Symbolic Logic

In mathematics, a proof is a deductive argument for a mathematical statement. In the argument, other previously established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms  Mathematical proof

Direct proof
Proof by mathematical induction-
Proof by [S]contraposition[/S]/transposition (P → Q) \Leftrightarrow (¬ Q → ¬ P)
Proof by construction
Proof by exhaustion
Probabilistic proof
Combinatorial proof
Nonconstructive proof
Statistical proofs in pure mathematics

----------------------------------------------------


Modus Ponens-   
                                                      p → q
                                                      p
                                                      _____
                                                      q   

Modus Tollens-   
                                                       p → q
                                                       ~q
                                                       _____
                                                       ~p   

Hypothetical Syllogism-   
                                                       p → q
                                                       q → r
                                                       _____
                                                       p → r   

Disjunctive Syllogism-   
                                                       p ∨ q
                                                       ~ p
                                                       _____
                                                       q   

Constructive Dilemma-   
                                                      (p → q) • (r → s)
                                                      p ∨ r
                                                      ______
                                                      Q ∨ S
   
Destructive Dilemma-   
                                                      (p → q) • (r → s)
                                                      ~q ∨ ~S
                                                      ______
                                                      ~P v ~R
   
Conjunction    -
                                                       p
                                                       q
                                                       _____
                                                       p • q   

Simplification-   
                                                       p • q
                                                       ____
                                                       p   

Addition   
p
_____
p ∨ q


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¬     negation (NOT)                     The tilde ( ˜ ) is also often used.
∧     conjunction (AND)             The ampersand ( & ) or dot ( · ) are also often used.
∨     disjunction (OR)                     This is the inclusive disjunction, equivalent to and/or in                                              English.
⊕     exclusive disjunction (XOR)     ⊕ means that only one of the connected propositions 
                                                is true, equivalent to either…or. Sometimes ⊻ is used.
|     alternative denial (NAND)     Means “not both”. Sometimes written as ↑
↓     joint denial (NOR)             Means “neither/nor”.
→     conditional (if/then)             Many logicians use the symbol ⊃ instead. This is also 
                                                known as material implication.
↔     biconditional (iff)                     Means “if and only if” ≡ is sometimes used, but this site
                                                reserves that symbol for equivalence.

Quantifiers

∀     universal quantifier             Means “for all”, so ∀xPx means that Px is true for every x.
∃     existential quantifier             Means “there exists”, so ∃xPx means that Px is true for at
                                                least one x.
Relations

⊨     implication                             α ⊨ β means that β follows from α
≡     equivalence                     Also ⇔. Equivalence is two-way implication, so α ≡ β
                                                means α implies β and β implies α.
⊢     provability                             Shows provable inference. α is provable β means that
                                                from α we can prove that β.
∴     therefore                             Used to signify the conclusion of an argument. Usually
                                                taken to mean implication, but often used to present
                                                arguments in which the premises do not deductively imply
                                                the conclusion.
⊩     forces                            A relationship between possible worlds and sentences in
                                               modal logic.
Truth-Values

⊤     tautology                            May be used to replace any tautologous (always true)
                                               formula.
⊥     contradiction                    May be used to replace any contradictory (always false)
                                              formula. Sometimes “F” is used.

Parentheses

( )     parentheses                   Used to group expressions to show precedence of
                                             operations.
Square brackets

[ ]                                          are sometimes used to clarify groupings.
Set Theory

∈     membership                   Denotes membership in a set. If a ∈ Γ, then a is a member
                                             (or an element) of set Γ.
∪     union                          Used to join sets. If S and T are sets of formula, S ∪ T is a
                                             set containing all members of both.
∩     intersection                  The overlap between sets. If S and T are sets of formula, S
                                             ∩ T is a set containing those elemenets that are members
                                             of both.
⊆     subset                          A subset is a set containing some or all elements of another
                                             set.
⊂     proper subset                  A proper subset contains some, but not all, elements of
                                             another set.
=     set equality                  Two sets are equal if they contain exactly the same
                                             elements.
∁     absolute complement          ∁(S) is the set of all things that are not in the set S.
                                             Sometimes written as C(S), S or SC.
-     relative complement          T - S is the set of all elements in T that are not also in S.
                                             Sometimes written as T \ S.
∅     empty set                          The set containing no elements.

