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#topology

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Answer

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any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation.

es in Euclidean spaces, each output coordinate of an affine map is a linear function (in the sense of calculus) of all input coordinates. Another way to deal with affine transformations systematically is to select a point as the origin; then, <span>any affine transformation is equivalent to a linear transformation (of position vectors) followed by a translation. Contents [hide] 1 Mathematical definition 1.1 Alternative definition 2 Representation 2.1 Augmented matrix 2.1.1 Example augmented matrix 3 Properties 3.1 Properti

#french #verbs

Verbs ending in -cer add a cedilla to the c before the letters a or o in order to retain the soft c sound. avancer to advance nous avanc¸ ons

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#french #verbs

Verbs ending in -ger add an e after the g before the letters a and o in order to maintain the soft g sound. changer to change nous changeons

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#french #verbs

Verbs which have -é- in the penultimate syllable of the infinitive change -é- to -è- in all forms except the nous and vous forms.

**compléter**: je complète; tu complètes; il, elle, on complète; nous complétons; vous complétez; ils, elles complètent

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#french #verbs

In some verbs which contain -e- in the next to the last syllable of the infinitive, the -e- changes to -e ` - in all forms except the nous and vous forms. Study the following. lever je le ` ve tu le ` ves il, elle, on le ` ve nous levons vous levez ils, elles le ` vent

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#french #verbs

In other verbs with -e- in the infinitive, the final consonant is doubled in all but the nous and vous forms.

**jeter:** je jette; tu jettes; il, elle, on jette; nous jetons; vous jetez; ils, elles jettent

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#french #verbs

Verbs whose infinitive ends in -oyer, -uyer and -ayer change -y- to -i- in all but the nous and vous forms.

**nettoyer:** je nettoie; tu nettoies; il, elle, on nettoie; nous nettoyons; vous nettoyez; ils, elles nettoient

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#french #verbs

Verbs whose infinitive ends in -ir add -is, -is, -it, -issons, -issez, -issent to the stem.

**finir**: je finis; tu finis; il, elle, on finit; nous finissons; vous finissez; ils, elles finissent

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#french #verbs

Verbs whose infinitive ends in -re add -s, -s, -, -ons, -ez, -ent to the stem.

**répondre**: je réponds; tu réponds; il, elle, on répond; nous répondons; vous répondez; ils, elles répondent

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#french #verbs

Some verbs like **ouvrir**, **offrir**, and **cueillir**, although the infinitive ends in -ir, are conjugated like regular -er verbs.

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#french #verbs

Question

Verbs ending in -cer add [...] to the c before the letters a or o in order to retain the soft c sound.

Answer

a cedilla

**avancer** to advance, nous avançons

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Verbs ending in -cer add a cedilla to the c before the letters a or o in order to retain the soft c sound. avancer to advance nous avanc¸ ons

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#french #verbs

Question

Verbs ending in -ger add [...] after the g before the letters a and o in order to maintain the soft g sound.

Answer

an **e**

**changer** to change, nous changeons

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scheduled repetition interval | last repetition or drill |

Verbs ending in -ger add an e after the g before the letters a and o in order to maintain the soft g sound. changer to change nous changeons

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#french #verbs

Question

Verbs which have -é- in [...] of the infinitive change -é- to -è- in all forms except the nous and vous forms.

Answer

the penultimate syllable

**compléter**: je complète; tu complètes; il, elle, on complète; nous complétons; vous complétez; ils, elles complètent

The two are pronounced differently:

The two are pronounced differently:

é | [e] | café [kafe] |

è | [ɛ] | père [père] |

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Verbs which have -é- in the penultimate syllable of the infinitive change -é- to -è- in all forms except the nous and vous forms. compléter : je complète; tu complètes; il, elle, on complète; nous complétons; vous compléte

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#french #verbs

Question

Verbs which have -é- in the penultimate syllable of the infinitive change -é- to -è- in all forms except [...].

