A note on linking ideas in academic writing

Flow of information

Sentences and ideas should be ordered and linked to present information in a reasonable manner. Here reasonable means the reader can easily find what he expects to see as he reads. The following example shows how sentences can be re-ordered to improve readability. It is extracted from the first paragraph of an essay with title "The main sources of diversity in work values":

Original		Revised
The importance of education should be pointed out as one of the crucial sources of diversity in attitudes to work. The rising educational level has been perceived as one of the most significant investments in the general quality of life. Tertiary education in particular plays a very special role in this respect.		One of the crucial sources of diversity in attitudes to work is education. The rising educational level has been perceived as one of the most significant investments in the general quality of life. In this respect, tertiary education in particular plays a very special role.

Comments. It is preferred that the first sentence connects directly to the title, as that is what a reader expects when he starts reading the paragraph. The importance of education is redundant here because education is already pointed out as a crucial source. In this respect should be put at the beginning of a sentence since it serves as a linking phrase.

Text cohesion

Cohesion refers to the way in which elements of language are linked together to make a text. The sequence of given information followed by new information can contribute to the cohesion of a text because the new information of one sentence often becomes the given information of the next sentence. The following is an extract from a thesis about the construction of stance in academic writing. Words are colourised to show how old information is rephrased or repeated to improve smooth transitions to new information.

Passage		Comments
For the thesis writer, the construction of stance takes a distinctive form which derives from the nature of the thesis itself.		Clarify the subject of this passage and identify the expected target readers—the thesis writers.
As noted above, writers have to convince the supervisors and examiners to accept their research, which means that the thesis is inherently persuasive.		Clarify the goal of thesis writers to bring up the key characteristic of a thesis: being persuasive.
However, acceptance depends upon satisfying two quite distinct requirements.		The subject acceptance (of persuasion) follows logically from the key old info "being persuasive".
The first is that the thesis should make an original contribution to knowledge.		Introduce a claim.
In creating new knowledge and conveying it to other members of the discipline, the writer’s stance is that of a professional.		Rephrase the claim and reach a partial conclusion.
The second requirement is that the thesis must show mastery of the field.		Make another claim.
The display of established knowledge is required because the work will be assessed and thus the writer must also take a stance as a candidate.		Rephrase the claim and make another partial conclusion.
However combining the roles of candidate and professional can lead to considerable conflict and uncertainty.		Summarise the partial conclusions and start a new discussion.

As this example demonstrates, the new information of one sentence often becomes the given information of the next sentence. This sequence of given information → new information links the ideas of the text together and creates cohesion.

Devices to link ideas

We present more examples showing how ideas can be linked between sentences grammatically. In the second sentences of each example, the given information are referred using three kinds of grammar patterns. New ideas can be introduced by adding extra information to the given information.

Examples		Comments
Contrast from differently doped regions immediately appears when the in-lens detector is switched on. This huge effect is believed to hold some important implications.		A comment (ie. the effect is huge) is implicitly introduced when the old information, which is a neutral description of a fact, is referred. It is preferable to separate description and evaluation clearly in academic writing.
It will be seen in the sample that the simulation is able to adapt again. The ability of this model to emulate changes in the characteristics of different sample types is cited as strong support for its validity.		The old information is rephrased by changing part of speech and inverting subject and object. By such rephrasing we avoid using "..., which ..." repeatedly for further explanation/elaboration.
In this analysis, "norm" refers to a prescribed or proscribed standard of behaviour. Such a definition is therefore broader than rules alone.		Using such a instead of this means that the author is going to generalise. In this case, the author intends to discuss a class of definitions instead of the definition of "norm".
The fact that the same British forces had to cover various commitments at once did stretch them to the limit, and perhaps beyond. The problem of over-stretch had emerged in the late 1950s.		The use of problem and over indicates the author's position in the following discussion on the aforementioned fact. Description and evaluation is cleanly separated.
For example, it would be interesting to find out if the active region contains more crystalline polymer as a result of operation. Such an investigation would involve using staining techniques.		The noun investigation is used to encapsulate the text to find out .... The use of such a indicates that the following discussion would not be restricted to that particular example.

Remark. The main device for linking ideas in these examples is called nominalisation, which is the use of a noun to express an idea that is previously expressed in the form of a clause or a sentence. Nominalisation is very useful in the construction of argument, as we can present what we already said in a shortened form and then move on to the new point following from that information. Here is an example:

I explained the results as such and such. This explanation is in agreement with earlier work.

We can add more details to the noun using prepositional phrases and clauses:

What I have sought to disprove is that there is an incompatibility between a "bad" nationalism and a "good" patriotism. The assertion that nationalism and patriotism are incompatible causes complete confusion when historians, even great ones, are confronted by past thinkers.

A note on useful decidable logics & theories

Note that this post discusses decidable fragments that allows at least some degree of quantifications, not decidable theories of ground formulas used in SMT solvers.

