, say the constructivists — we neither have a proof, nor do we know that a proof is impossible. The mathematical philosophy called intuitionism can be summarised as saying that mathematics is all made up by humans. Our job is to find out truth, not to invent it. Most working mathematicians are Platonists. Either you love something or you hate it already, whether you know it or not.
But then you love fishing, so by the price-of-tomatoes trick ouch! And the other way round if you try to deny the second implication. We can leave them squabbling: each, on their own ground, is certain that they are right. On the other hand, classicists accept everything that constructivists do, and more besides.
The overall situation is summarised in figure 3. Everything that can be claimed is inside the box. Everything in the circle is classically provable; nothing outside it is.
Everything in the blob is constructively provable; nothing outside it is. Conjectures outside the circle — most conjectures, in fact — are neither classically and constructively provable.
Conjectures inside the blob are both classically and constructively provable. You can look at this situation in two ways. From the classical point of view, constructivists are too fussy. There are proofs in the grey region which classicists can easily make, but constructivists just refuse to accept. So, you might say, classical logic is just more powerful, more useful. So, you might say, constructive logic is safer.
There is a debate about what mathematics really means, and no important philosophical question ever gets a final answer. This one will run and run. If you want to join in you can, or you can just let them get on with it. You ought to know, as a user, that there are different logics, because they have different characteristics and different uses.
Logic, for the practising computer scientist, is a playground: we can play with any ball that we can pick up. This book deals with three logics constructive and classical Natural deduction in parts II and III, Hoare logic for programs in part IV , but there are lots more out there. You pick one to suit your purposes.
Constructive logic has a particular link to formal calculation, and that makes it relevant to reasoning about declarative programs. Classical logic is usually used to reason about imperative programs see part IV. Since constructive formal proof is easier than classical formal proof it really is! Jape includes both constructive and classical versions of the contradiction rule.
It lets you swing both ways and build classical proofs and constructive disproofs see chapter 9 , both at the same time, of claims that fall in the disputed region of figure 3. What can you do with confu- sion? Surprisingly, quite a bit. If a contradiction arises in a proof, we are in an impossible situation.
We know that we accept either A or B or both. So the only reasonable possibility is to accept A. This happens quite a bit when reasoning by cases: one of the cases is impossible, and need not be taken seriously.
When we translate this into a formal disjunction step, we seem to be stuck: A.. But how does A follow from B?
The technical way in which Natural Deduction deals with this situation is to say that in an impossible situation, you can conclude whatever you like in order to tidy things up. In short, if you meet a contradiction, write the subproof off. Both constructivists and classicists accept it, and both can use it.
We can derive any conclusion from a contradiction, then. Classicists also do it the other way round, in the classical contra rule of table 3.
Assumptions are resources you can use to reach a conclusion. So, given what you know about logic already, you should be able to understand that: If you accept the classical treatment of contradiction, you accept the constructive treatment as well — but not vice-versa. That means that you can imitate any constructive contra step in classical logic. It needs only ten rules for the logical connectives — table 3.
Contradiction comes in two forms; the classical includes the constructive. The simplicity is real: those rules, plus four more for the quantifiers see table 6. Of course you have to learn how to use the rules You use a rule scheme to make an instance that suits your needs.
The names A, B and C are parameters of the rule schemes of chapter 3. You make an instance by replacing each of the parameters of a scheme by a formula that you choose. Of course you must replace every occurrence of A by a copy of the A-formula, and likewise the Bs must all be replaced by copies of the B-formula, and so on, but otherwise you have complete freedom: you can use the same formula for A as for B, or a different one, just as you wish.
The formulae that replace the parameters to make an instance are called arguments. Formula arguments replace scheme parameters to make instances. Figure 4. So is.. And so on — we can build up formulae of any size we like using whatever symbols and connectives we like.
But it seems we must use rather a lot of brackets! Formulae fit schemes; schemes match formulae. Once you know what formula scheme a formula fits, you immediately know what rule schemes it fits. A conjunction, for example, fits the conjunction rules. Shape- matching is the basis of proof search. Brackets in a formula perfectly delineate its shape, but brackets are serious visual clutter. They make the formula harder to read, even though they make it easier to explain what it means. In essence, the convention is a means of telling you where the brackets would have to go if you bothered to put them in.