Answer

the nous and vous forms

**compléter**: je complète; tu complètes; il, elle, on complète; nous complétons; vous complétez; ils, elles complètent

The changes are used to indicate the stress of the word; the nous and vous don't change because they already have their stress

The changes are used to indicate the stress of the word; the nous and vous don't change because they already have their stress

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Verbs which have -é- in the penultimate syllable of the infinitive change -é- to -è- in all forms except the nous and vous forms. compléter : je complète; tu complètes; il, elle, on complète; nous complétons; vous complétez; ils, elles complètent

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#french #verbs

Question

In other verbs with -e- in the infinitive, [...] is doubled in all but the nous and vous forms.

Answer

the final consonant

**jeter:** je jette; tu jettes; il, elle, on jette; nous jetons; vous jetez; ils, elles jettent

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

In other verbs with -e- in the infinitive, the final consonant is doubled in all but the nous and vous forms. jeter: je jette; tu jettes; il, elle, on jette; nous jetons; vous jetez; ils, elles jettent

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#french #verbs

Question

In other verbs with -e- in the infinitive, the final consonant is doubled in all but [...] forms.

Answer

the nous and vous

**jeter:** je jette; tu jettes; il, elle, on jette; nous jetons; vous jetez; ils, elles jettent

These are the shorter words, one syllable.

These are the shorter words, one syllable.

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

In other verbs with -e- in the infinitive, the final consonant is doubled in all but the nous and vous forms. jeter: je jette; tu jettes; il, elle, on jette; nous jetons; vous jetez; ils, elles jettent

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#french #verbs

Question

Verbs whose infinitive ends in [...] change -y- to -i- in all but the nous and vous forms.

Answer

-oyer, -uyer and -ayer

**nettoyer:** je nettoie; tu nettoies; il, elle, on nettoie; nous nettoyons; vous nettoyez; ils, elles nettoient

This change avoids adding unnecessary trailing sound to the verbs

This change avoids adding unnecessary trailing sound to the verbs

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Verbs whose infinitive ends in -oyer, -uyer and -ayer change -y- to -i- in all but the nous and vous forms. nettoyer: je nettoie; tu nettoies; il, elle, on nettoie; nous nettoyons; vous nettoyez; ils, elles nettoient

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#french #verbs

Question

In present tense, verbs whose infinitive ends in -ir add [...] to the stem.

Answer

-is, -is, -it, -issons, -issez, -issent

**finir**: je finis; tu finis; il, elle, on finit; nous finissons; vous finissez; ils, elles finissent

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Verbs whose infinitive ends in -ir add -is, -is, -it, -issons, -issez, -issent to the stem. finir : je finis; tu finis; il, elle, on finit; nous finissons; vous finissez; ils, elles finissent

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#french #verbs

Question

Verbs whose infinitive ends in -re add [...] to the stem.

Answer

-s, -s, -, -ons, -ez, -ent

**répondre**: je réponds; tu réponds; il, elle, on répond; nous répondons; vous répondez; ils, elles répondent

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repetition number in this series | 0 | memorised on | scheduled repetition | ||||

scheduled repetition interval | last repetition or drill |

Verbs whose infinitive ends in -re add -s, -s, -, -ons, -ez, -ent to the stem. répondre : je réponds; tu réponds; il, elle, on répond; nous répondons; vous répondez; ils, elles répondent

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#french #verbs

Question

Some verbs like **[...]**, although the infinitive ends in -ir, are conjugated like regular -er verbs.

Answer

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scheduled repetition interval | last repetition or drill |

Some verbs like ouvrir , offrir , and cueillir , although the infinitive ends in -ir, are conjugated like regular -er verbs.

** **

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#french #verbs

Question

Some verbs like **ouvrir**, **offrir**, and **cueillir**, although the infinitive ends in -ir, are conjugated like [...] verbs.

