The term ‘information’ can mean many things. Here information is understood as factual semantic content, where if s is an instance of semantic content then:

s consists of one or more data

‘Factual’ simply means that the semantic content is about some state of affairs associated with a fact. So entries in Encyclopedias, road signs and maps are some examples of semantic information. Furthermore, such structured data count as pieces of semantic information relative to systems or languages in which they are well-formed and meaningful. For more on this standard definition see [1].

It is important to emphasise that the term ‘information’ is a polysemantic one, applied to a range of phenomena across a range of disciplines. The semantic conception of information used in this paper is but one of many conceptions, drawn from the philosophy of information field. For some alternative views on and surveys of information, including criticism of the view that information is a kind of data, see [2], [3] and [4].

In this paper we consider only semantic information represented by propositions [5], and the inspected approaches can be seen to fall under the logical approach to semantic information theory [4] (p. 315). Hence, semantic information is understood in terms of propositions and throughout this paper represented using statements in a basic propositional logic (i.e., the information that p). Finally, as will become clearer, a veridicality thesis for semantic information is endorsed, where if i counts as an instance of semantic information then it is semantic content that is also true [6].

So information is carried by statements/propositions and judgements such as ‘statement A yields more information or is more informative than statement B’ are naturally made [7]. The motivations for and the aims of this enterprise to develop ways to quantify semantic information can be straightforwardly appreciated. The following cases and considerations will serve to illustrate this:

Imagine a situation in which a six-sided die is rolled and it lands on 4. Given the following collection of statements describing the outcome of the roll, which is the most informative? Which is the least informative? –

The die landed on 1

A formal framework for the quantification of semantic information could assign a numerical measure to each statement and rank accordingly.

Now, there is a coherent sense in which the following ranking of answer statements in terms of highest to lowest informativeness is right:

A ∧ B

A ∧ B precisely describes the truth of the domain, it gives the correct answer to both questions. A is also true but only provides information about one of the items, giving the correct answer to the first question. A ∨ B only tells us that the answer to at least one of the questions is ‘yes’, but not which one. ¬A ∧ ¬B on the other hand is false and would provide an incorrect answer to both questions.

What is a suitable formal framework for the quantification of semantic information that can rigorously capture these intuitions and provide a way to measure the complete range of statements within a domain of inquiry?

Thus motivations and aims for the task of this paper are clear.

Bar-Hillel and Carnap's seminal account of semantic information [8,9] measures the information yield of a statement within a given language in terms of the set of possible states it rules out and a logical probability space over those states.

The general idea is based on the Inverse Relationship Principle, according to which the amount of information associated with a proposition is inversely related to the probability associated with that proposition. This account will henceforth be referred to as the theory of Classical Semantic Information (CSI) [10]. Using some a priori logical probability measure pr on the space of possible states, two measures of information (cont and inf) are provided, such that: cont ( A ) = d f 1 − pr ( A )and inf ( A ) = d f − log 2 ( pr ( A ) )

In order to demonstrate these definitions, take a very simple propositional logical space consisting of three atoms [11]. The space concerns a weather framework, where the three properties of a situation being considered are whether or not it will be (1) hot (h), (2) rainy (r) and (3) windy (w) [12]. Since there are three variable propositions involved, there are eight logically possible ways things can be, there are eight possible states. Here is a truth table depicting this:

State	h	r	w
w1	T	T	T
w2	T	T	F
w3	T	F	T
w4	T	F	F
w5	F	T	T
w6	F	T	F
w7	F	F	T
w8	F	F	F

Now, each possible state can be represented using a state description, a conjunction of atomic statements consisting of each atom in the logical space or its negation, but never both. For example, the state description for w₁ is h ∧ r ∧ w and the state description for w₈ is ¬h ∧ ¬r ∧ ¬w. For the sake of example, the probability distribution used will be a simple uniform one; that is, since there are eight possible states, each state has a probability of 1 8 that it will obtain or be actual [13]. Using these parameters, here are some results:

