PROPOSITIONAL LOGIC 2

English for ScientistsMaria Cristina Teodorani

IMPLICATIONS

• p: hypothesis ; q: conclusion• If p, then/therefore q• p implies q (p à q) • Reversed if both denied (⌐q à ⌐p)• X will go to lab if Y is not there• If Y is there, then X won’t go to lab• Y is not at the lab, therefore Y is there

IMPLICATIONS• Biconditional: compound statement formed by a

combination under an "and" condition: they are both true at the same time

• p ó q (p if and only if q / p IFF q)• We essentially use it for definitions and vice-versa

statements• A triangle is right if its angle measures 90°• A triangle is right if and only if one of its angles

measures 90°• A triangle is equilateral if and only if (iff) its angles all

measure 60°.

IMPLICATIONS: TEXT BUILDING

• Conceptually, deep inelastic scattering is based on (àimplies) two fundamental principles of modern Physics. On the one hand, special relativity establishes(àimplies) the equivalence between mass and energy. Each mass value corresponds to (à implies) a certain energy value. Based on that equivalence (àas a consequence of that / it implies that…), physical processes can involve (àimply) the transformation of massive objects into radiation and vice versa (póq) in agreement with (àbecause of / as an implication of) the principle of energy conservation.

IMPLICATIONS: TEXT BUILDINGQuantum mechanics, on the other hand, introduces (à implies the introduction of) an irreducibly stochastic element to Physics. It impliesthat particles can decay into other particles in agreement with (as an implication of / in that it implies the) energy conservation and the conservation of quantum numbers. The probabilities for such processes depend on (àimply being a function of) coupling constants, which characterize (àimply the definition of) interaction strengths between the involved particles, and on (àand they also imply) the spectrum of possibilities to realize the process in phase space, i.e. (“id est”àwhich in turn implies) the space of the involved particles’ locations and momenta. Joining special relativity and quantum mechanics then implies that the collision of highly accelerated, and thus (àas a result of this implication) highly energetic, particles can turn (àimply the arrangement of) the initial particles into all possible particle combinations whose production is consistent with (àcoherent with the implication of) the valid conservation laws.

IMPLICATIONS: TEXT BUILDING

• The experimental physicist therefore (àas a result of all the implications) can determine (àconclude he can determine) the spectrum of particles that can in principle exist in (are implicit in) our world by carrying out (with the implication that he he has to carry out) the appropriate scattering experiments. (Adapted from R. Dawid, String Theory and The Scientific Method, CUP 2013 , pp. 75-76).

• Writing a scientific text means essentially produce implications.

• Implications produce other connectives (as well as suitable verbs in the context) that make the text coherent and cohesive (in bold).

IMPLICATIONS: BICONDITIONALS

• “A triangle is equilateral if and only if (iff) its angles all measure 60°”.

• means both "If a triangle is equilateral then its angles all measure 60°" and "If all the angles of a triangle measure 60° then the triangle is equilateral".

• A triangle is equilateral ó its angles all measure 60°

• p ó q

TEXT BUILDING• Definition. Let R be a commutative ring. A nonempty subset I of R is

called an ideal of R if (i) a ± b ∈ I for all a,b ∈ I and(ii) ra ∈ I, for all a ∈ I and r ∈ R.

• Proposition. Let R be a commutative ring with identity. Then R is a field if and only if it has no proper nontrivial ideals.

• Definition Let I be a proper ideal of the commutative ring R. Then I is said to be a prime ideal of R if for all a,b ∈ R it is true that ab ∈ I implies a ∈ I or b ∈ I. The ideal I is said to be a maximal ideal of R if for all ideals J of R such that I ⊆ J ⊆ R, either J = I or J = R. (Adapted from http://www.math.niu.edu)

• A definition is made of the statements after a logical ‘then’. Propositional contradictions may disprove it.

CONTEXT IMPLICATIONS

• The term implicature is used by Grice* to account for what a speaker can imply as distinct from what he literally says.

• He is an Englishman, he is, therefore brave• If it turns out that he is English and not brave,

the “conventional implicature” is mistaken but the utterance need not be false.**

* Grice, H.P., ‘Logic and conversation’ in (eds.) P. Cole & J. Morgan Syntax ans Semantics 3 : Speech Acts New York: Academic Press 1975 and ‘Presupposition and Conversational Implicature’ in (ed.) P. Cole 1981.