Answer

regular -er

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Some verbs like ouvrir , offrir , and cueillir , although the infinitive ends in -ir, are conjugated like regular -er verbs.

** **

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#functional-analysis

Question

Functional analysis studies

**[...]**that are endowed with some kind of limit-related structure and- the linear functions defined on these spaces and respecting these structures in a suitable sense.

Answer

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, et

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

** **

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#functional-analysis

Question

Functional analysis studies

- vector spaces that are endowed with some kind of limit-related structure and
- the
**[...]**defined on these spaces and respecting these structures in a suitable sense.

Answer

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Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, et

analysis (psychology). [imagelink] One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis. <span>Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations. The usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function and the name was first used in Hadamard's 1910 book on that

** **

#jaynes #plausible-reasoning

the logical product or the conjunction, denotes the proposition ‘both A and B are true

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#jaynes #plausible-reasoning

the logical sum or disjunction, stands for ‘at least one of the propositions, A, B is true’

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#jaynes #plausible-reasoning

These symbols are only a shorthand way of writing propositions, and do not stand for numerical values

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#jaynes #plausible-reasoning

that A and B have the same truth value, then they are logically equivalent propositions, in the sense that any evidence concerning the truth of one pertains equally well to the truth of the other, and they have the same implications for any further reasoning

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#jaynes #plausible-reasoning

The relation between a proposition and its denial is reciprocal

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[unknown IMAGE 1803198532876]

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#has-images #jaynes #plausible-reasoning

Question

In Boolean logic, the idempotence property means **[...]**

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[unknown IMAGE 1803200105740]

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#has-images #jaynes #plausible-reasoning

Question

In Boolean logic, the commutativity property means **[...]**

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[unknown IMAGE 1803201678604]

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#has-images #jaynes #plausible-reasoning

Question

In Boolean logic, the associativity property means **[...]**

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[unknown IMAGE 1803203251468]

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#has-images #jaynes #plausible-reasoning

Question

In Boolean logic, the distributivity property means **[...]**

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[unknown IMAGE 1803204824332]

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#has-images #jaynes #plausible-reasoning

Question

In Boolean logic, the duality property means **[...]**

Answer

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[unknown IMAGE 1803207970060]

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#has-images #jaynes #plausible-reasoning

Question

In Boolean logic, the implication proposition can be represented as logical equation of **[...]**

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[unknown IMAGE 1803207970060]

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#has-images #jaynes #plausible-reasoning

Question

In Boolean logic, the implication proposition means **[...]** is true or equally **[...]** is false

Answer

\(\overline{A} + B\), \(A \overline{B}\)

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#jaynes #plausible-reasoning

Question

the logical product denotes the proposition ‘[...]

Answer

both A and B are true

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the logical product or the conjunction, denotes the proposition ‘both A and B are true

** **

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#jaynes #plausible-reasoning

Question

the logical sum stands for [...]

Answer

at least one of A, B is true

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the logical sum or disjunction, stands for ‘at least one of the propositions, A, B is true’

** **

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#jaynes #plausible-reasoning

Question

propositions with the same truth value are [...] propositions

Answer

logically equivalent

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that A and B have the same truth value, then they are logically equivalent propositions, in the sense that any evidence concerning the truth of one pertains equally well to the truth of the other, and they have the same implications for any further reasoning

** **

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#jaynes #plausible-reasoning

Question

The relation between a proposition and its denial is [...]

Answer

reciprocal

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The relation between a proposition and its denial is reciprocal

[unknown IMAGE 1803207970060]

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#has-images #jaynes #plausible-reasoning

Question

the strong syllogisms can be represented in Boolean logic as **[...]**

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#jaynes #plausible-reasoning

The theory of plausible reasoning based on weak syllogisms is not a ‘weakened’ form of logic; it is an extension of logic with new content not present at all in conventional deductive logic.

