# Measures of Information

## Abstract

**:**

## 1. Introduction

- evidence from psychology indicates that humans have inherent difficulties in interpreting information;
- information has a geometry that constrains the possible approaches to truth;
- the interpretation of truth, like all concepts, is Ecosystem-dependent so we should expect different approaches to truth in different Ecosystems.

## 2. Summary of the Model for Information (MfI)

- how well does x represent what it is intended to represent?
- how useful is x in what an IE needs to achieve?

- properties: each slice has a set of properties and for each property there is one or more physical processes (called measurement) that takes the slice as input and generates the value of the property for the slice;
- entity: a type of slice corresponding to a substance;
- interaction: a physical process that relates the properties of two entities.

- the symbols that are used;
- the structure of IAs and the rules that apply to them;
- the ways in which concepts are connected;
- the ways in which IAs are created and parsed (embodying the rules for creating any compatible IA);
- the channels that are used to interact.

## 3. Interpretation and Connection Strategy

#### 3.1. Connection Strategy

- (a)
- Context: as well as the IA under consideration, how much else of the external environment is taken into account?
- (b)
- Search space: there are choices about how widely to look. An IE can look in various subsets of content memory or event memory. It can look elsewhere in the Ecosystem or more generally in the environment.
- (c)
- Options: how many options will be entertained? Is the first good enough or are all of them needed?
- (d)
- Success threshold: when is a connection good enough to qualify for the interpretation? Is there any kind of quality threshold that must be met?

“…most of what you (your System 2) think and do originates in your System 1, but System 2 takes over when things get difficult, and it normally has the last word. … System 1 has biases, however, systematic errors that it is prone to make in specified circumstances.”

As a result, we cannot reliably trust our System 1 (but often do so without realising it).“System 1 is radically insensitive to both the quality and quantity of information that gives rise to impressions and intuitions.”

- the abstraction pattern of an IE (e.g., DNA, system architecture);
- the history of interactions of an IE (e.g., education, market success);
- the elements of an IA.

- questions and answers (in conversation);
- cross-examination (in the legal process);
- peer review (in the academic world);
- thought and introspection.

#### 3.2. Derivation

- (a)
- Measurement process: possibly at many different times and using many different measurement processes relating to different Ecosystems a number of properties of slices are measured;
- (b)
- Processing by an IE: each IE (potentially amongst many) senses and analyses the input and then creates one or more IAs;
- (c)
- Transmission: the IAs can be transmitted on various channels and in various modes (e.g., one-to-one or one-to-many);
- (d)
- Processing: intermediate IEs may convert between Ecosystems, combine IAs or process them in other ways;
- (e)
- Combination: one such IE produces the final IA (f).

- in IT, there are rigorous processes for designing, building and testing systems [10]—ensuring that the subsequent derivation of information is robust;
- in legal processes there are rules for handling evidence so that information presented in court has a reliable derivation;
- scientists pay considerable attention to the design of experiments so that measurement is as reliable as possible.

## 4. Measures

#### 4.1. Definitions

- atom: the fundamental indivisible level of content in the Modelling Tool (note that this definition is different from the definition of “atom” in [11]);
- chunk: groups of atoms;
- assertion: the smallest piece of content that is a piece of information—containing two chunks in a manner that conforms to an assertion abstraction pattern;
- passage: a related sequence of assertions.

- M is a Modelling Tool (and all of the subsequent definitions are with respect to M);
- Γ
_{Μ}is the set of content in M; - Γ
_{γΜ }is the set of chunk content in M and γ, δ ∈ Γ_{γΜ}; - Γ
_{αΜ }is the set of assertion content in M and α, β ∈ Γ_{αΜ}where α = (γ, ρ, δ) and ρ is an assertion abstraction pattern; - Γ
_{πΜ }is the set of passage content in M and π ∈ Γ_{πΜ}; - Ξ is the set of set theoretic relationships;
- Σ is the set of all slices;
- T is a set of discrete times and t ∈ T;
- φ
_{γ}: Γ_{γΜ}× T → 2^{Σ}, is a chunk interpretation; Φ_{γ}is the set of all chunk interpretations (for M); - φ
_{α}: Γ_{αΜ}× T → 2^{Σ}× Ξ × 2^{Σ}, is an assertion interpretation; Φ_{α}is the set of all assertion interpretations (for M); - φ
_{π}: Γ_{πΜ}× T → f (2^{Σ}, …, 2^{Σ}), is a passage interpretation where f is a Boolean expression (containing ANDs); Φ_{π}is the set of all passage interpretations (for M); - φ
_{eγ, }φ_{eα, }φ_{eπ}are the corresponding the Ecosystem interpretations.