Modalities

□     necessarily                     Used only in modal logic systems. Sometimes expressed as []
                                            where the symbol is unavailable.
◊     possibly                         Used only in modal logic systems. Sometimes expressed as
                                           <> where the symbol is unavailable.

Propositions, Variables and Non-Logical Symbols

The use of variables in logic varies depending on the system and the author of the logic being presented. However, some common uses have emerged. For the sake of clarity, this site will use the system defined below.

Symbol             Meaning                     Notes

A, B, C … Z     propositions     Uppercase Roman letters signify individual propositions. For example, P may symbolize the proposition “Pat is ridiculous”. P and Q are traditionally used in most examples.

α, β, γ … ω     formulae     Lowercase Greek letters signify formulae, which may be themselves a proposition (P), a formula (P ∧ Q) or several connected formulae (φ ∧ ρ).

x, y, z             variables     Lowercase Roman letters towards the end of the alphabet are used to signify variables. In logical systems, these are usually coupled with a quantifier, ∀ or ∃, in order to signify some or all of some unspecified subject or object. By convention, these begin with x, but any other letter may be used if needed, so long as they are defined as a variable by a quantifier.

a, b, c, … z     constants           Lowercase Roman letters, when not assigned by a quantifier, signifiy a constant, usually a proper noun. For instance, the letter “j” may be used to signify “Jerry”. Constants are given a meaning before they are used in logical expressions.

Ax, Bx … Zx     predicate symbols     Uppercase Roman letters appear again to indicate predicate relationships between variables and/or constants, coupled with one or more variable places which may be filled by variables or constants. For instance, we may definite the relation “x is green” as Gx, and “x likes y” as Lxy. To differentiate them from propositions, they are often presented in italics, so while P may be a proposition, Px is a predicate relation for x. Predicate symbols are non-logical — they describe relations but have neither operational function nor truth value in themselves.

Γ, Δ, … Ω     sets of formulae     Uppercase Greek letters are used, by convention, to refer to sets of formulae. Γ is usually used to represent the first site, since it is the first that does not look like Roman letters. (For instance, the uppercase Alpha (Α) looks identical to the Roman letter “A”)

Γ, Δ, … Ω     possible worlds     In modal logic, uppercase greek letters are also used to represent possible worlds. Alternatively, an uppercase W with a subscript numeral is sometimes used, representing worlds as W0, W1, and so on.

{ }     sets     Curly brackets are generally used when detailing the contents of a set, such as a set of formulae, or a set of possible worlds in modal logic. For instance, Γ = { α, β, γ, δ }

Tradition Square of Opposition

Parsons, Terence, "The Traditional Square of Opposition", The Stanford Encyclopedia of Philosophy (Spring 2014 Edition), Edward N. Zalta (ed.), URL = .

Contrary- All S are P, No S is P All s is P is contrary to the claim NO S is P. 

----------------------- 

A contrary can be true as well as false. Contraries can both be false. Contraries can't both be true. 

The A and E forms entail each other's negations 

Subcontrary Some S are P, Some S are not P

 -------------------------------------------- 

Sub contraries can't both be false. Sub contraries can both be true. The negation of the I form entails the (unnegated) E form, and vice versa. 

Contradiction- All S are P, Some S are not P, Some S are P, No S are P

 ----------------------------------- 

For contradictions -Two propositions are contradictory if they cannot both be true and they cannot both be false. Contradictory means there is exactly one truth value and if one proposition is true the other MUST be false. If one is false the other MUST be true. 

The propositions can't both be true and the propositions can't both be false. 

The A and O forms entail each other's negations, as do the E and I forms. 

The negation of the A form entails the (unnegated) O form, and vice versa; likewise for the E and I forms.
 Super alteration[- Every S is P, implies Some S are P No S is P, implies Some S are not P 

-------------------------------------------- 

The two propositions can be true.