Statement (A)	cont(A)	inf(A)
h ∧ w ∧ r	0.875	3
¬h ∧ ¬w ∧ ¬r	0.875	3
h ∧ r	0.75	2
h ∨ w	0.25	0.415
h ∨ ¬h	0	0
h ∧ ¬h	1	∞

Note that with the formulas for cont() and inf(), it is clear that the following relationships hold between logical entailment and semantic information, simply due to the fact that A ⊢ B ⇒ pr(A) ≤ pr(B): A ⊢ B ⇒ cont ( A ) ≥ cont ( B ) A ⊢ B ⇒ inf ( A ) ≥ inf ( B )

Bar-Hillel and Carnap also provided some formulas to measure relative information, amongst them: cont ( A ∣ B ) = d f cont ( A ∧ B ) − cont ( B )and inf ( A ∣ B ) = d f inf ( A ∧ B ) − inf ( B )

Given these definitions, it so happens that cont(A∣B) = cont(B ⊃ A) and inf(A∣B) = −log₂(pr(A∣B)).

It is worth briefly noting that Bar-Hillel and Carnap's account was subsequently expanded upon by Jaakko Hintikka. Amongst the contributions made by Hintikka are:

Further investigation into varieties of relative information.

Readers further interested in Hintikka's contributions are advised to consult [14–16].

In part prompted by these considerations, Luciano Floridi has developed a Theory of Strongly Semantic Information (TSSI), which differs fundamentally to CSI. It is termed ‘strongly semantic’ in contrast to Bar-Hillel and Carnap's ‘weakly semantic’ theory because unlike the latter, where truth values do not play a role, with the former semantic information encapsulates truth.

Floridi's basic idea is that the more accurately a statement corresponds to the way things actually are, the more information it yields. Thus information is tied in with the notion of truthlikeness. If a statement perfectly corresponds to the way things actually are, if it completely describes the truth of a domain of inquiry, then it yields the most information. Then there are two extremes. On the one hand, if a statement is necessarily true in virtue of it being a tautology, it yields no information. On the other, if a statement is necessarily false in virtue of it being a contradiction, it also yields no information. Between these two extremes and perfect correspondence there are contingently true statements and contingently false statements, which have varying degrees of information yield.

Using the weather framework introduced above, let the actual state be w₁. The following ranking of propositions, from highest information yield to lowest information yield illustrates this idea:

h ∧ r ∧ w

It is time to take a look in more detail at Floridi's account as given in [17]. The informativeness of each statement s [18] is evaluated as a function of (1) the truth value of s and (2) the degree of semantic deviation (degree of discrepancy) between s and the actual situation w. This degree of discrepancy from the actual situation is measured by a function f, which takes the statement as input and outputs some value in the interval [-1, 1]. This allows for the expression of both positive (when s is true) and negative (when s is false) degrees of discrepancy. Basically, the more a statement deviates from 0, the less informative it is.

Floridi stipulates five conditions, that “any feasible and satisfactory metric will have to satisfy” [17] (p. 206):

Condition 1. For a true s that conforms most precisely and accurately to the actual situation w, f(s) = 0.

For cases that fall under the fourth condition, f measures degree of inaccuracy and is calculated as the negative ratio between the number of false atomic statements (e) in the given statement s and the length (l) of s.

To get values for e and l, it seems that some simplifying assumptions need to be made about a statement's form. Statements to be analysed in this way are in conjunctive normal form, with each conjunct being an atomic statement. e is the number of false conjuncts and l is the total number of conjuncts. This point will be discussed in more detail shortly.

For cases that fall under the fifth condition, f measures degree of vacuity and is calculated as the ratio between the number of situations, including the actual situation, with which s is consistent (n) and the total number of possible situations (S) or states.

