Download PROPOSITIONAL LOGIC 2

PROPOSITIONAL LOGIC 2
English for Scientists
Maria Cristina 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
• 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 (à
implies) 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 BUILDING
Quantum mechanics, on the other hand, introduces (à implies the
introduction of) an irreducibly stochastic element to Physics. It implies
that 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°
• póq
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.
• A definition is made of the statements after a logical ‘then’.
Propositional contradictions may disprove it.
(Adapted from http://www.math.niu.edu)
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:*
• 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.*
(Adapted from Brown, Yule, 1983, quoted)
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—whereas
dissolved 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
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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 particlespecific 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 nanoparticlespecific 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 nonkernel clauses) give extra information
MORE ON KERNEL
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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 D
consists 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
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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
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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
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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 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 http://biology.clc.uc.edu)
TEXT BUILDING
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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
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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
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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 RodeaPalomares, 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
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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
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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 (implication) and then q2 (logical connective with sequencing)
p1 à q1 à 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
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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 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 conjunctions,
disjunctions, negations, implications with
respect to the basic string
• Further compound them into a coherent and
cohesive text