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#jaynes #plausible-reasoning

a false proposition implies all propositions

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#jaynes #plausible-reasoning

merely knowing the truth values of propositions A and B does not provide any information on deducibility

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#jaynes #plausible-reasoning

An expression B = f (A 1 ,...,A n ) involving n propositions is a logic function on a space S of M = 2 n points; and there are exactly 2 M such functions.

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#jaynes #plausible-reasoning

Since we are, at this stage, restricting our attention to Aristotelian propositions, any logic function C = f (A, B) such as (1.15) has only two possible ‘values’, true and false; and likewise the ‘independent v ariables’ A and B can take on only those two values.

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#jaynes #plausible-reasoning

logic functions with only one point being true can be constructed simply by conjunctions of input propositions,

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#jaynes #plausible-reasoning

Any logic function that is true on at least one point can be constructed as logical sum of basic conjunctions

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#jaynes #plausible-reasoning

the two operations (AND, NOT) already constitute an adequate set for deductive logic

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#jaynes #plausible-reasoning

A set of operations suffice to generate all possible logic functions form an adequate set

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#jaynes #plausible-reasoning

The duality property shows the logic disjunction \( A + B \) is the same as denying that they are both false \( \overline{ \bar{A} \bar{B} } \)

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#jaynes #plausible-reasoning

The operation ‘NAND’ is defined as the negation of ‘AND’

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#jaynes #plausible-reasoning

the operation NOR defined by the negation of OR

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#jaynes #plausible-reasoning

besides a common ground, a logic gate has two input terminals and one output

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#jaynes #plausible-reasoning

We call them ‘desiderata’ rather than ‘axioms’ because they do not assert that anything is ‘true’ but only state what appear to be desirable goals.

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#jaynes #plausible-reasoning

We expect plausible reasoning to have no contradictions and be able to determine any unique extension of logic

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#jaynes #plausible-reasoning

A ⇒ B is false only if A is true and B is false

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#jaynes #plausible-reasoning

The question of logical deducibility of one proposition from a set of others arises in a crucial way in the Gödel theorem

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#jaynes #plausible-reasoning

This great difference in the meaning of the word *implies* in ordinary language and in formal logic is a tricky point that can lead to serious error if it is not properly understood

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#jaynes #plausible-reasoning

The theory of plausible reasoning is based on weak syllogisms

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The theory of plausible reasoning based on weak syllogisms is not a ‘weakened’ form of logic; it is an extension of logic with new content not present at all in conventional deductive logic.

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#jaynes #plausible-reasoning

Plausible reasoning is an extension of deductive logic because it also applies to weak syllogisms.

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The theory of plausible reasoning based on weak syllogisms is not a ‘weakened’ form of logic; it is an extension of logic with new content not present at all in conventional deductive logic.

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#jaynes #plausible-reasoning

Question

The theory of plausible reasoning is based on [...]

Answer

weak syllogisms

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The theory of plausible reasoning is based on weak syllogisms

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#jaynes #plausible-reasoning

Question

Plausible reasoning is an extension of [...] because it also applies to weak syllogisms.

Answer

deductive logic

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Plausible reasoning is an extension of deductive logic because it also applies to weak syllogisms.

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#jaynes #plausible-reasoning

Question

[...] is an extension of deductive logic because it also applies to weak syllogisms.

Answer

Plausible reasoning

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Plausible reasoning is an extension of deductive logic because it also applies to weak syllogisms.

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#jaynes #plausible-reasoning

Question

[...] implies all propositions

Answer

a false proposition

*If a proposition \( A \) is false, then \( A \) and \( A B \) always have the same truth value*

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a false proposition implies all propositions

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#jaynes #plausible-reasoning

Question

merely knowing the truth values of propositions A and B does not provide any information on [...]