The convention I use in this book is quite standard, and is based on giving each connective a binding priority and a binding direction. We learn calculation slogans: I was taught the ones in table 4. Table 4. The list of slogans lengthens, and calculation gets more intricate, when we learn that you can do additions and multiplications in any old order — e. It gets more intricate still when we learn syntactic equivalences which convert additions into subtractions — e.
When we have mixed operators the slogans tell us which way to work. The main difference is the use we make of the shapes: when analysing a formula we work from the outside inwards; when calculating we work from the inside outwards. Bracket negations first none in this example, so nothing to do. Last, additions and subtractions: there is one of each, and they do overlap, so binding power comes into play.
Exercise 4. Bracket each of the following formulae, using the priority scheme of table 4. Once you can spot a principal operator you can fit a formula to a formula scheme. In each of the following formulae, circle the principal operator, using the priorities and binding directions of table 4. Avoid bracketing the formula if you can. The ordering of the connectives in the table is arbitrary and the binding directions, apart from negation which, as a prefix connective, necessarily binds right-to-left, are arbitrary too.
Bracket each of the following formulae, using the priorities and binding directions of table 4. In each of the following formulae, circle the principal connective, using the priorities and binding directions of table 4. Rule instantiation needs a bit of care. What we do instead is fit formulae to formula schemes by finding their principal operator. Then they fit any rule that depends on that formula scheme. Proofs which use only the connectives and the constant symbols are pretty straightforward.
One of the legacies of his work is a notation for stating logical claims. A conjecture expresses the claim that you can prove a conclusion C from some premises A1 , A2 ,.
Theorems can be used as auxiliary rules of inference, as we shall see. Nowadays Natural Deduction is seen as a classification of those logics which are defined by intro and elim rules. There are lots of alternative Natural Deduction systems, distinguished by choice of rules. Even in this book there are two: the rules of table 3. When it matters, I shall point out claims that are provable classically but not constructively.
Each proof step uses a rule to make a proof out of one or more subproofs or, in the case of the truth rule, no subproofs at all ; in doing so it makes a little tree of deductions, with the consequent of the step at the root of the tree, and its antecedents at the roots of its subtrees. That tree can be plugged as a subproof into another step to help make a larger tree, and so on.
The conclusion of the whole proof is the root of the whole tree. The leaves of the tree are the premises of the proof: accept the claims made in the leaves, and the proof shows you how to persuade yourself, or someone else, to accept the conclusion written at the root. Larger tree proofs are wider still and much harder to read. Luckily, there is an easier way.
Each line consists of a logical formula, numbered on the left and justified on the right. Each line is either a premise or a deduction from a previous line or lines. The justification either says that this line is a premise, or it names the rule used to make the deduction step and the number s of the previous line s.
By convention premises come first, deductions later, and necessarily the conclusion comes last. For example, a line proof of 5. I am a cabbage premise 2. On the other hand, as you will see, when we make proofs we often work from the bottom upwards. The proof of 5. I am a Martian premise 2.
We must not refer to later deductions, because that can produce a circular argument, one which has no proper support. Definition 5. In a line proof every line must be justified either as a premise or by use of a rule appealing to previous lines. Just as the proof rule uses a box, so must the presentation.
The components of our proofs can be lines or boxes, the components of the boxes can be lines or boxes, the components of those boxes can be lines or boxes, and so on down, as deep as we wish to go. In box-and-line proofs, similarly, a deduction can refer to the whole of a box but not to part of it. Steps inside the box, however, can look outside and appeal to earlier boxes and earlier lines. Here, for example, is an argument justifying my belief that when I get into a bath I get wet.
Then the entire box from lines 4 to 7 is cited in the justification for the deduction on line 8. This is the box-and-line condition. In a box-and-line proof 1. The condition can be summarised in the slogans of table 5. Table 5. Figure 5. Note that the extra assumption, on line 5, is not appealed to anywhere. Justifications of deduced lines must respect the antecedent-ordering of their corresponding rules. On line 5 of figure 5. Some of the reasoning which follows is good, and some is bad.
I often squabble with Bernard Sufrin, the colleague and friend with whom I first developed Jape, because he likes things to be cooler. The argument has been going on for decades. Please leave the thermostat alone! Relying on the conclusion of that argument, I shamelessly try to impose my will on Bernard, in his own room. Looking at the rules summarised in table 3. Real-world arguments introduce them as and when they are needed. My dispute with Bernard rests partly on a claim about my weakness: I do tend to fall asleep when I feel warm.