#### 4.2. Chunks, Assertions, Passages and Interpretations

^{c}(where A

^{c}is the absolute complement of A). By contrast, interpretations of assertions use set containment and equality comparisons (⊆, ⊇, ⊂, ⊃, =, ≠).

- the relations implicit in the assertion;
- synonyms (i.e., chunks for which the interpretation is the same);
- discrimination (i.e., chunks for which the interpretation is different).

- the φ-closure of γ, σ
_{φ}(γ) = {δ ∈ Γ_{γΜ}: φ (δ, t) = φ (γ, t)}; - the closure of γ, σ(γ) = {δ ∈ Γ
_{γΜ}: φ_{e}(δ, t) = φ_{e}(γ, t)}; - Γ
_{σΜ}= {σ(γ): γ ∈ Γ_{γΜ}};

- φ is MT-closed if σ(γ) = σ
_{φ}(γ) for all γ ∈ Γ_{γΜ}; - φ is MT-structured if φ maps to the same relation as φ
_{e}for all passages; - φ is MT-consistent if φ is MT-closed and MT-structured.

#### 4.3. Boolean Passages

- α OR β;
- IF α THEN β.

_{βΜ}.

_{1}) for the assertions (perhaps a language) and another (M

_{2}) that includes the Boolean operations and Boolean expressions. Such an IA is described as a Composite IA [1]. In this case, the complexity raised above does not arise. In M

_{2}, the chunks are M

_{1}assertions (as in “IF P AND Q THEN R” in which P, Q and R are chunks for M

_{2}). In this case, there is a chance that the two Modelling Tools are interpreted independently and the results combined separately. This distinction corresponds to a question raised by Popper in [12] that is discussed in Section 5. In terms of the connection strategy discussed above, the search space may well be different if two Modelling Tools are involved rather than just one.

_{n}is a set of Modelling Tools that support a range of non-Boolean operations and expressions. Composite IAs produced from M

_{n}will allow different overall interpretations depending on the degree to which the interpretations from the different Modelling Tools are combined (or not). This leads to the question: what is the effect on IQ if the different interpretations are independent or in different combinations? This approach may enable an analysis of complex, recursive passages or generalisations of Popper’s question. This is a topic for further research.

#### 4.4. Measures for Chunks—Accuracy

^{Σ}induced by set inclusion and the corresponding partial order on Σ’.

_{e}is the Ecosystem interpretation and φ, φ' are chunk interpretations. For chunks, we can define the following:

- φ is γ-accurate (with respect to t) if φ (γ, t) = φ
_{e}(γ, t) ≠ ∅; - φ is γ-inaccurate (with respect to t) if φ (γ, t) ∩ φ
_{e}(λ, t) = ∅ ; - φ is more γ-⊂-accurate than φ' (with respect to t) if φ' (γ, t) ⊂ φ (γ, t) ⊆ φ
_{e}(γ, t); - φ is more γ-⊃-accurate than φ' (with respect to t) if φ
_{e}(γ, t) ⊆ φ (γ, t) ⊂ φ' (γ, t); - φ is more γ-accurate than φ' (with respect to t) if it is more γ-⊂-accurate or more γ-⊃-accurate (note that it cannot be both);
- φ is γ-partially accurate (with respect to t) if φ (γ, t) ∩ φ
_{e}(λ, t) ≠ ∅ and φ (γ, t) φ_{e}(λ, t) ≠ ∅.

_{γΜ}. Figure 5 shows some of these definitions diagrammatically. We can use these definitions to compare interpretations. Define:

- φ <
_{acc,t}φ' if φ' is more accurate than φ; - φ =
_{acc,t}φ' if φ = φ'; - ≤
_{acc,t}, >_{acc,t}, ≥_{acc,t}correspondingly.

_{acc,t}is anti-symmetric—suppose that

- φ <
_{acc,t}φ' and φ' ≤_{acc,t}φ; then - φ ≠
_{acc,t}φ' and there is some γ for which (depending on the type of relative accuracy) either- φ (γ, t) ⊂ φ' (γ, t) ⊆ φ
_{e}(γ, t), or - φ
_{e}(γ, t) ⊆ φ' (γ, t) ⊂ φ (γ, t).

_{acc,t}φ. So ≤

_{acc,t}provides a partial order for Φ

_{γ}for any time t (other partial order requirements are guaranteed by set inclusion).

