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PROPOSITIONAL LOGIC 2 English for Scien/sts Maria Cris/na Teodorani IMPLICATIONS • 
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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 •  BicondiPonal: compound statement formed by a combinaPon under an "and" condiPon: they are both true at the same Pme •  p ó q (p if and only if q / p IFF q) •  We essenPally use it for definiPons 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 inelasPc sca_ering is based on (à implies) two fundamental principles of modern Physics. On the one hand, special relaPvity establishes (àimplies) the equivalence between mass and energy. Each mass value corresponds to (à implies) a certain energy value. Based on that equivalence (àas a consequence of that / it implies that…), physical processes can involve (àimply) the transformaPon of massive objects into radiaPon and vice versa (póq) in agreement with (àbecause of / as an implicaPon of) the principle of energy conservaPon. IMPLICATIONS: TEXT BUILDING Quantum mechanics, on the other hand, introduces (à implies the introducPon of) an irreducibly stochasPc element to Physics. It implies that parPcles can decay into other parPcles in agreement with (as an implicaPon of / in that it implies the) energy conservaPon and the conservaPon of quantum numbers. The probabiliPes for such processes depend on (àimply being a funcPon of) coupling constants, which characterize (àimply the definiPon of) interacPon strengths between the involved parPcles, and on (àand they also imply) the spectrum of possibiliPes to realize the process in phase space, i.e. (“id est”àwhich in turn implies) the space of the involved parPcles’ locaPons and momenta. Joining special relaPvity and quantum mechanics then implies that the collision of highly accelerated, and thus (àas a result of this implicaPon) highly energePc, parPcles can turn (àimply the arrangement of) the iniPal parPcles into all possible parPcle combinaPons whose producPon is consistent with (àcoherent with the implicaPon of) the valid conservaPon laws. IMPLICATIONS: TEXT BUILDING •  The experimental physicist therefore (àas a result of all the implicaPons) can determine (àconclude he can determine) the spectrum of parPcles that can in principle exist in (are implicit in) our world by carrying out (with the implicaPon that he he has to carry out) the appropriate sca_ering experiments. (Adapted from R. Dawid, String Theory and The Scien/fic Method, CUP 2013 , pp. 75-­‐76). •  WriPng a scienPfic text means essenPally produce implicaPons. •  ImplicaPons produce other connecPves (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 •  Defini=on. Let R be a commutaPve 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. •  Proposi=on. Let R be a commutaPve ring with idenPty. Then R is a field if and only if it has no proper nontrivial ideals. •  Defini=on Let I be a proper ideal of the commutaPve 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. •  A definiPon is made of the statements aoer a logical ‘then’. ProposiPonal contradicPons may disprove it. (Adapted from h_p://www.math.niu.edu)
CONTEXT IMPLICATIONS •  The term implicature is used by Grice* to account for what a speaker can imply as disPnct 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 “convenPonal implicature” is mistaken but the u_erance need not be false.** * Grice, H.P., ‘Logic and conversaPon’ in (eds.) P. Cole & J. Morgan Syntax ans Seman/cs 3 : Speech Acts New York: Academic Press 1975 and ‘PresupposiPon and ConversaPonal Implicature’ in (ed.) P. Cole 1981. ** Adapted from G. Brown, G. Yule, Discourse Analysis Cambridge University Press 1983. CONTEXT IMPLICATIONS •  There are also conversa=onal implicatures:* •  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 descripPon of a parPcular state of affairs or a request of help. •  Implicatures are pragmaPc aspects of meaning •  They not only depend on the convenPonal meaning but also on the shared context in which they are produced.* (Adapted from Brown, Yule, 1983, quoted)
IMPLICATIONS AND RELATIVES •  RelaPve clauses are introduced by relaPve pronouns (that, which, whose, where, who, etc.) •  They imply something or are implicit into something else •  They supply the omi_ed implicatures •  They imply the subject (you cannot omit them) or the object (you can omit them) •  They add extra implicaPons (and so informaPon) •  They can be turned into or used as pàq, póq statements •  They also imply a “pragmaPc” extra knowledge TEXT BUILDING: IMPLICATIONS AND RELATIVES •  The intrinsic characterisPcs of nanomaterials imply a mulP-­‐variable complexity, which affects (àwhich in turn implies the fact that it affects) their toxicological potenPal. In many aspects, this complexity is related to (àimplicit to) their colloidal nature that disPnguishes them radically from (àthat is implied in making them set apart from) dissolved chemicals—