** Adapted from G. Brown, G. Yule, Discourse Analysis Cambridge University Press 1983.

CONTEXT IMPLICATIONS• There are also conversational implicatures:* (Adapted from Brown, Yule, 1983, quoted)

• A: I am out of petrol.B: There is a garage round the corner.

• The implicature is that the garage is not only round the corner, but also will be open and selling petrol.

• To get the implicature we have to know certain facts about the world, that garages sell petrol, and that “round the corner” is not a great distance away; we have to interpret A’s remark as a description of a particular state of affairs or a request of help.

• Implicatures are pragmatic aspects of meaning • They not only depend on the conventional meaning but

also on the shared context in which they are produced.*

IMPLICATIONS AND RELATIVES• Relative clauses are introduced by relative pronouns

(that, which, whose, where, who, etc.)• They imply something or are implicit into something

else• They supply the omitted implicatures• They imply the subject (you cannot omit them) or the

object (you can omit them)• They add extra implications (and so information) • They can be turned into or used as pàq, póq

statements• They also imply a “pragmatic” extra knowledge

TEXT BUILDING: IMPLICATIONS AND RELATIVES

• The intrinsic characteristics of nanomaterials imply a multi-variable complexity, which affects (àwhich in turn implies the fact that it affects) their toxicological potential. In many aspects, this complexity is related to (àimplicit to) their colloidal nature that distinguishes them radically from (àthat is implied in making them set apart from) dissolved chemicals—whereasdissolved chemicals undergo chemical speciation, colloids are affected by both chemical and physical speciation (àthe reason lies in/implies the fact that). This fact has specific implications to their toxicological examination and their risk assessment, especially in aquatic systems.

TEXT BUILDING: IMPLICATIONS AND RELATIVES

• Toxicological properties of nanomaterials are studied by applying different novel approaches, including (à which include / which imply the inclusion of ) methods to untangle chemical effects mediated by ions and physical effects (nanoparticle-specific effects)*, methods to address and understand the issue of agglomeration and physical speciation and their implications for (à which are used in testing) toxicity, and methods to untangle indirect effects, such as (àwhose one among them is / implies the use of ) theshading of algae by particle suspensions from particle-specific and ion-mediated effects. An interesting approach included a change in perspective, studying the implications of (à which implies the study of ) biological traits of the organisms (e.g., size, aspect ratio, biovolume, etc.) in the toxicity of nanomaterials as a tool to perform effective interspecies extrapolations of nanomaterial bioactivity.* the brackests here stand for “which are based on the nanoparticle-specific effects / which imply the use of…”

KERNEL/NON-KERNEL CLAUSES

• Kernel clauses are defining relative clauses adding functional information

• If the relative pronoun is followed by a verb, then the relative pronoun is a subject pronoun and cannot be omitted.

• If the relative pronoun is followed by a noun or pronoun , then the relative pronoun is an object pronoun and can be omitted.

• Non-defining relative clauses (also called non-kernel clauses) give extra information

MORE ON KERNEL • In linear algebra and functional analysis, the kernel (also null space or nullspace) of

a linear map L : V → W between two vector spaces or two modules V and W is the set of all elements v of V for which L(v) = 0, where 0 denotes the zero vector in W

• The kernel of an m × n matrix A with coefficients in a field K (typically the field of the real numbers or of the complex numbers) is the set of xs such that Ax = 0, where 0 denotes the zero vector with m components. The matrix equation Ax = 0 is equivalent to a homogeneous system of linear equations. From this viewpoint, the null space of A is the same as the solution set to the corresponding homogeneous system of equations.

• Let C∞(R) be the vector space of all infinitely differentiable functions R → R, and let D: C∞(R) → C∞(R) be the differentiation operator D(f)=df/dx, then the kernel of Dconsists of all functions in C∞(R) whose derivatives are zero, i.e. the set of all constant functions.

