3.1. Bar-Hillel/Carnap’s Theory of Information

The Bar-Hillel/Carnap theory treats information as “synonymous with the content of [a] sentence, normalized in a certain way” (Bar-Hillel and Carnap Reference Bar-Hillel and Carnap1952, 1). If we ask what carries information, the answer is sentences, statements, and propositions. The work that this theory does is explaining what information is associated with which statements. Information is analyzed relative to some logical probability space, where statements describe actual and possible worlds. It is assigned to statements about events, and the selected information measure depends on the logical probability of events or some properties of an object the statement is about. Every statement is a tautology, a contradiction, or logically indeterminate (or contingent; Bar-Hillel and Carnap Reference Bar-Hillel and Carnap1953, 156–57). According to IRP, which is at the heart of this theory, the less likely a message is, the more informative it is (Barwise Reference Barwise1997, 14).Footnote ⁴ The notion of likelihood here is a logical one that is independent of actual commitments. It is best thought of as a measure of the extent of logical space a theory occupies; the more space it occupies, the less informational it is.

To see the criticality of IRP for the Bar-Hillel/Carnap theory, we briefly introduce their two main measures of information. The first is Cont(p) and it is formed by the set of all possible worlds excluded by the statement p. It is defined as the complement of the a priori probability of p:

The second measure, Inf(p), is analogous to the definition of the amount of information in the Mathematical Theory of Communication,Footnote ⁵ only that it is based on inductive probability rather than statistical probability (Bar-Hillel and Carnap Reference Bar-Hillel and Carnap1953, 151):

A curious result of the IRP is that contradictions contain maximal information. According to the Bar-Hillel/Carnap theory, since a tautology does not exclude any possible world and corresponds to the logical probability 1, it does not contain any information. But, a self-contradictory statement excludes all possible worlds and corresponds to the logical probability 0 and, thus, contains infinite information.Footnote ⁶ It follows that a noncontradictory statement can never be more informational than a contradictory statement. By making a statement less and less probable, according to the IRP, it gradually increases its informational content to the point where it is self-contradictory and “too informative to be true” (Floridi Reference Floridi2011, 109; italics added).

BCP is presented in the context of a probabilistic theory of information. Nevertheless, this so-called paradox would arise for the same reason for any theory that shares an algebraic structure with the Bar-Hillel/Carnap theory but is not associated with the measure(s) above that would make it a probabilistic theory. Any classical possible worlds account of information, in which the worlds are both consistent and complete, would be similarly vulnerable to BCP.

3.2. The Theory of Strongly Semantic Information

Information, on TSSI, is carried by data rather than just statements as in Bar-Hillel and Carnap’s theory, for maps, formulas, diagrams, pictures, and other semiotic constructs can also convey information. For simplicity, a datum is taken here as lack of uniformity in the real world. Floridi’s examples of data include a black dot on a white page, the presence of some noise, a light in the dark, or a binary 0 as opposed to a 1 (Floridi Reference Floridi2010a, 22–23). He defines a datum as two distinct uninterpreted variables in a domain that is left open to further interpretation (Floridi Reference Floridi2011, 85). When data are processed by some “information agent,” they can be viewed as “constraining affordances” (87). Data are affordances for the processing agent insofar as she can capitalize on them in forming information. They are constraints depending on the interaction with, and the nature of, the agent.

Floridi argues that for nonempty data to qualify as semantic content they need to have two additional properties: (1) be well formed and (2) be meaningful (Reference Floridi2010a, 20). Well-formedness means that the “data are rightly put together, according to the rules (syntax) that govern the chosen system, code or language being used. Syntax here must be understood broadly, not just linguistically” (20–21). Meaningfulness means that the “data must comply with the meanings (semantics) of the chosen system, code or language in question. … The data constituting information can be meaningful independently of an informee” (21–22). This definition faces serious problems. However, since they are less critical for our current argument, we accept Floridi’s definition of semantic content as meaningful structured data.Footnote ⁷