#### 4.5. Measures for Chunks—Precision

- the ∩
_{T’}-range of φ with respect to γ, ∩_{T’}-ran (φ, γ) = {σ: σ ∈ φ (γ, t) ∀ t ∈ T’}; - the ∪
_{T’}-range of φ with respect to γ, ∪_{T’}-ran (φ, γ) = {σ: ∃ t ∈ T’ such that σ ∈ φ (γ, t)}; - φ is γ-precise (with respect to T’) if ∩
_{T’}-ran (φ, γ) = ∪_{T’}-ran (φ, γ); - φ is γ-imprecise (with respect to T’) if ∩
_{T’}-ran (φ, γ) = ∅; - φ is more γ-precise than φ' (with respect to T’) if
- ∩
_{T’}-ran (φ, γ) ⊃ ∩_{T’}-ran (φ', γ) and ∪_{T’}-ran (φ, γ) ⊆ ∪_{T’}-ran (φ', γ), or - ∩
_{T’}-ran (φ, γ) ⊇ ∩_{T’}-ran (φ', γ) and ∪_{T’}-ran (φ, γ) ⊂ ∪_{T’}-ran (φ', γ);

- φ is as γ-precise as φ' (with respect to T’) if
- ∩
_{T’}-ran (φ, γ) ⊇ ∩_{T’}-ran (φ', γ) and ∪_{T’}-ran (φ, γ) ⊆ ∪_{T’}-ran (φ', γ).

_{γΜ. }Again we can compare interpretations. Define:

- φ <
_{pre,T’}φ' if φ' is more precise than φ' for time period T’; - φ =
_{pre,T’}φ' if φ = φ'; - ≤
_{pre,T’}, >_{pre,T’}, ≥_{pre,T’}correspondingly.

_{pre,T’}is anti-symmetric because if φ is more γ-precise than φ' then it cannot be true that φ' is as γ-precise as φ. So ≤

_{pre,T’}is a partial order for Φ

_{γ}.

_{pre,[t]}instead of ≤

_{pre,T’}.

#### 4.6. Measures for Chunks—Coverage

- γ >
_{cov}γ’ if γ has greater coverage than γ'; - ≥
_{cov}, ≤_{cov}, <_{cov}correspondingly.

_{cov}γ’ and γ’ ≤

_{cov}γ and we cannot deduce that γ = γ’. But if we define the set

- σ’(γ) = {δ ∈ Γ
_{γΜ}: γ ≤_{cov}δ and δ ≤_{cov}γ}

_{cov}is a partial order for Γ

_{σ’Μ}= {σ’(γ): γ ∈ Γ

_{γΜ}}. In this case we clearly have σ’(γ) ⊆ σ(γ). But the definition of exact synonyms is that they are interpreted in the same way so σ᾽(γ) = σ(γ).

#### 4.7. Measures for Chunks—Resolution

- φ resolves as well as φ' (with respect to t) if for all γ ∈ Γ
_{γΜ}, φ (γ, t) ⊆ φ' (γ, t); (note that this is a different definition from that in [1] but it expresses the concept more clearly).

- φ ≤
_{res,t}φ' if φ resolves as well as φ' (for t); - ≥
_{res,t}, <_{res,t}, >_{res,t}correspondingly.

_{res,t}is a partial order for Φ

_{γ}. (As before, by insisting that resolution applies to all chunks we can guarantee that ≤

_{res,t}is anti-symmetric.)

#### 4.8. Measures for Chunks—Timeliness

_{tim}.

#### 4.9. Measures for Chunks—Overall

_{a1}and ≤

_{a2}are partial orders on A, ≤

_{b}is a partial order on B and C = A × B. Define:

- a ≤
_{a}a’ if a ≤_{a1}a’ and a ≤_{a2}a’ for a, a’ ∈ A; - c ≤
_{c}c’ if a ≤_{a1}a’ and b ≤_{b}b’ for c = (a, b), c’ = (a’, b’), a, a’ ∈ A, b, b’ ∈ B.

_{a}is the sum of ≤

_{a1}and ≤

_{a2}and ≤

_{c}is the product of ≤

_{a1}and ≤

_{b}and both are partial orders. Table 1 lists the orders described above and some combinations.