whereas dissolved chemicals undergo chemical speciaPon, colloids are affected by both chemical and physical speciaPon (àthe reason lies in/implies the fact that). This fact has specific implica=ons to their toxicological examinaPon and their risk assessment, especially in aquaPc systems. TEXT BUILDING: IMPLICATIONS AND RELATIVES •  Toxicological properPes 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 (nanoparPcle-­‐specific effects)*, methods to address and understand the issue of agglomeraPon and physical speciaPon and their implica=ons for (à which are used in tesPng) toxicity, and methods to untangle indirect effects, such as (àwhose one among them is / implies the use of ) theshading of algae by parPcle suspensions from parPcle-­‐
specific and ion-­‐mediated effects. An interesPng approach included a change in perspecPve, studying the implica=ons of (à which implies the study of ) biological traits of the organisms (e.g., size, aspect raPo, biovolume, etc.) in the toxicity of nanomaterials as a tool to perform effecPve interspecies extrapolaPons of nanomaterial bioacPvity. * the brackests here stand for “which are based on the nanoparPcle-­‐
specific effects / which imply the use of…” KERNEL/NON-­‐KERNEL CLAUSES •  Kernel clauses are defining relaPve clauses adding funcPonal informaPon •  If the relaPve pronoun is followed by a verb, then the relaPve pronoun is a subject pronoun and cannot be omi_ed. •  If the relaPve pronoun is followed by a noun or pronoun , then the relaPve pronoun is an object pronoun and can be omi_ed. •  Non-­‐defining relaPve clauses (also called non-­‐
kernel clauses) give extra informaPon MORE ON KERNEL • 
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In linear algebra and funcPonal 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 equaPon Ax = 0 is equivalent to a homogeneous system of linear equaPons. From this viewpoint, the null space of A is the same as the soluPon set to the corresponding homogeneous system of equaPons. Let C∞(R) be the vector space of all infinitely differenPable funcPons R → R, and let D: C∞(R) → C∞(R) be the differenPaPon operator D(f)=df/dx, then the kernel of D consists of all funcPons in C∞(R) whose derivaPves are zero, i.e. the set of all constant funcPons. (Adapted from h_p://en.wikipedia.org/wiki/Kernel_(linear_algebra) ) A kernel statement being a defining asserPon obviously deals with atomic, or elementary, proposiPons –the term kernel also standing for nucleus An atomic proposiPon produces sense as well as a soluPon set of equaPons MORE ON KERNEL •  Programming languages work thanks to a series of kernel statements that consPtute their peculiar syntax •  SEMANTICS UNDERLYING THE KERNEL STETEMENT: sequencing or condiPoning, looping, declaring, wriPng, reading, etc. •  KERNEL: THE “VERBAL SHELL” (OR SYNTAX): while(){} ; if () {} else {}; do () {}; write () {} , etc. •  Machine code à formal codeànatural code MORE ON KERNEL: AN EXAMPLE • 
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program projecPle 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 Pme t =’, t, ‘x=’, x, ‘y =’, y end if end do end Hey dear PC, 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 codeànatural code • 
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TEXT BUILDING: KERNEL/NON-­‐KERNEL CLAUSES • 
Photosynthesis, which is the process of converPng light energy into chemical energy and storing it in the bonds of sugar, occurs in plants and some algae (Kingdom ProPsta), 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 li_le to none occurs in stems, etc. The upper and lower epidermal cells, which serve primarily as protecPon 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 transportaPon system, moving water and nutrients around the plant as needed. The mesophyll cells have chloroplasts and this is where photosynthesis 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 h_p://biology.clc.uc.edu) TEXT BUILDING: KERNEL /NON-­‐KERNEL CLAUSES •  The overall chemical reacPon 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 deforestaPon. There are two parts to photosynthesis, which are called the light reacPon and the dark reacPon •  The light reacPon, which happens in the thylakoid membrane, converts light energy into chemical energy. This chemical reacPon 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 reacPon. 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 h_p://biology.clc.uc.edu) TEXT BUILDING: KERNEL /NON-­‐KERNEL CLAUSES • 