I am awake premise 3. Or perhaps I am not as awake as line 2 claims. When we hear a rustle in the undergrowth we have to jump to conclusions before the tiger jumps on us: jumping too soon may be a logical error, but jumping too late can be fatal.
It seems he must have been drowned, and that looks like a logical conclusion from the evidence. Unfortunately for our pride in our deductive skills, we are told in the last chapter that the Major was poisoned by the butler and then thrown in the lake. Gaps and slips in reasoning often happen when people are trying to reason semantically, using mean- ings instead of formula shapes.
We know, or we think we know, a good deal about the likely causes of death in imaginary s country houses, and we correspondingly rush to judgement. Semantic reasoning is useful and sometimes essential in real life tigers and all that but formal shapewise reasoning is more reliable when we can use it. The proof, if it is a proof, should start: 1. Line 3 of 5. This not a single-step deduction according to the rules, but it is logically valid. Because it uses generic formula-names E, F and G rather than specific formulae like drowning, submersion and death ugh!
Three lines are all that we need: two premises, one straightforward deduction, a valid and convincing argument. Although the original argument was padded out with too many premises and some unnecessary deductions, the conclusion seems to be a logical necessity. What went wrong? The argument should have been stated like this: 1. I and several million others live in London, where the traffic problem is specially acute despite the fact that it has the best public transport provision in the country.
This sparks lots of hot political argument, with alternative causes constantly being suggested and novel remedies put forward. During the school holidays, and at half-term breaks, there is much less congestion. Clearly school-run journeys are causing congestion. I still think London has too many cars. Evidence about the relationship between school runs and congestion is non-controversial.
The Greens assume that everybody listening to them would realise that school-run journeys take place only on school days. There are lots of journeys of the second kind on school days, surely none at all on non-school days.
A tiny minority of absent-minded people will do daft things every day, and we can safely ignore them. Given those four premises, the Greens might expand their argument as follows. So: is it a valid argument? Perhaps surprisingly, it is! Formal reasoning mimics this argument. At this point I draw your attention to something going wrong with the attempt to prove a cause by logical reasoning. School runs do not cause school days!
Nevertheless, I press on. With the theorem of 5. Certainly not. To prove a scientific implication you have to demonstrate an association of cause and effect and describe a convincing mechanism which connects the two.
But the effect is caused by school holidays. Parents go on holiday when their children are on holiday. There are fewer drivers travelling to work, and there is therefore less congestion, during the school holidays.
So nobody really knows whether the school-holiday argument is a better explanation for congestion than the school-run argument. This is a real example, straight off the radio, honestly it is! In road traffic we find back-breaking straws everywhere,4 but I set that objection aside for the time being. The claim is that school-run journeys are a small proportion of travel-to-school journeys.
What is the relationship of school-run journeys to morning rush-hour car journeys? There are hidden premises here. The presenter of this argument may have thought that it was obvious that, since children are a minority of the population agreed and since most journeys to work are by car not agreed — certainly untrue in London where public transport takes almost all the strain , travel-to-school journeys must be a small proportion of rush-hour car-to- work journeys.
It seems possible that in London school-run journeys are a significant proportion of rush-hour car journeys, and if so the counter-argument would be demolished. Logical reasoning, however careful, 4 The whole of central and east London was once paralysed for several hours because a single lorry broke down in a river tunnel. The police had to smuggle the lorry driver away and keep his name and address secret for fear of reprisals.
I got home smug and warm as quickly as usual on my bike, of course. It is about making proofs, as well as reading them. Formal proofs are quite easy to read, once you understand the logical rules, but they do look very inventive: when you first see one you wonder how the steps were chosen and how the assumptions were dreamed up. The proof calculator Jape, available on the internet for free see www. And the strategy is really, really, really easy to use. That strategy is summarised in table 5.
Proof search is driven by the shapes of formulae. There are further slogans about particular connectives and their rules, but these are the core of what you need to know. Only slogan 8 looks weird classical contradiction gets special treatment in section 5.