• A kernel statement being a defining assertion obviously deals with atomic, or elementary, propositions –the term kernel also standing for nucleus

• An atomic proposition produces sense as well as a solution sets of equations

MORE ON KERNEL

• Programming languages work thanks to a series of kernel statements that constitute their peculiar syntax

• SEMANTICS UNDERLYING THE KERNEL STETEMENT: sequencing or conditioning, looping, declaring, writing, reading, etc.

• KERNEL: THE “VERBAL SHELL” (OR SYNTAX): while(){} ; if () {} else {}; do () {}; write () {} , etc.

• Machine code à formal codeànatural code

MORE ON KERNEL: AN EXAMPLE

• program projectile• implicit none • real:: g, t, vx, vy, x, y • x=0 y=0 t=0 g=9.8 vx=5 vy=5• do while (y>=0) • t=t+0.01 • x = vx*t • y = Vx * t-1./2. *g*t**2

MORE ON KERNEL: AN EXAMPLE• if (y >= 0) then • write(*,*) ‘at the time t =’, t, ‘x=’, x, ‘y =’, y • end if • end do • end• Hey dear PCC, it’s me who says what to do, I’m giving

you a bunch of variables, then while y≥0 do your Maths, then if (and only if) y is either 0 or greater than 0 then (it implies the fact that you have to) write the result.

• Machine code à formal codeànatural code

TEXT BUILDING• Photosynthesis, which is the process of converting light energy into chemical energy and storing it

in the bonds of sugar, occurs in plants and some algae (Kingdom Protista), which need only light energy, CO2, and H2O to make sugar. The process of photosynthesis takes place in the chloroplasts that use chlorophyll, the green pigment involved in photosynthesis. A plant leaf, whose parts include the upper and lower epidermis, the mesophyll, the vascular bundle(s) (veins), and the stomates, is the place where photosynthesis typically happens, while little to none occurs in stems, etc. The upper and lower epidermal cells, which serve primarily as protection for the rest of the leaf, do not have chloroplasts, thus photosynthesis does not occur there. The stomates are holes which occur primarily in the lower epidermis and are for air exchange: they let CO2 in and O2 out. The vascular bundles or veins in a leaf are part of the plant’s transportation system, moving water and nutrients around the plant as needed. The mesophyll cells have chloroplasts and this is wherephotosynthesis occurs. The parts of a chloroplast include the outer and inner membranes, intermembrane space, stroma, and thylakoids stacked in grana. The chlorophyll is built into the membranes of the thylakoids. Chlorophyll looks green because it absorbs red and blue light, making these colors unavailable to be seen by our eyes. It is the green light which is NOT absorbed that finally reaches our eyes, making chlorophyll appear green. However, it is the energy from the absorbed red and blue light that is, thereby, able to be used to do photosynthesis. The green light (that) we can see is not and cannot be absorbed by the plant, and thus cannot be used to do photosynthesis. (Adapted from http://biology.clc.uc.edu)

TEXT BUILDING• The overall chemical reaction involved in photosynthesis is:

6CO2 + 6H2O (+ light energy) → C6H12O6 + 6O2.This is the source of the O2 we breathe, whose amount is a significant factor in the concerns about deforestation. There are two parts to photosynthesis, which are called the light reaction and the dark reaction

• The light reaction, which happens in the thylakoid membrane, converts light energy into chemical energy. This chemical reaction must, therefore, take place in the light. Any biologists who is specialized in this sector can tell us that chlorophyll and several other pigments such as beta-carotene are organized in clusters in the thylakoid membrane and are involved in the light reaction. Each of these differently-colored pigments can absorb a slightly different color of light and pass its energy to the central chlorphyll molecule to do photosynthesis. The central part of the chemical structure of a chlorophyll molecule is a porphyrin ring, which consists of several fused rings of carbon and nitrogen with a magnesium ion in the center. (Adapted from http://biology.clc.uc.edu)