Of the inaccuracy metric Floridi writes, “[it] allows us to partition s = s^l into l disjoint classes of inaccurate s {Inac₁, …, Inac_l} and map each class to its corresponding degree of inaccuracy” [17] (p. 208).

The model Floridi uses to illustrate his account, denoted E, has two predicates (G and H) and three objects (a, b and c). So in all there is a set of 64 possible states W = w₁, …, w₆₄. An application of the inaccuracy metric to the model E is presented in Table 1 (degree of informativeness calculations will be explained below).

For vacuous statements, Floridi writes “[the vacuity metric] and the previous method of semantic weakening allow us to partition s into l − 1 disjoint classes Vac = {Vac₁, …, Vac _l−1}, and map each class to its corresponding degree of vacuity” [17] (p. 209). The semantic weakening method referred to consists of generating a set of statements by the following process. In this case, the number of all atomic propositions to be used in a statement is 6. Start with a statement consisting of 5 disjunctions, such as G a ∨ G b ∨ G c ∨ H a ∨ H b ∨ H c

This is the weakest type of statement and corresponds to Vac₁. Next, replace one of the disjunctions with a conjunction, to get something like ( G a ∨ G b ∨ G c ∨ H a ∨ H b ) ∧ H c

This is the second weakest type of statement and corresponds to Vac₂. Continue this generation process until only one disjunction remains. Table 2 summarises an application of this process to the model E:

Say that the actual situation corresponds to the state description Ga ∧ Ha ∧ Gb ∧ Hb ∧ Gc ∧ Hc. Table 3 summarises an example member of each class:

With a way to calculate degrees of vacuity and inaccuracy at hand, Floridi then provides a straightforward way to calculate degrees of informativeness (g), by using the following formula, where once again f stands for the degree of vacuity/inaccuracy function: g ( s ) = 1 − f ( s ) 2

Thus we have the following three conditions:

(f(s) = 0) ⇒ (g(s) = 1)

Furthermore, this is extended and an extra way to measure amounts of semantic information is provided. As this extension is simply derivative and not essential, I will not go into it here. Suffice it to say, naturally, the higher the informativeness of s, the larger the quantity of semantic information it contains; the lower the informativeness of s, the smaller the quantity of semantic information it contains. To calculate this quantity of semantic information contained in s relative to g(s), Floridi makes use of integrals and the area delimited by the equation given for degree of informativeness.

As we have seen so far, the CSI approach to quantifying semantic information does not stipulate a veridicality condition for information. Floridi's TSSI on the other hand holds that semantic information should be meant as implying truth. This paves the way for an alternative approach to quantifying semantic information; rather than measuring information in terms of probability, the information given by a statement is to be measured in terms of its truthlikeness. Discussion in Section 3.1 suggested however that Floridi's contribution in [17] is perhaps of more conceptual rather than technical value, as the metrics provided can make way for some more detailed and refined improvements.

At any rate, there already is a mass of formal work on truthlikeness to draw from [20]. Indeed the truthlikeness research enterprise has existed for at least a few decades now, far before any idea of utilising it to quantify semantic information. Whilst there are quite a few approaches to truthlikeness in the literature, to keep things simple and in some cases for certain technical reasons, I will look at two of them. The presentation given here of these two accounts is kept relatively simple and guided by the task at hand, which is to investigate their applicability to semantic information quantification. While seemingly a simple concept in essence, the notion of truthlikeness has resisted a straightforward formal characterisation. Over the last few decades of research, a variety of rival approaches have developed, each with their own pros and cons. In turn it follows that information quantification via truthlikeness measures is also not going to be a straightforward matter with a definitive account.

We start with a prominent and elegant approach to truthlikeness first proposed by Pavel Tichy and expanded upon by Graham Oddie [21] (p. 44). This approach is an example of approaches that measure the truthlikeness of a statement A by firstly calculating its distance from a statement T using some distance metric Δ, where T is a state description of the actual state (so T is in a sense the truth). Actually, the distance metric is ultimately a function that operates on states. Δ(w_i, w_j) measures the distance between states w_i and w_j and we use the notation Δ(A, T) in effect as shorthand for an operation that reduces to Δ operation on states corresponding to A and T.

