Aristotle-The Square of Opposition (Whiteboard Animation)

Aristotle laid out the principles of his logic in his writing Περὶ Ἑρμηνείας, in Latin De Interpretatione, in English On Exposition. It is a graphical representation of the relations between propositions that guarantee their truth. If philosophers and scientists would internalise the logical rules in Aristotle's square of opposition, a lot of misunderstandings would be prevented. SEE: The Square of Opposition as a Whiteboard animation

 Basics of the Square of Opposition of Aristotle

0:06 A proposition (e.g. "All Greeks are men.") consists of a subject ("Greeks") and a predicate ("men").
The four types of propositions are:
Universal positive ("All Greeks are men.", abbreviated "aGM"),
Universal negative ("No Greeks are men.", abbreviated "nGM"),
Particular positive ("Some Greeks are men.", abbreviated "sGM") and
Particular negative ("Some Greeks are not men", abbreviated "sGnM").

1:38 Contradiction (Aristotle)

Universal positive and particular negative, as well as universal negative and particular positive are contradictory. They can't both be true and can't both be false at the same time.

1:59 Contraries (Aristotle)

Universal positive and Universal negative propositions are contraries. They can't both be true, but can both be false at the same time.

2:15 Subcontraries (Aristotle)

Particular positive and Particular negative propositions are subcontraries. They can't both be false, but can both be true at the same time.

2:35 Implication (Aristotle)

Implied propositions (particular positive and particular negative) are true, when their implying propositions (universal positive and universal negative) are true.

2:55 Counter Indication (Aristotle)

Universal propositions (positive and negative respectively) are false, when their particular propositions (positive and negative respectively) are false.

3:19 Converse propositions (Aristotle)

In converse propositions, subjects (e.g. Greeks) and predicates (e.g. men) can be switched without altering the proposition's truth.

Converse Propositions are:
"No Greeks are men" and
"Some Greeks are men".
so it is also true that
"No men are Greeks" as well as
"Some men are Greeks".

3:44 Complements (Aristotle)

A complement of a subject or predicate is everything that it is not.
E.g. "all that is not a man" and "all that is not a Greek".

3:58 Contrapositive propositions (Aristotle)

In contrapositive propositions ("all Greeks are men" and "some Greeks are not men"), if the subjects' and predicates' complements are switched, the proposition retains its truth.

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See Also:

• Traditional Square of Opposition

The Question Regarding the Nature of Time

Discussions of the nature of time, and of various issues related to time, have always featured prominently in philosophy, but they have been especially important since the beginning of the 20th Century. This article contains a brief overview of some of the main topics in the philosophy of time — Fatalism; Reductionism and Platonism with respect to time; the topology of time; McTaggart's arguments; The A Theory and The B Theory; Presentism, Eternalism, and The Growing Universe Theory; time travel; and the 3D/4D controversy — together with some suggestions for further reading on each topic, and a bibliography. Time Markosian, Ned, "Time", The Stanford Encyclopedia of Philosophy (Spring 2014 Edition), Edward N. Zalta (ed.), URL = .

If we believe time to be in relation to space-time, then the parameters of our thinking have a distinction about how we look at time?

Time is often referred to as the fourth dimension, along with the spatial dimensions. Time

So you see, one is being selective about the parameters they give them self with which to regard time.

Investigations of a single continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy. Time

If we are part and parcel, then what said that any idea of continuity can express itself. One would have to believe there is a perfect symmetry in existence that is expressed as an asymmetric example of such perfection, and maybe defined as the matters?

Immanuel Kant, in the Critique of Pure Reason, described time as an a priori intuition that allows us (together with the other a priori intuition, space) to comprehend sense experience.[60] With Kant, neither space nor time are conceived as substances, but rather both are elements of a systematic mental framework that necessarily structures the experiences of any rational agent, or observing subject. Kant thought of time as a fundamental part of an abstract conceptual framework, together with space and number, within which we sequence events, quantify their duration, and compare the motions of objects. In this view, time does not refer to any kind of entity that "flows," that objects "move through," or that is a "container" for events. Spatial measurements are used to quantify the extent of and distances between objects, and temporal measurements are used to quantify the durations of and between events. Time was designated by Kant as the purest possible schema of a pure concept or category. Time

All bold added for emphasis by me.

So, we may come to believe something about yourself that was not quite evident before until we acquiescent to the question regarding the nature of time as we have come to know them.

 If you come to believe there are limits in terms of the reductionist efforts regarding measure, so as to be limited in our perceptions, then what lies beyond, that what we may measure? Do you know how to measure a thought?

 But perhaps most significant is that all their observations are compatible with relativity. At no point does the time machine-simulator lead to grandfather-type paradoxes, regardless of the tricks it plays with causality. That’s just as Deutsch predicted. See: The Quantum Experiment That Simulates A Time Machine

How would you perceive Time Dilation. How would you then perceive Time Travel. How would you perceive time variable measure? Given the constraints of such a measure, we have come "to believe" something in science.