Order | Definition | Assumes | Applies to |

≤_{acc,t} | As above | Fixed t | Φ_{γ} |

≤_{pre,T’} | As above—precision applies to interpretations of all instances of a chunk in the time period in question | Range T’ | Φ_{γ} |

≤_{res,t} | As above | Fixed t | Φ_{γ} |

≤_{cov} | As above | - | Γ_{σ} |

≤_{tim} | As above | - | T |

≤_{cq,t} | sum (≤_{acc,t}, ≤_{res,t}) | Fixed t | Φ_{γ} |

≤_{ci,t} | product (≤_{cov}, ≤_{res,t}) | Fixed t | Γ_{σ}Φ_{γ} = Γ_{σ} × Φ_{γ} |

_{cq,t }measures the quality of the interpretation of a chunk. ≤

_{ci,t}measures both the coverage of a chunk (≤

_{cov}) and how much of that potential has been converted in the interpretation (≤

_{res,t}).

#### 4.10. Measures for Assertions

- φ (α, t) = (φ (γ, t), r, φ (δ, t)).

- α is (not) φ-plausible (with respect to t) if r (φ (γ, t), φ (δ, t)) is (not) satisfied;
- α is (not) plausible (with respect to t) if r (φ
_{e }(γ, t), φ_{e}(δ, t)) is (not) satisfied.

Order | Definition | Applies to |

≤_{cq,[t]} | sum (≤_{pre,[t]}, ≤_{acc,t}, ≤_{res,t}) | Φ_{γ} |

≤_{aq,[t]} | product (≤_{cq,[t]}, ≤_{cq,[t]}) | Φ_{γ} × Φ_{γ} |

≤_{ai,[t]} | product (≤_{ci,t}, ≤_{ci,t}) | Γ_{σ}Φ_{γ} × Γ_{σ}Φ_{γ} |

_{cq,[t]}in the table, the [t] subscript in ≤

_{pre,[t]}denotes that precision is measured over all of the chunks in a group in the assertion. This measure is of relatively little value for assertions but will assume a greater importance when we consider passages below.

_{aq,[t]}) measures the quality of the chunks in the assertion but still just works in the direction of the y-axis in Figure 7. The final order (≤

_{ai,[t]}) measures the overall coverage and resolution of the chunks in the assertion.

- P = Γ
_{αΜ}× Φ_{α}× T; - P
_{αt0}= {φ ∈ Φ_{α}: α is not φ-plausible (with respect to t)} and similarly P_{αt1}for which α is φ-plausible; - P
_{αt}= P_{αt0 }∪P_{αt1}.

_{αt0}and P

_{αt1}are disjoint, so define ≤

_{ar,α[t]}as follows:

- if φ, φ' ∈ P
_{αt0}, then φ ≤_{ar,α[t]}φ' if φ ≥_{aq,[t]}φ'; - if φ, φ' ∈ P
_{αt1}, then φ ≤_{ar,α[t]}φ' if φ ≤_{aq,[t]}φ'; - if φ ∈ P
_{αt0}and φ' ∈ P_{αt1}, then φ <_{ar,α[t]}φ'.

_{ar,α[t]}is a partial order that enables the comparison of the reliability of plausibility between different interpretations of the same assertion.

#### 4.11. Measures for Boolean Passages

- β
_{1}= f (α_{1i}) is a passage and α_{1i}= (γ_{1i}, ρ_{1i}, δ_{1i}); - β
_{2}= f (α_{2j}) is a passage and α_{2j}= (γ_{2j}, ρ_{2j}, δ_{2j});

- β
_{1}≤_{na}β_{2}if α_{1i}∈ {α_{2j}for all j} for all i.

_{2}is constructed from a superset of the assertions in β

_{1}.

Order | Definition | Applies to |

≤_{na} | As above | Γ_{β} |

≤_{pcov} | product (≤_{cov}, …, ≤_{cov}) | Γ_{β} |

≤_{pc} | sum (≤_{na}, ≤_{pcov}) | Γ_{β} |

≤_{pq,[t]} | product (≤_{cq,[t]}, …, ≤_{cq,[t]}) | Φ_{β} |

≤_{pi,[t]} | product (≤_{ci,[t]}, …, ≤_{ci,[t]}) | Γ_{σ}_{n}Φ_{β}_{n} = Γ_{σ}Φ_{γ}× … × Γ_{σ}Φ_{γ} |

≤_{pi2,[t]} | product (≤_{na}, ≤_{pi,[t]}) | Γ_{βn}× Γ_{σn}Φ_{βn} |

≤_{pu,[t]} | product (≤_{pc}, ≤_{pq,[t]}) | Γ_{βn}× Φ_{βn} |

_{pcov}applies to Γ

_{β}, but how can this be if two Boolean passages are of different sizes? Strictly, ≤