Semantic content qualifies as Floridian information only when it is also veridical. The added value, according to Floridi, is that information takes us one step closer to knowledge—as justified true belief. “For semantic information to be upgraded to knowledge, it is necessary and sufficient to be embedded in a network of questions and answers that correctly accounts for it” (Floridi Reference Floridi2011, 268). Also, since a self-contradictory statement is necessarily false, it cannot qualify as information, thereby avoiding BCP. Contrary to the Bar-Hillel/Carnap theory, TSSI is based on truth-values and degrees of discrepancy of semantic content with respect to a given situation. The informational content of p is calculated in terms of p’s distance from some event or situation S that is described by p (119). ‘Total distance’ is equivalent to p being true in all possible worlds; that is, p is a tautology and is minimally informative. ‘Maximum closeness’ is equivalent to q giving the precise description of S (at the relevant level of abstraction) and is, thus, maximally informative.

Floridi argues that two modifications in the understanding of Cont(p) above are required for avoiding BCP. First, Cont(p) is not a measure of the quantity of information but rather of data (or uninterpreted information) in p. And Cont(p) can be correctly construed as a measure of the quantity of interpretable data in p only when p is construed as “syntactically well-formed strings of data [that] are not already known to be either necessarily true or false a priori” (Floridi Reference Floridi2011, 129). Second, given that Bar-Hillel and Carnap’s theory only deals with interpretable data and their possible combinations a priori, the general equation of Cont(p) “cannot be assumed to have a precise value independently of its frame of reference” (129).

4.1. Informativeness and Information

Should ‘informativeness’ and ‘information’ be treated interchangeably as though they are synonymous? It might seem, at least prima facie, that they are treated as separate notions on TSSI. Floridi argues, for example, that “silence may be very informative. This is a peculiarity of information: its absence may also be informative” (Floridi Reference Floridi2011, 88). If the absence of information can be very informative, then ‘information’ should not be equated with ‘informative’. But Floridi also argues that “the amount of informativeness of each [message] can be evaluated absolutely, as a function of (a) … the alethic value possessed by [the message] and (b) the degree of discrepancy … between [the message] and a given state of the world” (118).

Yet, TSSI treats ‘informativeness’ and ‘information’ synonymously. For Floridi further argues that if a message “has a very high degree of informativeness, … we want to be able to say that it contains a large quantity of semantic information and, vice versa, the lower the degree of informativeness of [the message] is, the smaller the quantity of semantic information conveyed by [it] should be” (Reference Floridi2011, 125). Moreover, he claims that, on TSSI, information need not be connected “to utility-functions for decision-making processes” (83). False information can support decision-making processes, since it “can still be pragmatically interesting” (94). Floridi seems to adopt the distinction between pragmatic and semantic information offered in (Sneed Reference Sneed1967). Accordingly, the latter concerns “truth-conditions” and the former concerns “certain properties possessed by people [such as the] property which a proposition has in virtue of the usefulness to some individual of knowing or believing” (393; italics added).

Like ‘information’, ‘informative’ is an ambiguous notion. It can mean ‘information’ but also ‘useful information’ (or ‘pragmatic information’ in Sneed’s terms). For our purposes here, ‘informative’ is interpreted as ‘useful information’. Once we make that move, (1) BCP should not be seen as paradoxical, and (2) ‘information’ can earn its explanatory keep in cognitive science.Footnote ⁸ In the brief discussion that follows we defend claims 1 and 2. What counts as information depends on the kind of cognitive agent the receiver is. Information is a relational notion that requires a receiver (any instance of signification/information contains an object and a signal but also an “interpretant”; Peirce Reference Peirce1868). Whether semantic content is also informative depends on the prior informational state of the receiver upon receiving it—as is shown by the following two examples.

Consider a scenario in which the same message is sent twice by the same information source: a message and its faithful copy. The two messages carry the same information content. Nevertheless, only the message that is successfully received by the receiver first is informative. Of course, receiving the second message—with the very same information content—can still be useful. For example, in the presence of noise, the first message could have been distorted during transmission.Footnote ⁹ Moreover, in some contexts, each of the two messages, arguably, carries additional meta-information that is its temporal indexing: message 1 was sent at T_x and the second at T_y. This temporal indexing might also be pragmatically significant: it may tell the receiver that some state of the information source has remained unchanged. (Note that if each message includes a timestamp as part of its content, the information content of the two messages is different.) Nevertheless, all this is meta-information in addition to the information content of each of the individual messages.