The result of this distance calculation is then used to calculate the statement's truthlikeness (Tr); the greater this distance, the less truthlike the statement and vice versa. This inverse relation is simply achieved with the following formula: Tr ( A , T ) = 1 − Δ ( A , T )

To see this approach at work, consider again the canonical weather framework, with w₁ being the actual state (throughout the remainder of this paper w₁ is assumed to be the actual state in all examples):

Before continuing with some examples, the introduction of some terminology is in order. We firstly recall that a state description is a conjunction of atomic statements consisting of each atom in the logical space or its negation, but never both and that each state description corresponds to a state. In our example T = h ∧ r ∧ w.

A statement is in distributive normal form if it consists of a disjunction of the state descriptions of states in which it is true. For example, h ∧ r is true in states w₁ and w₂, so its distributive normal form is (h ∧ r ∧ w) ∨ (h ∧ r ∧ ¬w). For notational convenience throughout the remainder of this paper, when listing a statement in its distributive normal form, state descriptions may be substituted by the states they correspond to. For example, h ∧ r may be represented with the term w₁ ∨ w₂.

Now, take the possible states w₁, w₈ and w₅, which correspond to the state descriptions h ∧ r ∧ w, ¬h ∧ ¬r ∧ ¬w and ¬h ∧ r ∧ w respectively. The difference between w₁ and w₈ is {h, r, w}; they differ in every atomic assertion. w₅ and w₁ on the other hand differ by only {h}. So ¬h ∧ r ∧ w is more truthlike than ¬h ∧ ¬r ∧ ¬w, because the distance between w₅ and the actual state w₁ is less than the distance between w₈ and w₁.

The general formula to calculate the distance between A and T is: (1) Δ ( A , T ) = d | W A |where

Let w_T stand for the state that corresponds to T.

So the statement ¬h∧r∧w (the state description for w₅) has a truthlikeness (information measurement for our purposes) of 2 3 and the statement ¬h ∧ ¬r ∧ ¬w (the state description for w₈) has a truthlikeness (information measurement) of 0.

This approach extends to the complete range of statements involving h, r and w. According to Formula 1, the distance of a statement from the truth is defined as the average distance between each of the states in which the state is true and the actual state. Take the statement h ∧ ¬r, which makes assertions about only 2 of the 3 atomic states. It is true in both w₃ and w₄, or {h, ¬r, w} and {h, ¬r, ¬w} respectively. w₃ has a distance of 0.33 and w₄ has a distance of 0.67 from w₁ so the average distance is 0.33 + 0.67 2 = 0.5. Note that henceforth the truthlikeness function (Tr) will be replaced by an information yield function (info), so that info ( A , T ) = 1 − Δ ( A , T )

Also, given that T is set as w₁ throughout this paper, info(A, T) will generally be abbreviated to info(A).

Table 5 lists some results using this method. How do these results fare as measures of information yield? Statement #1 (complete truth) clearly yields the most information and #21 (complete falsity) clearly yields the least. #10, #11, #15 and #16 indicate that for a disjunctive statement, the more false constituents it has the less information it yields. In general, the more false constituents contained in a formula the more likely a decrease in information yield, as indicated by the difference between #5 and #9.