 Sub alteration- All S are P, Some S are P No S are P, Some S are not P 

---------------------------------- 

A proposition is a subaltern of another if it must be true The A form entails the I form, and the E form entails the O form. 

"The 'I' proposition, the particular affirmative (particularis affirmativa), Latin 'quoddam S est P', usually translated as 'some S are P'" . 

As in the first(Proposition 1) or the "I" "To be clear the I proposition is SOME S is P. This is what is meant by a I proposition. Well you can certainly infer if an I proposition is true that an E proposition is false because they are contradictory. Unfortunately there is NOTHING else to infer with certainty. That is there will be times where the proposition will be true and different times it will be false. This is called contingent truths. That is the proposition is not true 100% of the time. It has false cases. Deductive logic tries to stay away from contingent truths." 

"The 'I' proposition, the particular affirmative (particularis affirmativa), Latin 'quoddam S est P', usually translated as 'some S are P'" 

Summary

Universal statements are contraries: 'every man is just' and 'no man is just' cannot be true together, although one may be true and the other false, and also both may be false (if at least one man is just, and at least one man is not just).

Particular statements are subcontraries. 'Some man is just' and 'some man is not just' cannot be false together

The particular statement of one quality is the subaltern of the universal statement of that same quality, which is the superaltern of the particular statement, because in Aristotelian semantics 'every A is B' implies 'some A is B' and 'no A is B' implies 'some A is not B'. Note that modern formal interpretations of English sentences interpret 'every A is B' as 'for any x, x is A implies x is B', which does not imply 'some x is A'. This is a matter of semantic interpretation, however, and does not mean, as is sometimes claimed, that Aristotelian logic is 'wrong'.

The universal affirmative and the particular negative are contradictories. If some A is not B, not every A is B. Conversely, though this is not the case in modern semantics, it was thought that if every A is not B, some A is not B. This interpretation has caused difficulties (see below). While Aristotle's Greek does not represent the particular negative as 'some A is not B', but as 'not every A is B', someone in his commentary on the Peri hermaneias, renders the particular negative as 'quoddam A non est B', literally 'a certain A is not a B', and in all medieval writing on logic it is customary to represent the particular proposition in this way.

These relationships became the basis of a diagram originating with Boethius and used by medieval logicians to classify the logical relationships. The propositions are placed in the four corners of a square, and the relations represented as lines drawn between them, whence the name 'The Square of Opposition'.

Dark Matter Research

An overview of how the LHC at CERN can look for dark matter. (Credit: STFC/Ben Gilliland)
(Click on Image for larger viewing)

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See Also:

• The Dark Matter Hunt

Inaugural Symposium of the Hyper-Kamiokande Proto-Collaboration

The Hyper-Kamiokande project aims to address the mysteries of the origin and evolution of the Universe's matter and to confront theories of elementary particle unification. To realize these goals the project will combine a high intensity neutrino beam from the Japan Proton Accelerator Research Complex (J-PARC) with a new detector based upon precision neutrino experimental techniques developed in Japan. The Hyper-Kamiokande project will be about 25 times larger than Super-Kamiokande, the research facility that was first to discover evidence for neutrino mass in 1998. On this occasion, a research proto-collaboration will be formed to advance the Hyper-Kamiokande project internationally and a symposium will be held to commemorate and promote the event. In addition, a signing ceremony marking an agreement for the promotion of the project between the University of Tokyo Institute for Cosmic Ray Research (ICRR) and the High Energy Accelerator Research Organization (KEK) Institute of Particle and Nuclear Studies will take place during the symposium. See: Hyper-Kamiokande

Superfluidity and the Roton

University of Chicago scientists can create an exotic, particle-like excitation called a roton in superfluids with the tabletop apparatus pictured here. Posing left to right are graduate students Li-Chung Ha and Logan Clark, and Prof. Cheng Chin.