The energy harvested via the light reacPon 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 nucleoPde 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 reacPon takes place in the stroma within the chloroplast, and converts CO2 to sugar. This reacPon does not directly need light in order to occur, but it does need the products of the light reacPon (ATP and another chemical called NADPH). The dark reacPon involves a cycle called the Calvin cycle in which CO2 and energy from ATP are used to form sugar. Actually, noPce 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 Pny bit of CO2 leo, 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 modificaPon 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 be_er, 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 h_p://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 “parPcle-­‐specific effects.”, Therefore (à because of these implicaPons we can conclude that) the contribuPon of free ion must always be under control. However, parPcle-­‐specific effects were also evident even when the contribuPon of free ion was clearly determined (à new implicaPons). When speaking of parPcle-­‐specific effects, internalizaPon of nanoparPcles seems not always required to induce toxicity (à does not seem to always imply toxicity inducPon). Therefore, it raises the ques=on (à the conclusion raises a new hypothesis/implicaPon): Is internalizaPon a prerequisite for parPcle-­‐specific toxicity or is just surface adsorpPon enough? A step forward from observa=on to understanding (à a new implicaPon) is required in this area. In addiPon, evidence presented showed that media composiPon and coaPngs may influence or even totally change (àmay imply a total changement) the intrinsic toxicity of the nanomaterials. Therefore, further understanding of physical interacPons and speciaPon of nanomaterials at the bio-­‐interfaces is necessary, which may shed light on (àimplying a be_er understanding of) the definiPon of the “correct” condiPons to properly perform and understand results from exposure experiments. (Adapted from Biophysical InteracPons at the Bio-­‐nano Interface: Relevance for AquaPc NanotoxicologyIsmael Rodea-­‐
Palomares, Universidad de Alcalá, h_p://globe.setac.org/2014/june/basel-­‐aquaPc-­‐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 connecPves, implicaPons, if clauses. Then contextualize your discourse using relaPve clauses. Choose as atomic proposiPons something dealing with the scienPfic subject relevant to your area. •  [p ⋀ (pàq)] à q •  P à (p ⋁ q) •  p ⋀ (⌐p ⋀ q) EXAMPLE (Compounding statements) •  ⌐ (p ⋁ ⌐q) à ⌐p •  If I neither get the spring Δl (dilataPon) 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, oxidaPon, acetyl CoA, compound, enzymes, proteins, oxaloacePc 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 spli_ed. •  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 informaPon and to supply implicaPons and omi_ed implicatures •  HINT: create logical statements labelling pn and qn the elementary statements, then connect them using the given connecPves. You should create a table before wriPng the text. TEXT BUILDING: EXERCISE • 
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p1= start with glucose molecule; q1= glycolysis splits glucose in a half; q2= end up with pyruvates If p1 then q1 (implicaPon) and then q2 (logical connecPve with sequencing) p1 à q1 à q2 A non-­‐kernel relaPve clause (introduced by which/who) essenPally focuses on further implicaPons and informaPon: it needs be used in tutorials; somePmes in papers or theses; it can be omi_ed in conversaPon 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 spli_ed 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 consequenPal point. NoPce that a further (untold) implicaPon sets logically forth by the 2 relaPve clauses, for glycolysis is the metabolic pathway converPng glucose into pyruvates. RelaPve clause 1 and 3, therefore, imply a possible relaPve 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 essenPally get spli_ed 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’ oxidaPon . It essenPally 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 consPtuents in order for them to react in the right way. In this way the acetyl CoA merges with oxaloacePc 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 oxaloacePc 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 iniPal payoff, then we end up with 4 ATPs, 10 NADHs and 2 FADH2, the la_er being the inputs in the ETC (electrons transport chain), 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 conjuncPons, disjuncPons, negaPons, implicaPons with respect to the basic word order •  Define arguments and add extra informaPon using appropriate relaPve clauses •  Further compound the obtained sentences into a coherent and cohesive text