Statements #7, #17 and #21 make sense; the greater the number of false conjuncts a statement has, the less information it yields, with the upper bound being a statement consisting of nothing but false conjuncts, which is accordingly assigned a measure of 0. Although #17 and #19 have the same measure, #17 has one true conjunct out of three atomic conjuncts and #19 has zero true atomic statements out of one atomic statement. From this it can be said that the assertion of a false statement detracts more from information yield than the absence of an assertion or denial of that statement. Half of #14 is true, so it is appropriately assigned a measure of 0.5. #20 further shows that falsity proportionally detracts from information yield. The difference between #18 and #16 is perhaps the most interesting. Although #18 has one true conjunct out of two and #16 contains no true atoms, #16 comes out as yielding more information, further suggesting the price paid for asserting falsity in a conjunction. Also, note that some false statements yield more information than some true statements. An interpretation of the Tichy/Oddie method, which will provide a way to further understand some of these results, is given in Section 4.6.

One possible issue with this method is that tautologies are not assigned a maximum distance and hence are assigned a non-zero, positive measure. Without going into detail, it seems that this is a more significant issue for a quantitative account of information than it is for an account of truthlikeness. The implication that tautologies have a middle degree of information yield strongly conflicts with the intuition and widely accepted position that tautologies are not informative. Whilst an explanation of this result will be discussed in Section 4.6, the simplest and perhaps best way to get around this issue would be to just exclude tautologies from this metric and assign them a predefined distance of 1 hence information yield of 0. Although an expedient, this move would put the tautology alongside its extreme counterpart the contradiction, which is also excluded. In the case of contradictions (#22), these calculations do not apply, because since they are true in no states this would mean division by zero.

In this section a new, computationally elegant method to measure information along truthlikeness lines is proposed. As will be seen, this value aggregate approach differs to both the Tichy/Oddie and Niiniluoto approaches, lying somewhere in between.

As has been well established by now, each statement corresponds to a set of states W in a logical space. Each of the states in this logical space can be valued, with the actual state being the uniquely highest valued. The rough idea behind the value aggregate approach is a simple one, with two factors concerning state distribution in W involved in determining the information yield of a statement:

the greater the highest valued state of W the better for information yield.

This determination is a two-tier process; first rank a statement according to the first factor and then apply the second. For example, the state description corresponding to the actual state (in our case h ∧ r ∧ w) ranks highest relative to the first factor. It then also ranks highest relative to the second factor because there are no other states of lesser value than w₁ in its state collection. Accordingly, its information yield is maximal.

As another example, the statement h ∨ r, which corresponds to states w₁ - w₆, yields more information than the statement h ∨ ¬r, which corresponds to states w₁ - w₄, w₇, w₈, because although both have the same highest valued state and the same number of lesser valued states, the lesser valued states for h ∨ r are closer to the highest valued state.

Whilst this informal idea is still perhaps somewhat loose, it motivates and can be linked to the formal method that will now be detailed. To begin with, each state is once again assigned a value and ranked, where the value for a state w (val(w)) is determined using the following method:

Let n stand for the number of propositional variables in the logical space

So in the case of our 3-atom logical space with w₁ being the actual state, each state is valued as follows:

val ( w 1 ) = 3 24

Now, given a statement A, its information yield is measured using the following algorithm:

Determine the set of states W in which A holds (W = {w | w |= A})

Following is an example of this value aggregate method.

As mentioned earlier in Section 4.4, Niiniluoto states a number of adequacy conditions “which an explicate of the concept of truthlikeness should satisfy” [26] (p. 232). An investigation into the applicability of these conditions to an account of semantic information quantification (whether all or only some apply) will not be pursued here; apart from perhaps an occasional comment, I do not intend to discuss matter. Suffice it to say, at the least most of them seem applicable.

Nonetheless, in order to initiate this new method I here include a list of these conditions plus a summary of how the value aggregate approach, along with the Tichy/Oddie and Niiniluoto approaches, fare against them. This summary confirms that the value aggregate approach is similar but not equivalent to either of the other two approaches.

The presentation of these conditions will largely conform to their original form, so bear in mind for our purposes that Tr() corresponds to info() and as Δ decreases/increases info() increases/decreases. Before listing the conditions, some terms need to be (re)established:

Niiniluoto uses the term constituent as a more general term for state descriptions.