Plato's Theory of Forms

The Golden Mean (philosophy)

Ancient Greek Philosophers

In philosophy, especially that of Aristotle, the golden mean is the desirable middle between two extremes, one of excess and the other of deficiency. -The Golden Mean (philosophy)

In the Eudemian Ethics, Aristotle writes on the virtues. Aristotle’s theory on virtue ethics is one that does not see a person’s actions as a reflection of their ethics but rather looks into the character of a person as the reason behind their ethics. His constant phrase is, "… is the Middle state between …". His psychology of the soul and its virtues is based on the golden mean between the extremes. In the Politics, Aristotle criticizes the Spartan Polity by critiquing the disproportionate elements of the constitution; e.g., they trained the men and not the women, and they trained for war but not peace. This disharmony produced difficulties which he elaborates on in his work. See also the discussion in the Nicomachean Ethics of the golden mean, and Aristotelian ethics in general.-http://en.wikipedia.org/wiki/Golden_mean_(philosophy)#Aristotle

Book VI: Intellectual virtue 

 Book VI of the Nicomachean Ethics is identical to Book V of the Eudemian Ethics. Earlier in both works, both the Nicomachean Ethics Book IV, and the equivalent book in the Eudemian Ethics (Book III), though different, ended by stating that the next step was to discuss justice. Indeed in Book I Aristotle set out his justification for beginning with particulars and building up to the highest things. Character virtues (apart from justice perhaps) were already discussed in an approximate way, as like achieving at middle point between two extreme options, but this now raises the question of how we know and recognize the things we aim at or avoid. Recognizing the mean recognizing the correct boundary-marker (horos) which defines the frontier of the mean. And so practical ethics, having a good character, requires knowledge.-

Acquiring Knowledge

A priori and a posteriori knowledge

The nature of this distinction has been disputed by various philosophers; however, the terms may be roughly defined as follows:

A priori knowledge is knowledge that is known independently of experience (that is, it is non-empirical, or arrived at beforehand, usually by reason). It will henceforth be acquired through anything that is independent from experience.
A posteriori knowledge is knowledge that is known by experience (that is, it is empirical, or arrived at afterward).

A priori knowledge is a way of gaining knowledge without the need of experience. In Bruce Russell's article "A Priori Justification and Knowledge"[19] he says that it is "knowledge based on a priori justification," (1) which relies on intuition and the nature of these intuitions. A priori knowledge is often contrasted with posteriori knowledge, which is knowledge gained by experience. A way to look at the difference between the two is through an example. Bruce Russell give two proposition in which the reader decides which one he believes more. Option A: All crows are birds. Option B: All crows are black. If you believe option A, then you are a priori justified in believing it because you don't have to see a crow to know it's a bird. If you believe in option B, then you are posteriori justified to believe it because you have seen many crows therefore knowing they are black. He goes on to say that it doesn't matter if the statement is true or not, only that if you believe in one or the other that matters.

The idea of a priori knowledge is that it is based on intuition or rational insights. Laurence BonJour says in his article "The Structure of Empirical Knowledge",[20] that a "rational insight is an immediate, non-inferential grasp, apprehension or 'seeing' that some proposition is necessarily true." (3) Going back to the crow example, by Laurence BonJour's definition the reason you would believe in option A is because you have an immediate knowledge that a crow is a bird, without ever experiencing one.- Acquiring Knowledge

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In epistemology, rationalism is the view that "regards reason as the chief source and test of knowledge"[1] or "any view appealing to reason as a source of knowledge or justification".[2] More formally, rationalism is defined as a methodology or a theory "in which the criterion of the truth is not sensory but intellectual and deductive".[3] Rationalists believe reality has an intrinsically logical structure. Because of this, rationalists argue that certain truths exist and that the intellect can directly grasp these truths. That is to say, rationalists assert that certain rational principles exist in logic, mathematics, ethics, and metaphysics that are so fundamentally true that denying them causes one to fall into contradiction. Rationalists have such a high confidence in reason that proof and physical evidence are unnecessary to ascertain truth – in other words, "there are significant ways in which our concepts and knowledge are gained independently of sense experience".[4] Because of this belief, empiricism is one of rationalism's greatest rivals. -Rationalism

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Empiricism is a theory which states that knowledge comes only or primarily from sensory experience.^[1] One of several views of epistemology, the study of human knowledge, along with rationalism and skepticism, empiricism emphasizes the role of experience and evidence, especially sensory experience, in the formation of ideas, over the notion of innate ideas or traditions;^[2] empiricists may argue however that traditions (or customs) arise due to relations of previous sense experiences.^[3]

Empiricism in the philosophy of science emphasizes evidence, especially as discovered in experiments. It is a fundamental part of the scientific method that all hypotheses and theories must be tested against observations of the natural world rather than resting solely on a priori reasoning, intuition, or revelation.

Empiricism, often used by natural scientists, says that "knowledge is based on experience" and that "knowledge is tentative and probabilistic, subject to continued revision and falsification."^[4] One of the epistemological tenets is that sensory experience creates knowledge. The scientific method, including experiments and validated measurement tools, guides empirical research.

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


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

¬     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