Answer

deducibility

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merely knowing the truth values of propositions A and B does not provide any information on deducibility

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#jaynes #plausible-reasoning

Question

merely knowing the [...] of propositions A and B does not provide any information on deducibility

Answer

truth values

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merely knowing the truth values of propositions A and B does not provide any information on deducibility

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Tags

#jaynes #plausible-reasoning

Question

logic functions with only one point being true can be constructed simply by [...],

Answer

conjunctions of input propositions

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logic functions with only one point being true can be constructed simply by conjunctions of input propositions,

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#jaynes #plausible-reasoning

Question

Any logic function that is true on at least one point can be constructed as [...]

Answer

logical sum of basic conjunctions

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Any logic function that is true on at least one point can be constructed as logical sum of basic conjunctions

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#jaynes #plausible-reasoning

Question

The operation ‘NAND’ is defined as [...]

Answer

the negation of ‘AND’

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The operation ‘NAND’ is defined as the negation of ‘AND’

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#jaynes #plausible-reasoning

Question

the operation NOR defined by [...]

Answer

the negation of OR

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the operation NOR defined by the negation of OR

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#jaynes #plausible-reasoning

Question

the two operations [...] already constitute an adequate set for deductive logic

Answer

(AND, NOT)

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the two operations (AND, NOT) already constitute an adequate set for deductive logic

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#jaynes #plausible-reasoning

Question

A set of operations suffice to [...] form an adequate set

Answer

generate all possible logic functions

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A set of operations suffice to generate all possible logic functions form an adequate set

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#jaynes #plausible-reasoning

Question

the two operations (AND, NOT) already constitute an adequate set for [...]

Answer

deductive logic

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the two operations (AND, NOT) already constitute an adequate set for deductive logic

** **

Tags

#jaynes #plausible-reasoning

Question

A set of operations suffice to generate all possible logic functions form an [...]

Answer

adequate set

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A set of operations suffice to generate all possible logic functions form an adequate set

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#jaynes #plausible-reasoning

Question

besides a common ground, a logic gate has [...]

Answer

two input terminals and one output

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besides a common ground, a logic gate has two input terminals and one output

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#jaynes #plausible-reasoning

Question

We expect plausible reasoning to have no [...] and be able to determine any unique extension of logic

Answer

contradictions

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We expect plausible reasoning to have no contradictions and be able to determine any unique extension of logic

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Tags

#jaynes #plausible-reasoning

Question

We expect plausible reasoning to have no contradictions and be able to determine the plausibility of [...]

Answer

any unique extension of logic

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We expect plausible reasoning to have no contradictions and be able to determine any unique extension of logic

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#jaynes #plausible-reasoning

Question

A ⇒ B is false only if [...]

Answer

A is true and B is false

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A ⇒ B is false only if A is true and B is false

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#jaynes #plausible-reasoning

Question

The question of [...] arises in a crucial way in the Gödel theorem

Answer

logical deducibility of one proposition from a set of others

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The question of logical deducibility of one proposition from a set of others arises in a crucial way in the Gödel theorem

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#jaynes #plausible-reasoning

Question

The duality property shows **[...]** is the same as denying that they are both false \( \overline{ \bar{A} \bar{B} } \)

Answer

the logic disjunction \( A + B \)

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The duality property shows the logic disjunction A+B is the same as denying that they are both false A¯B¯⎯⎯⎯⎯⎯⎯⎯

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#jaynes #plausible-reasoning

Question

This great difference in the meaning of the word *[...]* in ordinary language and in formal logic is a tricky point that can lead to serious error if it is not properly understood

Answer

implies

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This great difference in the meaning of the word implies in ordinary language and in formal logic is a tricky point that can lead to serious error if it is not properly understood

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Tags

#jaynes #plausible-reasoning

Question

Saying at least one proposition is true \( A + B \) is the same as **[...]**

Answer

denying both are false \( \overline{ \bar{A} \bar{B} } \)

*The duality property*

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The duality property shows the logic disjunction A+B is the same as denying that they are both false A¯B¯⎯⎯⎯⎯⎯⎯⎯

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#has-images

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