Here are the conditions:

(M1) (Range) 0 ≤ Tr(A, S_*) ≤ 1.

Table 11 gives a summary of the measures against the adequacy conditions, where +(γ) means that the measure satisfies the given condition with some restriction on the value of γ and (λ) [29]:

The value aggregate method will satisfy those seeking a covariation of information and logical strength. As can be seen, it fails only M11. For example, Δ_*2 = Δ_*3 = Δ_*5, yet info(w₂ ∨ w₃ ∨ w₅) = info(w₂ ∨ w₃) = info(w₂). The value aggregate method, like the Tichy/Oddie method, does not differentiate sets of states such as these. In terms of information, it could be argued that satisfaction of this condition is not essential; removing or adding a state description disjunct in this example does not inform an agent any more or less of the true state.

Using the value aggregate method, a metric for misinformation measurements can simply be obtained by inverting the utility values for each state; where u is original utility of a state, its new utility becomes 1 − u. As with the Tichy/Oddie method, this is equivalent to misinfo(A) = 1 − info(A). Unlike the Tichy/Oddie method though, where there is a perfect symmetry between truth and falsity, the accumulation of values with the aggregate approach results in a skew towards the information measure. As a consequence there are significant differences between the two in terms of misinformation value calculations. For example, whilst the Tichy/Oddie approach gives misinfo(w₁ ∨ w₈) = misinfo(w₄∨w₅) = 0.5, this new approach gives misinfo(w₁∨w₈) = 0.125 < misinfo(w₄∨w₅) = 0.375.

Also, the value aggregate method can be extended to incorporate different utility functions for different purposes, in the same way as discussed in Section 4.7. As described in earlier Section 5, with the value aggregate approach the last element in the array X₁ is the highest valued and is used to fill in the remaining positions of X₂. In the case of w₁ in particular, this means that a collection of states consisting of w₁ and relatively few other states will jump ahead quite quickly given the continued addition of the highest valued element.

The adoption of a non-linear utility function could be used to regulate this. For example, a simple logarithmic utility function such as y = 20log₂(x + 1) (x being the value of the standard utility) places w₁, w₈ below w₁, w₃, w₅, w₇, whereas with the original standard linear utility this order is reversed. Also to note, as long as the actual state is assigned the highest utility, the information measure will covary with logical strength for true statements, whatever utility function is used.

I would like to now briefly discuss the possibilities of (1) using a CSI-style inverse probabilistic approach as a basis for a quantitative account of semantic information (2) combining the CSI approach and truthlikeness approaches.

With regards to (1), there seems to be inherent issues with the CSI inverse probabilistic approach. Thinking about the two approaches to semantic information quantification we have looked at (CSI and truthlikeness), Table 12 depicts a rough conceptual principle that can be extracted. The four possible ways of combining the two factors under consideration are represented using a standard 2×2 matrix, with each possible way being assigned a unique number between 1 and 4. 1 represents ‘best’ for information yield and 4 represents ‘worst’.

Therefore, narrowing down the range of possibilities is a good thing for information yield when done truthfully. On the other hand, if done falsely it is a bad thing. Borrowing an idea from decision theory, this type of reward/punishment system can be formally captured with a simple scoring rule. Take the following, where for a statement A:

if A is true then give it an information measure of cont(A)

Whilst this approach would give acceptable results for true statements, when it comes to false statements it is too coarse and does not make the required distinctions between the different classes of false statements. A state description in which every atom is false is rightly assigned the lowest measure. Furthermore, other false statements which are true in the state corresponding to that false state description will be assigned acceptable measures.

But this approach also has it that any false state description will be accorded the lowest measure. So the state description h ∧ r ∧ ¬w (w₂) would be assigned the same measure as ¬h ∧ ¬r ∧ ¬w (w₈), which is clearly inappropriate. Furthermore, equal magnitude supersets of the states correlating to these state descriptions (e.g., w₂, w₄ and w₄, w₈) would also be assigned equal measures.