Decidable fragments of first-order logic

Monadic first-order logic. Also known as the relational monadic fragment (RMF) or the Löwenheim class [2], this fragment is a classical example of decidable first-order logic without equality. The RMF consists of the first-order formulas where all relation symbols are unary:$$\phi ::= R(x) \mid \neg \phi \mid \phi \vee \phi \mid \phi \wedge \phi \mid \exists x.\phi \mid \forall x.\phi.$$Note that equality is not allowed in this logic and there are no constant symbols. Satisfiability of the logic is first proved to be NE-complete [2], where NE = $\bigcup_c NTIME(2^{cn})$. Later it is shown that NE = NEXPTIME [1], and thus the problem is NEXPTIME-complete. The Löb-Gurevich class (cf. [8]) is a decidable extension of the RMF where unary function symbols are also allowed. Both of the classes enjoy the finite model property, cf [8].

Effectively propositional logic (EPR). [5] Also known as the Bernays-Schönfinkel class, this fragment consists of first-order formulas such that:
1. The vocabulary is restricted to constant and relation symbols.
2. The quantifier prefix is restricted to $\exists^*\forall^*$ for formulas in prenex normal form.
Condition 2 implies that EPR is not closed under negation. EPR enjoys the finite model property, meaning that a satisfiable formula is guaranteed to have a finite model. The satisfiability problem of EPR is NEXPTIME-complete.
We can extend EPR to allow quantifier alternation and function symbols while maintaining the same properties, as long as the formula in the extended logic is "stratified" [11]. More precisely, we define the quantifier alternation graph $G(\phi)$ for a formula $\phi$ as a directed graph where the set of vertices is the set of sorts, and for each function symbol $f : (s_1,\ldots, s_n) \to s$ in the skolem normal form of $\phi$, there is an edge $s_i \to s$ for $1 \le i \le n$. $\phi$ is called stratified if $G(\phi)$ does not contain a cycle through the sort of some universally quantified variable. For example, $\phi := \forall x. \exists y. f(x) = y$ is skolemized to $\forall x. f(x) = g(y)$ with $f,g : A \to B$. Hence $G(\phi)$ is acyclic and thus $\phi$ is in the decidable fragment.
Stratified formulas are also decidable, for there is no way to generate an infinite sequence of instantiation terms. By contrast, $\psi := \forall x. \exists y. f(y) = x$ is skolemized to $\forall x. f(g(x)) = x$ where $f: A \to B$ and $g: B \to A$. Hence $G(\psi)$ contains a cycle through the sort of $x$, and thus $\psi$ is not in the decidable fragment.
Formulas in the extended EPR can be effectively translated into propositional logic formulas through a instantiation process over a finite Henken domain, see e.g., [12,13]. Extended EPR is supported by first-order logic provers such as iProver (which is known as the most efficient solver for EPR), and by modern SMT solvers such as Z3.

Presburger arithmetic. The first-order theory of structure $(\mathbb N, +)$ can be decided by encoding $n\in \mathbb N$ in binary (LSB first), such that the atomic ternary-relation $+$ corresponds to a regular relation $+_2$ over this coding and $\mathbb N$ corresponds to the $\omega$-regular language $(0+1)^*0^\omega$. Buchi has shown in 1962 that given any formula $\phi(x_1,...,x_n)$ over ${\rm FO}(\mathbb N, +)$, one can effectively compute an $\omega$-automaton $A(x_1,...,x_n)$ that defines the same relation modulo convolution $\otimes$, namely, $\phi(x_1,...,x_n)$ holds iff $A$ accepts the $\omega$-word $x_1 \otimes \cdots \otimes x_n.$ Satisfiability problem for ${\rm FO}(\mathbb N, +)$ thus reduces to the emptiness problem for $\omega$-automata and is thus decidable. See [9] for a review of the algorithmic and computational complexity aspects of PA. Also see a recent historical review in this paper and its presentation.

A list of decidable theories can be found on Wikipedia.

References

1. Structural properties of complete problems for exponential time. 1997.
2. Complexity results for classes of quantification formulas. 2001.
3. Verification of randomised security protocols.
4. Decidability of second-order theories and automata on infinite trees. 1969.
5. Complexity results for classes of quantificational formulas. 1980.
6. The theory of ends, pushdown automata, and second-order logic. 1985.
7. Bisimulation through Probabilistic Testing. 1991.
8. The Classical Decision Problem. 2001.
9. A Survival Guide to Presburger Arithmetic. 2018.
10. Automata: From Logics to Algorithms. 2007.
11. Complete instantiation for quantified formulas in satisfiabiliby modulo theories. 2009.
12. Deciding effectively propositional logic using DPLL and substitution sets. 2008.
13. Proof systems for effectively propositional logic. 2008.