TEXT BUILDING• The energy harvested via the light reaction is stored by forming a chemical called ATP (adenosine

triphosphate), which is a compound used by cells for energy storage. This chemical is made of the nucleotide adenine bonded to a ribose sugar, which in turn is bonded to three phosphate groups. This molecule (that) we’re considering is very similar to the building blocks for our DNA. The dark reaction takes place in the stroma within the chloroplast, and converts CO2 to sugar. This reaction does not directly need light in order to occur, but it does need the products of the light reaction (ATP and another chemical called NADPH). The dark reaction involves a cycle called the Calvin cycle in which CO2 and energy from ATP are used to form sugar. Actually, notice that the first product of photosynthesis is a three-carbon compound called glyceraldehyde 3-phosphate. Almost immediately, two of these join to form a glucose molecule. Most plants put CO2 directly into the Calvin cycle. Thus the first stable organic compound formed is the glyceraldehyde 3-phosphate. Since that molecule contains three carbon atoms, these plants are called C3 plants. For all plants, hot summer weather increases the amount of water that evaporates from the plant. Plants lessen the amount of water that evaporates by keeping their stomates closed during hot, dry weather. Unfortunately, this means (that) once the CO2 in their leaves reaches a low level, they must stop doing photosynthesis. Even if there is a tiny bit of CO2 left, the enzymes used to grab it and put it into the Calvin cycle just do not have enough CO2 to use. Typically the grass in our yards just turns brown and goes dormant. Some plants like crabgrass, corn, and sugar cane have a special modification to conserve water. These plants capture CO2 in a different way: they do an extra step first, before doing the Calvin cycle. These plants have a special enzyme that can work better, even at very low CO2 levels, to grab CO2 and turn it first into oxaloacetate, which contains four carbons. Thus, these plants are called C4 plants. The oxaloacetate, which the CO2 is released from, is vital: it is this CO2 that is put into the Calvin cycle. This is why crabgrass can stay green and keep growing when all the rest of your grass is dried up and brown. (Adapted from http://biology.clc.uc.edu)

TEXT BUILDING: IMPLICATIONS AND RELATIVES

• In metal-based nanomaterials, free ion can be a determinant in the toxicity and may be behind some apparently “particle-specific effects.”, Therefore (à because of these implications we can conclude that) the contribution of free ion must always be under control. However, particle-specific effects were also evident even when the contribution of free ion was clearly determined (à new implications). When speaking of particle-specific effects, internalization of nanoparticles seems not always required to induce toxicity (à does not seem to always imply toxicity induction). Therefore, it raises the question (à the conclusion raises a new hypothesis/implication): Is internalization a prerequisite for particle-specific toxicity or is just surface adsorption enough? A step forward from observation to understanding (à a new implication) is required in this area. In addition, evidence presented showed that media composition and coatings may influence or even totally change (àmay imply a total changement) the intrinsic toxicity of the nanomaterials. Therefore, further understanding of physical interactions and speciation of nanomaterials at the bio-interfaces is necessary, which may shed light on (àimplying a better understanding of) the definition of the “correct” conditions to properly perform and understand results from exposure experiments. (Adapted from Biophysical Interactions at the Bio-nano Interface: Relevance for Aquatic NanotoxicologyIsmael Rodea-Palomares, Universidad de Alcalá, http://globe.setac.org/2014/june/basel-aquatic-nanotechnology.html)

EXERCISES• Build up suitable statements and their truth table

according to the compounds below, then connect them into a coherent text using basic strings, logical connectives, implications, if clauses. Then contextualize your discourse using relative clauses. Choose as atomic propositions something dealing with the scientific subject relevant to your area.

• [p ⋀ (pàq)] à q• P à (p ⋁ q)• p ⋀ (⌐p ⋀ q)

EXAMPLE(Compounding statements)

• ⌐ (p ⋁ ⌐q) à ⌐p• If I neither get the spring Δl (dilatation) nor

the mass then I don’t get the Δl• p=I get the Δl q=I don’t get the mass

p q ⌐q p ⋁ ⌐q ⌐ (p ⋁⌐q) ⌐p ⌐ (p ⋁ ⌐q) à ⌐p

0 0 1 1 0 1 1

0 1 0 0 1 1 1

1 0 1 1 0 0 1

1 1 0 1 0 0 1

TEXT BUILDING: EXERCISE• TITLE: The citric acid cycle (or Krebs’s cycle)• NOUNS: molecule, glucose, glycolysis, pyruvates, carbon, oxygen, carbon

dioxide, ATP, NADH, NAD+, cell, cytoplasm, acid, cycle, membrane, mitochondria, cristae, matrix, oxidation, acetyl CoA, compound, enzymes, proteins, oxaloacetic acid, ETC, ADP, ATP, FAD, FADH2.