_{pcov}applies to Γ

_{σ}× … × Γ

_{σ }but two projections map Γ

_{β }first to Γ

_{γ}× … × Γ

_{γ}then to Γ

_{σ}× … × Γ

_{σ}so we can apply ≤

_{pcov}to Γ

_{β}. Now suppose that β

_{1}, β

_{2}are Boolean passages with β

_{1}larger than β

_{2}(in the sense that it contains more assertions) and β

_{11}is the initial piece of β

_{1}that corresponds to the size of β

_{2}. Then define ≤

_{pcov}for β

_{1}, β

_{2}as follows:

- if β
_{11}≥_{pcov}β_{2}then β_{1}>_{pcov}β_{2}; - if β
_{11}<_{pcov}β_{2}then β_{1}and β_{2}are not comparable; - if β
_{11}and β_{2}are not comparable then β_{1}and β_{2}are not comparable.

_{pq,[t]}for β

_{1}, β

_{2}as follows:

- if β
_{11}≤_{pq,[t]}β_{2}then β_{1}<_{pq,[t]}β_{2}; - if β
_{11}>_{pq,[t]}β_{2}then β_{1}and β_{2}are not comparable; - if β
_{11}and β_{2}are not comparable then β_{1}and β_{2}are not comparable.

_{pi,[t]}and ≤

_{pu,[t]}? In this case, coverage increases with the number of chunks but quality does not, so we need to restrict comparison to Boolean passages of the same size (in terms of assertions). This is denoted by the n suffix in the third column of Table 3.

- β = f (α
_{i}); - α
_{i}= (γ_{i}, ρ_{i}, δ_{i}); - φ (α
_{i}, t) = (φ (γ_{i}, t), r_{i}, φ (δ_{i}, t)), where r_{i}corresponds to ρ_{i};

- β is (not) φ-plausible (with respect to t) if f (r
_{i}(φ (γ_{i}, t), φ (δ_{i}, t))) is (not) satisfied; - β is (not) plausible (with respect to t) if f (r
_{i}(φ_{e}(γ_{i}, t), φ_{e}(δ_{i}, t))) is (not) satisfied.

- Q = Γ
_{βΜ}× Φ_{β}× T; - Q
_{βt0}= {φ ∈ Φ_{β}: β is not φ-plausible (with respect to t)} and similarly Q_{βt1}; - Q
_{βt}= Q_{βt0}∪ Q_{βt1}.

_{βt0}and Q

_{βt1}are disjoint, so define ≤

_{pr,β[t]}as follows:

- if φ, φ' ∈ Q
_{βt0}, then φ ≤_{pr,β[t]}φ' if φ ≥_{pq,[t]}φ'; - if φ, φ' ∈ Q
_{βt1}, then φ ≤_{pr,β[t]}φ' if φ ≤_{pq,[t]}φ'; - if φ ∈ Q
_{βt0}and φ' ∈ Q_{pt1}, then φ <_{pr,β[t]}φ'.

_{pr,β[t]}is a partial order that enables the comparison of the reliability of plausibility between elements of Q

_{βt}.

#### 4.12. Restricting Measures

#### 4.13. Summary of Orders

## 5. Truth and Truthlikeness

“Whether an informational theory could explain truth more satisfactorily than other current approaches” and

MfI allows truth to be examined from a number of different viewpoints and supplies some constraints (of the type that Floridi refers to). Truth is expressed using IAs and IAs are interpreted by IEs. So, truth (for an IE) is related to both an IA and its interpretation and the nature of truth cannot be disentangled from the quality of information. There are several implications of this.“if that is answered in the negative, whether an informational approach could at least help to clarify the theoretical constraints to be satisfied by other approaches.”

The “accepted examples” and “coherent traditions” he mentions correspond to Information Ecosystems.“I mean to suggest that some accepted examples of scientific practice—examples which include law, theory, application, and instrumentation together—provide models from which spring coherent traditions of scientific research.”

#### 5.1. Measures of Information—Truth

_{pr,β[t]}). This criterion corresponds to those used in different Ecosystems including, for example:

- the law—“beyond reasonable doubt”;
- science—for example, the six-sigma criterion used in particle physics experiments like the test for the Higgs boson.

_{tim}or some variant of it.

_{pi2,[t]}.