Work on formulae with connectives. Shape-match with formula-schemes in rules. Fit hypothesis formulae to antecedents of elim rules; fit conclusion for- mulae to consequents of intro rules. Elim steps usually go forward, intro steps nearly always go backward. Prefer rules that generate assumptions. Believe in slogan 1. If all else fails, classical contradiction might be worth a try.
That means it makes proof steps, just as an arithmetic calculator makes arithmetic steps. But, unlike an arithmetic calculator, it can undo a step or several steps, so you can use it to search through the maze of possible proof developments.
And, as a calculator, it guarantees accuracy, so you can use it to find out just what the effect of a proof step would be. What makes Jape useful is that it can help you to understand and learn a strategy for finding proofs. Jape deals with complete proofs and also with proof attempts: box-and-line structures in which all the deductions that have been made are valid logical steps, but which need more deductions to complete them. What you see, as for example in figure 5.
To save precious screen space, Jape usually puts as many premises as it can on a single line. The priority rules of table 4. We might hope to be able to make a formal proof which supports that intuition. If you accept one formula you can prove the other; if you accept the other you can prove the one; so they are equivalent. Slogan 3 tells us to shape-match; slogan 4 tells us what formula-schemes to match.
Despite slogan 1, most people would make a forward step in this situation just because they can, and figure 5. Line 2 of figure 5. The effect so far is to have extracted all the component parts of the premise. Now line 7 is no longer open, and all there is to work on is the new line 6. Slogan 4 is the only one we need in this case.
More often we need a mixture of strategies, as in figure 5. Certainly, it makes 5 Why is it a dead end? More on this in chapter 8. Presented with the problem in figure 5. There are two versions of the step: figure 5. The rule has a slogan all of its own. The search has been split into two cases, shown by the two boxes each with its own lines of dots.
The proof in one case is immediate figure 5. In figure 5. Once both those steps are complete figure 5. The second case is similar, and not shown. Constructive contradiction is really easy to use. Classical proof needs the classical contra rule. The principles of using classical contra are 1. Whereas every other intro rule has a consequent which will only match a formula with an appropriate principal operator, the consequent A of classical contra will match any conclusion formula at all.
That means that you can always make a classical contradiction step backwards and you never have a clue, either from the shape of the conclusion you are trying to prove or the hypotheses you are trying to prove it from, just when to do it.
Instead, you usually backtrack and try it earlier. As for how far to backtrack — experience and reflection is the only way to find out. Backward steps using the slogans get us to 5. Trying the assumption we already used the premise gives 5.
The other arm works in a similar way. The end result in 5. The proof looks very surprising if you forget the search process and read figure 5. Imagine how difficult it must be to make proofs forwards-only, when you have to try to make assumptions without knowing what assumptions to make! Even deciding when to make an assumption, in such a cock-eyed scheme, would seem difficult. By contrast, constructive proof search is straighforward.
Even when classical contradiction has to be used, proof search is still highly rational. We ought to be able to prove it, and from what you know already you should realise how: start with a classical contradiction step figure 5. Weird, or what? Read forward the proof seems quite marvellous, but apart from the first step it was an entirely connective-driven business.
Jape puts hyp steps in automatically. On paper and on the blackboard you have to put them in for yourself. Natural Deduction deals with generalisations, specialisations, search for an example, and all the other ways that a claim can be about a collection of things or about a thing chosen from a collection, by using quantified formulae.
In this book there are only two quantifiers: Table 6. In order to reason, we must be precise and fundamental and, especially with the quantifiers, we have to avoid falling into paradox-traps. The collection we generalise over is our logical universe, and we can make claims about the whole universe, or some sub-universe, or about particular individuals in the universe.
If I tried to be precise about such an unconstrained universe I would risk falling into the set-theoretic hole that Russell pointed out to Frege see chapter 1.
My desk has perhaps one of these properties; each of my granddaughters has at some time had all three; the logic we are studying has one of them and so, occasionally, does my wastebasket. Similarly, Good i is supposed to mean i is good.
But the claim I started with was a generalisation, a claim about every good baby and its deserts. I generalise my claim about the individual i by crossing out the name i wherever it occurs and replacing it with a variable name x.
It means precisely that every good baby in the universe deserves favour. The reverse of generalisation is specialisation. A universal claim applies to every individual, so we can replace the variable with the name of any individual, get rid of the quantifier, and there we are.