• VERBS: have, produce, oxidize, get oxidized, reduce, get reduced, be, consider, clear off, get back, merge, react, bring, canalize, jump, call, know, take place, occur, happen, start off, end up, split, get splitted.

• CONNECTIVES if, if …then, so, so that, since (implied that/due to the fact that), and, therefore.

• RELATIVES: use which and where to give extra information and to supply implications and omitted implicatures

• HINT: create logical statements labelling pn and qn the elementary statements, then connect them using the given connectives. You should create a table before writing the text.

TEXT BUILDING: EXERCISE• p1= start with glucose molecule; q1= glycolysis splits glucose in a half; q2= end up

with pyruvates• If p1 then q1 (implication) and then q2 (logical connective with sequencing)• p1à q1à q2• A non-kernel relative clause (introduced by which/who) essentially focuses on

further implications and information: it needs be used in tutorials; sometimes in papers or theses; it can be omitted in conversation where it works as an (untold) implicature

• If we start with a glucose molecule, (which is a 6-carbon molecule,)1 then it will get splitted in half by glycolysis, (which is…)2, so that we end up with 2 pyruvates, (which are 3-carbon molecules)3.

• In this case ‘so that’ means ‘and then’, reinforcing the consequential point.• Notice that a further (untold) implication sets logically forth by the 2 relative

clauses, for glycolysis is the metabolic pathway converting glucose into pyruvates. Relative clause 1 and 3, therefore, imply a possible relative clause 3, that would have been redundant: it would have worked as a tautology.

TEXT BUILDING: a possible layout• If we started off with a glucose molecule (C6H12O6), which is a 6-carbon molecule, then we know it

would essentially get splitted in half by glycolysis, so that (=therefore)we would end up with 2 pyruvates (CH3COCOO−), which are 3-carbon molecules. This actually happens in glycolysis, in the presence or in the absence of oxygen, so that we have a net payoff of 2 ATPs and 2 NADHs, together with carbon dioxide emission. Glycolysis occurs in the cytoplasm of the cell. The citric acid cycle (or Krebs’s cycle) takes place in the inner membrane, (which is) the inner space of the mitochondria, (which is) a compartment inside the cristae (that separates it from the outer space), which is known as matrix. The pyruvates are not quite ready for the Krebs’s cycle, since they are not oxidized yet . The preparatory step for the Krebs’s cycle is indeed the pyruvates’ oxidation . It essentially clears one of its carbons off the pyruvates, so that we end up with a 2-carbon compound called acetyl CoA and also reduces some NAD+ to NADHs. Once we have the acetyl CoA we are ready to jump into the citric acid cycle. The acetyl CoA is canalized by enzymes, which are proteins that bring together the constituents in order for them to react in the right way. In this way the acetyl CoA merges with oxaloacetic acid, which is a 4-carbon molecule, forming citric acid, which is again a 6-carbon molecule. Then the citric acid gets oxidized, over a bunch of steps, to get back to oxaloacetic acid, so that two carbons are once more cleared off forming carbon dioxide, while some NAD+ gets reduced to NADHs, some ADP turns into ATP and some FAD gets reduced into FADH2. We have exactly 3 NADHs x 2 (there are 2 pyruvates) = 6 NADHs (plus 2 from the preparatory state), 2 ATPs and 2 FADH2. If we consider the glycolysis’s initial payoff, then we end up with 4 ATPs, 10 NADHs and 2 FADH2, the latter being the inputs in the ETC (electrons transportchain), where they get oxidized. Every NADH in this chain produces 3 ATPs, so the 10 NADHs will produce 30 ATPs in the ETC, while the 2 FADH2 will produce 4 ATPs in the ETC. Therefore we end up with 38 ATPs.

EXERCISES

• Choose any subjects of your area• Create any p, q atomic statements• Compound them using conjunctions,

disjunctions, negations, implications with respect to the basic string

• Further compound them into a coherent and cohesive text