Given the aim to factor in considerations of truth value, the problem with any account of information quantification based on a CSI-style inverse probabilistic approach is related to the failure of Popper's content approach to truthlikeness. As Niiniluoto nicely puts it:

among false propositions, increase or decrease of logical strength is neither a sufficient nor necessary condition for increase of truthlikeness. Therefore, any attempt to define truthlikeness merely in terms of truth value and logical deduction fails. More generally, the same holds for definitions in terms of logical probability and information content [30] (p. 296).

With regards to (2), one option is to adopt a hybrid system. As was seen, the CSI approach works well when its application is confined solely to true statements. So perhaps a system which applies the CSI approach to true statements and a truthlikeness approach to false statements could be used. Furthermore, apart from such hybrid systems, there is the possibility of combining calculations from both approaches into one metric. For example, an incorporation of the CSI approach into the value aggregate approach could help distinguish between, say w₂ and w₂, w₃, which as we saw are given the same measure using the value aggregate approach.

Classes of update semantics for logical databases can be divided into formula-based and model-based approaches. Basically, with model-based approaches the semantics of an update for a database are based on the models of that database and with formula-based approaches statements in the database are operated on instead [31] (p. 12).

Borrowing this classification division for our investigation, it can be seen that the approaches discussed so far would fall under a model-based approach to semantic information quantification. Beyond suggesting the possibility of a formula-based approach, I will also outline a method that can be seen to fall on this side of the divide. This experimental outline is more so for the sake of illustration rather than anything else, though it could be useful.

To begin with, statements are first converted to conjunctive normal form. Although this means that logically equivalent statements are treated the same, they are still being dealt with directly and there is no analysis whatsoever of the models that they correspond to.

A logical statement is in conjunctive normal form (CNF) when it consists of a conjunction of disjunctive clauses, with each disjunct in each conjunction being a literal (either an atom or a negated atom). A ∧ B ∧ ¬C and (A ∨ ¬B) ∧ (A ∨ C) are two examples of statements in CNF. Also, here the normalised statements are fully converted to CNF, meaning that all redundancies are eliminated. Here is the procedure:

use equivalences to remove ↔ and →.

Given a normalised statement, how can its information yield be measured? Let us start with the simplest type of statement in CNF, one consisting of a conjunction of literals. Each literal has a value of magnitude 1. At this stage there are two ways to go about this. The first method (Method 1) involves just positively incrementing the total information yield of the statement as each true literal is encountered. The second method (Method 2) involves also decrementing the total information yield of the statement as each false literal is encountered.

Take the statement h ∧ r ∧ ¬w. It contains 2 true literals and 1 false literal. Going by Method 1, the total information yield of this statement is 2. Going by Method 2, where the misinformation instances of a statement count against its total information yield, the total information yield of this statement is 1.

This is all very simple and gives the right results. h ∧ r ∧ w has a maximal information yield of 3. h ∧ r ∧ ¬w and h ∧ r have the same information yield of 2 going by Method 1, but going by method 2 the former is ‘punished’ for its assertion of misinformation and goes down to 1.

The introduction of proper disjunctions makes things more complicated. How can the calculation method reflect the fact that h ∨ r yields less information than other statements such as h and h ∧ r. The more disjuncts a disjunction contains, the less information it yields; information yield is inversely related to the number of disjuncts. Hence, where n is the number of disjuncts in a disjunction, a simple yet potentially suitable measure of the maximum information yield of the disjunction is 1 n. With this way, each literal in the disjunction has a value of magnitude 1 n 2.

To illustrate all of this, take the statement h ∨ r ∨ w. The total information yield of this statement is 3 × 1 9 = 1 3. Given a disjunction with false literals, the results depend of whether Method 1 or Method 2 is adopted. For the statement h ∨ r ∨ ¬w, Method 1 gives a result of 1 3 + 1 3 = 2 3 and Method 2 gives a result of 1 3 + 1 3 − 1 3 = 1 3.