See: Cesium atoms shaken, not stirred, to create elusive excitation in superfluid 

We present experimental evidence showing that an interacting Bose condensate in a shaken optical lattice develops a roton-maxon excitation spectrum, a feature normally associated with superfluid helium. The roton-maxon feature originates from the double-well dispersion in the shaken lattice, and can be controlled by both the atomic interaction and the lattice modulation amplitude. We determine the excitation spectrum using Bragg spectroscopy and measure the critical velocity by dragging a weak speckle potential through the condensate—both techniques are based on a digital micromirror device. Our dispersion measurements are in good agreement with a modified Bogoliubov model. DOI: http://dx.doi.org/10.1103/PhysRevLett.114.055301

After Relativism

Watch more videos on iai.tv

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 "...underwriting the form languages of ever more domains of mathematics is a set of deep patterns which not only offer access to a kind of ideality that Plato claimed to see the universe as created with in the Timaeus; more than this, the realm of Platonic forms is itself subsumed in this new set of design elements-- and their most general instances are not the regular solids, but crystallographic reflection groups. You know, those things the non-professionals call . . . kaleidoscopes! * (In the next exciting episode, we'll see how Derrida claims mathematics is the key to freeing us from 'logocentrism'-- then ask him why, then, he jettisoned the deepest structures of mathematical patterning just to make his name...)

* H. S. M. Coxeter, Regular Polytopes (New York: Dover, 1973) is the great classic text by a great creative force in this beautiful area of geometry (A polytope is an n-dimensional analog of a polygon or polyhedron. Chapter V of this book is entitled 'The Kaleidoscope'....)"

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See Also:

• Answers to Question-Interesting
• Relativism
• Mind Maps: Mathematical Structures

Quantum Chromodynamics

Source - http://serious-science.org/videos/1060

Nobel Prize laureate David Gross on Rutherford experiments, asymptotic freedom, and the origin of the particle masses

The Dark Matter Hunt

Dark matter, the substance making up 85 percent of all the mass in the universe, is invisible. The goal of ADMX is to detect it by turning it into photons, particles of light. Dark matter was forged in the early universe, under conditions of extreme heat. ADMX, on the other hand, operates in extreme cold. Dark matter comprises most of the mass of a galaxy. To find it, ADMX will use sophisticated devices microscopic in size. 

Scientists on ADMX—short for the Axion Dark Matter eXperiment—are searching for hypothetical particles called axions. The axion is a dark matter candidate that is also a bit of a dark horse, even as this esoteric branch of physics goes. See:  Dark horse of the dark matter hunt

 http://www.phys.washington.edu/groups/admx/experiment.html

Rationalism vs Empiricism

The dispute between rationalism and empiricism concerns the extent to which we are dependent upon sense experience in our effort to gain knowledge. Rationalists claim that there are significant ways in which our concepts and knowledge are gained independently of sense experience. Empiricists claim that sense experience is the ultimate source of all our concepts and knowledge.

Rationalists generally develop their view in two ways. First, they argue that there are cases where the content of our concepts or knowledge outstrips the information that sense experience can provide. Second, they construct accounts of how reason in some form or other provides that additional information about the world. Empiricists present complementary lines of thought. First, they develop accounts of how experience provides the information that rationalists cite, insofar as we have it in the first place. (Empiricists will at times opt for skepticism as an alternative to rationalism: if experience cannot provide the concepts or knowledge the rationalists cite, then we don't have them.) Second, empiricists attack the rationalists' accounts of how reason is a source of concepts or knowledge. SEE: Markie, Peter, "Rationalism vs. Empiricism http://plato.stanford.edu/entries/rationalism-empiricism/", The Stanford Encyclopedia of Philosophy (Summer 2013 Edition), Edward N. Zalta (ed.),

 Long before I had come to understand this nature of rationalism there were already signs that such a journey was already being awakened. This was an understanding for me as to the nature of what could be gained from the ability to visualize beyond empirical nature of our journey into the sensible realm.

I guess in a such an awakening,  as to what we know,  there is the realization that what comes after helps to make that sense. So in a way one might like to see how rationalism together with Empiricism actually works. It is not in the sense that I might define one group of historical thinkers to contrast each other to say that one should excel over another, but to define how such a rationally sound person moves toward empiricism to understand the reality we created by experimentation and repeatability that empiricism enshrouds.