#### 5.2. Measures of Information—Truthlikeness

- (1)
- “the room contains 150 people” is plausible;
- (2)
- “the room contains about 150 people” is also plausible;
- (3)
- “the room contains 151 people” is implausible but close (in some sense).

- P = Γ
_{αΜ}× Φ_{α}× T; - pl: P → {0, 1} where “0” represents “not plausible” and “1” represents “plausible”;
- α
_{i}= (γ, ρ, δ_{i}) for 1 ≤ i ≤ n where the δ_{i}correspond to values for a single property and Α = {α_{i}}; - Q
_{t0}= {α_{i}∈ Γ_{αΜ}: pl (α_{i}, φ_{e}, t) = 0} and similarly Q_{t1}; - Q
_{t}= Q_{t0}∪ Q_{t1}.

_{1}, α’

_{2}, α’

_{2}}, Α’ ⊆ Α and Α’ has the following property:

- α’
_{1}>_{cov}α’_{2}<_{cov}α’_{3}; - α’
_{1}∈ Q_{t0}; - α’
_{2}, α’_{3}∈ Q_{t1}.

- α ∈ Q
_{t0}; - α’
_{1}>_{cov }α >_{cov }α’_{2 }<_{cov }α’_{3}.

_{1}. And a corresponding statement applies if:

- α ∈ Q
_{t1}; - α’
_{2}<_{cov}α <_{cov}α’_{3}.

## 6. The Amount of Information

#### 6.1. The Bar-Hillel Carnap Paradox (BCP)

So the BCP leads to the following questions:“According to the classic quantitative theory of semantic information, there is more information in a contradiction than in a contingently true statement.”

- what does “more information” refer to?
- what is a contradiction?
- what does it mean in terms of MfI and the measures above?

_{ci,t}), the amount of information relates to the degree to which the chunks constrain and how well the constraint is resolved in interpretation. Chunks can be combined into ever more constraining specifications. For assertions and Boolean passages it is also a question of volume—how many assertions form the Boolean passage (using the order ≤

_{na}). These can be combined for Boolean passages (in the order ≤

_{pi2,[t]}).

_{cq,t}) is low.

_{pu,[t]}defined in Table 3 includes these elements (although it uses ≤

_{pc}rather than ≤

_{pi2,[t]}because resolution is already taken into account in ≤

_{pq,[t]}).

#### 6.2. The Scandal of Deduction

This is discussed further in [18,19]. When the scandal of deduction is viewed through the lens of MfI it prompts a number of questions including, for example, the following:“According to the received view, logical deduction never increases (semantic) information. This tenet clashes with the intuitive idea that deductive arguments are useful just because, by their means, we obtain information that we did not possess before.”

- “Logical deduction never increases (semantic) information”—for whom?
- What does increased information mean?

α_{1} = “P” α _{2} = “P IMPLIES Q” α _{3} = “Q” | IE memory contains… | ||||

- | α_{2} | α_{3} | α_{2}, α_{3} | ||

IA contains… | α_{1} | 1 | 2 | 3 | 4 |

α_{1}, α_{2} | 5 | 6 | 7 | 8 |

- (a)
- the IE makes a connection with the content during recognition (depending on the connection strategy) but does not store the content in memory;
- (b)
- the content is stored in memory as a result of recognition (as a by-product of the Modelling Tool processing, again depending on connection strategy);
- (c)
- the content is stored in memory as a result of processing (again depending on the nature of the processing).

- the amount in the IA;
- how much is added to the memory of the IE and, in particular, whether either or both of the following are added (assuming that “P” is added in each case):
- ○
- “P IMPLIES Q”;
- ○
- “Q”.

Option | Amount of information in IA | Possibility that “P IMPLIES Q” added | Possibility that “Q” added |

1 | Less | No | No |

2 | Less | No | c |

3 | Less | No | No |

4 | Less | No | No |

5 | More | a, b | a, b, c |

6 | More | No | a, b, c |

7 | More | a, b | No |

8 | More | No | No |

_{1}, α

_{2}are not reliably plausible then α

_{3}is not a reliable deduction.

## 7. Conclusions

## Acknowledgments

## Conflicts of Interest

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Walton, P. Measures of Information. *Information* **2015**, *6*, 23-48.
https://doi.org/10.3390/info6010023

**AMA Style**

Walton P. Measures of Information. *Information*. 2015; 6(1):23-48.
https://doi.org/10.3390/info6010023

**Chicago/Turabian Style**

Walton, Paul. 2015. "Measures of Information" *Information* 6, no. 1: 23-48.
https://doi.org/10.3390/info6010023