The second is just like the first, a claim about the properties of individual j. The last looks a bit odd: richard was once a baby, but favour is surely a different kind of thing. Universal quantifications are universal claims, claims about every individual in the universe. We can anonymise it, make it say that there is some exceptional individual out there, but avoid naming that individual.
Since we already accept that i is such a baby, we must accept the anonymised claim. The reverse of anonymisation is nomination.
We name a particular individual as a witness to demon- strate an existence formula. In this case I could use i, because that is the example I anonymised to begin with.
We have to find one which satisfies the claim: that is, one that makes the claim provable in constructive logic or true in classical logic. When anonymising I cross out some or all or even none of the occurrences of a chosen name i in the example above ; when nominating I must replace all the occurrences of the quantified variable x in the example above with a chosen name.
Abstracting, Good and Baby — the same formulae but with holes in place of the name i — are templates for constructing remarks about arbitrary individuals, simple predicates which describe a particular property. Everybody, at some stage of their life, thinks the world is treating them like j and giving everything to some i.
You can even play the trick of leaving no holes at all. So a predicate formula with no holes is just a very peculiar predicate: every instance is the same as every other. Such a predicate is as tricky as a cunning uncle, and just as hard to legislate against. Composite predicates can be arbitrarily complicated formulae. We might want to say EBGD icecream , a claim which will gain the approval of most babies but fewer parents. To keep things straight you have to mark the holes to show which name went where.
EBGDr i, favour expresses the rela- tionship between good baby i and favour in 6. EBGDr richard , favour reproduces the vacuous remark of 6. In particular, it works even if the composite predicate contains quantifiers. Matters get more complicated if a formula contains quantifiers which share a variable.
To avoid difficulty I shall strictly avoid such practices. But what matches Q? It all fits. Table 6. There are lots of other non-empty universes: the universe of whole numbers, the universe of rational numbers, the universe of people who have won Olympic gold. Online support material includes a detailed student solutions manual with a running commentary on all starred exercises, and a set of editable slide presentations for course lectures.
Key Features Introduces an unusually broad range of topics, allowing instructors to craft courses to meet a range of various objectives Adopts a critical attitude to certain classical doctrines, exposing students to alternative ways to answer philosophical questions about logic Carefully considers the ways natural language both resists and lends itself to formalization Makes objectual semantics for quantified logic easy, with an incremental, rule-governed approach assisted by numerous simple exercises Makes important metatheoretical results accessible to introductory students through a discursive presentation of those results and by using simple case studies.
This new and revised edition includes substantial additions which make the text even more useful to students and instructors alike. Central to these changes is an Appendix, 'How to Learn Logic', which takes the student through fourteen compact and sharply directed lessons with exercises and answers"--Google books viewed Feb.
Score: 4. Starts with simple symbols and conventions and concludes with the Boole-Schroeder and Russell-Whitehead systems. No special knowledge of mathematics necessary. Each step in the development of the formal system is clearly motivated, with the relationship of formal logic to ordinary reasoning central. Hundreds of examples of formalizing based on criteria for what counts as a good formalization.
More than exercises with answers. This system, term logic, is different in a number of ways from the standard system employed in modern logic; most striking is its greater simplicity and naturalness. Based on a radically different theory of logical syntax than the one Frege used when initiating modern mathematical logic in the 19th Century, term logic borrows insights from Aristotle's syllogistic, Scholastic logicians, Leibniz, and the 19th century British algebraists.
Term logic takes its syntax directly from natural language, construing statements as combinations of pairs of terms, where complex terms are taken to have the same syntax as statements. Whereas standard logic requires extensive 'translation' from natural language to symbolic language, term logic requires only 'transcription' into the symbolic language. Its naturalness is the result of its ability to stay close to the forms of sentences usually found in every day discourse. Written by the founders of the term logic approach, An Invitation to Formal Reasoning is a unique introduction and exploration of this new system, offering numerous exercises and examples throughout the text.
Summarising the standard system of mathematical logic to set term logic in context, and showing how the two systems compare, this book presents an alternative approach to standard modern logic for those studying formal logic, philosophy of language or computer theory. Formal Logic Author : Paul A. Translations, tables, trees, natural deduction, and simple meta-proofs are taught through over exercises. A companion website offers supplemental practice software and tutorial videos.
This book arose out of a popular course that the author has taught to all types of undergraduate students at Loyola University Chicago.
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