So this awakening while slow to materialize, comes from understanding something about the logic of the world and the definitions and architecture of that logical approach. To me in this day and age it has lead to some theory about which computational view could hold the idea about how we might see this reality. I am reticence to view this  as a form of that reality. It is for what holds me back is a self evident moment using deducted features of our reasoning,  which could move us to that moment of clarity.

 The Empiricism Thesis: We have no source of knowledge in S or for the concepts we use in S other than sense experience Empircism -

 Empirical fact would not be the basis of reality for Nick Bostrum's simulation argument for instance. I hope to explain why.

 The basis of this association(Rationalist, or, a Empiricist) is whether one gains by a deductive method, or, an inductive method. A sense experience tells us, science as we know it, is inductive. We must garner repeatable experiments to verify reality, a rationalist, by logic and reason of theory alone. Verification, comes afterward. This for a rationalist is a deductive something which can be true, can be "innate" before we accept the inductive method means,  that is it can be rationally ascertained. It is only after ward that such a process could be said to be true or false.

If the late character of our sources may incite us to doubt the authenticity of this tradition, there remains that, in its spirit, it is in no way out of character, as can be seen by reading or rereading what Plato says about the sciences fit for the formation of philosophers in book VII of the Republic, and especially about geometry at Republic, VII, 526c8-527c11. We should only keep in mind that, for Plato, geometry, as well as all other mathematical sciences, is not an end in itself, but only a prerequisite meant to test and develop the power of abstraction in the student, that is, his ability to go beyond the level of sensible experience which keeps us within the "visible" realm, that of the material world, all the way to the pure intelligible. And geometry, as can be seen through the experiment with the slave boy in the Meno (Meno, 80d1-86d2), can also make us discover the existence of truths (that of a theorem of geometry such as, in the case of the Meno, the one about doubling a square) that may be said to be "transcendant" in that they don't depend upon what we may think about them, but have to be accepted by any reasonable being, which should lead us into wondering whether such transcendant truths might not exist as well in other areas, such as ethics and matters relating to men's ultimate happiness, whether we may be able to "demonstrate" them or not.See: Frequently Asked Questions about Plato by Bernard SUZANNE

When you examine deeply the very nature of your journey, then, you come to realize what is hidden underneath "experience." So while being an empiricist, it is necessary to know that such a joining with the rationalist correlates with the reasoned only after the mentioned experience. These are "corollary experiences," which serve to identify that which had been identified long before the sensible world had been made known.

Paradoxically, it was Einstein who reluctantly introduced the notion of spontaneous events, which might after all be the root of Bellʼs theorem. The lesson for the future could, however, be that we should build the notion of locality on the operationally clear 'no-signalling' condition—the impossibility of transferring information faster than light. After all, this is all that theory of relativity requires. The moral of the story is that Bellʼs theorem, in all its forms, tells us not what quantum mechanics is, but what quantum mechanics is not. Quantum non-locality—it ainʼt necessarily so... -

Empiricism then is to validate as a corollary that which had been cognate(maybe poor choice of word here but instead should use cognition). This does not mean you stop the process, but to extend the visionary possibility of that which can be cognitive....peering into the very nature of reality. Becomes the " we should build the notion of locality on the operationally clear 'no-signalling' condition."

Here the question of entanglement raises it's head to ask what is really being trasmitted as the corrallary of information,  as a direct physical connection in a computational system. In a quantum gravity scheme what is exchanged as a spin 2 graviton we might examine in the corollary of this no signalling condition but as a direct understanding of gravitational signalling.?

Such an examination reveals the Innate process with which we may already know "some thing,"  is awakened by moving into the world of science. While we consider such computational reality in context of a ontological question,  then,  such a journey may be represented as the geometry of the being which reveals a deeper question about the make-up of that reality.


Affective Field Theory of Emotion regarding sensory development may aid in the journey for understanding the place with which "the idea/form in expression arises," and that which is reasoned, beyond the related abstractions of "such a beginning," by becoming the ideal, in the empiricist world.

The Future in NanoTechnologies

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See Also:

• NanoTechnology
• The Opensource Handbook of Nanoscience and Nanotechnology
• Scienceshow/Nanotechnology