Table 13 lists results using Method 1 and results using Method 2. Clearly these methods are not as refined as the model-based approaches we have looked at, with the variation in assignments of measure values being significantly cruder. Method 1 is clearly just about aggregating truth. Interestingly, the results for Method 2 can be ranked in the same order as the results for the Tichy/Oddie approach in Table 5.

In closing I shall briefly introduce the idea of estimated information. Within the literature a distinction is made between the semantic and epistemic problems of truthlikeness [32] (p. 121):

The semantical problem: “What do we mean if we claim that the theory X is closer to the truth than the theory Y?”

The focus of this paper has been the semantical problem of information quantification; what do we mean when we say that statement A yields more information than statement B? The epistemic problem relates to estimating information yield; given some partial evidence E, are we to estimate that statement A yields more information or less information than statement B?

Of course, an account of the semantical problem is primary, with any account of the epistemic problem being secondary, based on a semantical foundation. Nonetheless a method to estimate information yield is of the highest importance.

In practice judgements of information yield are going to be made with limited evidence, without knowledge of the complete truth (the one true state description). If an agent already knows the complete truth in a domain of inquiry then there would be no new information for them to acquire anyway, no statement in the domain of inquiry could be informative for them. In such scenarios, the actual information yield of any statement is already known, or can be calculated.

In general though, calculations of information yield are of interest to agents who do not know the complete truth and who are seeking new information. When such agents must choose amongst a set of different statements, their aim is to choose the statement that they estimate will yield the most information relative to the actual state (which the agent has limited epistemic access to). In making this estimation and choice the agent will often already posses some evidence and this evidence will rule out certain possible states from consideration in the calculations.

The standard formula for estimated utility in decision theory can be used to calculate the expected information yield of a statement A given prior evidence E: info est ( A ∣ E ) = ∑ i = 1 n info ( A , S i ) × pr ( S i ∣ E )n stands for the number of possible states in the logical space and S_i stands for the state description corresponding to state i.

Given a certain piece of information as evidence, the difference between actual and estimated information yield calculations for a statement can be marked. For example, with info() as the Tichy/Oddie method:

info(h ∧ r ∧ w) = 1

Interestingly, although the completely true h ∧ r ∧ w has a maximum information yield of 1 and the completely false ¬h ∧ ¬r ∧ ¬w has a minimum information yield of 0, even certain true statements used as evidence in estimation calculations will give ¬h ∧ ¬r ∧ ¬w a higher estimated information measure than h ∧ r ∧ w.

The main point of this paper has been the advocacy of quantitative accounts of semantic information based on the notion of truthlikeness. Whilst this is in contrast to traditional inverse probabilistic approaches such as Bar-Hillel and Carnap's CSI, it is not to be seen as an attempt to completely dismiss their legitimacy. Such accounts prove useful in certain applications and effectively capture certain aspects of information, particularly the deep intuition that information is proportional to uncertainty reduction [33].

However there are certain features of an approach such as CSI which are at odds with both ordinary senses of information and criteria associated with certain applications of information. A truthlikeness approach on the other hand strongly resonates with these ordinary senses and satisfies these criteria. Furthermore, such an approach naturally accommodates the corresponding notion of misinformation. As was seen, fortunately there is already a significant body of work on truthlikeness to draw from. Yet also, as evidenced in this paper there is still room for further investigation, experimentation and the development of new quantification methods, with a specific focus on information.

Apart from any suggested or implied above, two possibilities for further research are:

The extension of information as truthlikeness approaches to richer systems beyond classical propositional logic. The methods in this paper that were drawn from the truthlikeness literature were actually extended to first order classical systems and even to systems of second order logic [21,26]. Also what, if any, are the possibilities for non-classical systems?