Next, consider another scenario in which a sender produces and transmits a message to a group of receivers and among them is the sender. Assume further that the communication channel is noiseless. Each of the receivers, but the sender, receiving the message for the first time learns a new fact about the situation described. The message is useful and, therefore, informative to all the receivers except for the sender. The informational content of the message is the same in every case, yet it is only informative to those receivers who were “surprised” by the receipt of the message. The sender already knew the relevant fact when she received the message. The message was uninformative, and at the same time informational, for the sender.

With this distinction in mind, we see that while BCP is counterintuitive at first blush, it is not paradoxical. A contradiction is maximally informational, rather than, contrary to the IRP, maximally informative. It contains maximum information, because it is a peculiar feature of Logic that everything follows from a contradiction. However, it is not maximally informative, because the receiver cannot extract the relevant true information from everything that follows absent a key for discriminating among the false and true consequents. Still, identifying a contradiction (either in a single statement or by accepting a statement that negates an old belief) is informative to the receiver to some degree (in a meta-informational sense), for she then realizes that she has to revise back.

How should we think of information given the role it plays in cognizing? TSSI offers a powerful framework for understanding information as a data-oriented concept. Under the right conditions, all perceived differences can give rise to some information for the perceiver. But clearly not all perceived differences are equally useful: there are degrees of usefulness relative to the informed agent. A dog visually identifying a juicy bone (perhaps, followed by a closer olfactory confirmation) among the many (distracting) toys on the floor will find the information about the bone very useful (especially if the dog is hungry). The dog’s owner might find the same data about the bone somewhat less—although still—useful. He is unlikely to eat the bone but would be wise to remove it from among the toys on the floor (unless they are all intended for his dog). The information about the presence of the bone is, arguably, more informative for the dog. While we are better off if the received information is true, veridicality is not a marker of information. Having useful information is crucial for a cognitive agent: the more useful the information is, the more informative it is for the agent.

IRP has value in explaining information processing in cognition and should, therefore, be preserved by accounts of information. When more and more evidence accumulates in support of a particular belief of an agent, it is informational for the agent. This cumulative evidence provides a justification for the agent to hold on to that belief. However, if less probable evidence is received that provides reason for her to retract an old belief, then that evidence is highly informative. Moreover, in trial-and-error learning, further instances of behavior that is found to be successful lead to the formation of a specific belief (e.g., Bradie Reference Bradie1986; Kingsbury Reference Kingsbury2008). The incentive to follow the same behavioral pattern becomes self-reinforcing.

4.2. Information and Rational Inference

A theory of semantic information is primarily about providing an account of the semantic content of an information-bearing state.Footnote ¹⁰ This naturally suggests a picture of inference in the manner in which we often speak to our students in class. “If you fully believe that p and you fully believe that if p then q, then you should infer q”—we tell them. While this is largely mistaken, it is what we tell them. Why do we do so? It is for the beliefs we do have that p and that if p then q would—if they were true—entail the truth of q. This cashes out the idea that these logical inferences spell out what our commitments are.

However, as Gilbert Harman pointed out, inference and Logic are not so closely aligned (Reference Harman1986). We are comfortable with the recognition that one might—on realizing that her commitments include q—revise to avoid that commitment. So, far from saying that if her explicit commitments entail something, q in this case, she should infer it, we point out that a possible response to the recognition of that commitment is to simply withdraw from some of the premises that led to that commitment. In that case, Logic is charting the statics, not the dynamics, of commitment. Logic tells us what the consequences of our commitments are, not how to update them or best bring them into cohesion. Despite the fact that contradictions are useless as premises on which to build an argument, we should not conclude that they have no logical role in reasoning. Reductio ad absurdum, for example, is a useful means of refuting theses.

Logic is a theory of consequence; it is a theory of what follows from what, not what a rational agent should infer.Footnote ¹¹ After all, were q believed to be false, a rational agent who sees that q follows from her commitments has a reason to revise her commitments. Similarly, not being inclined to infer anything from a contradiction is a good reason to revise back to a consistent kernel of beliefs. Accepting a contradiction commits the agent to accept everything that follows from it, including false information. We seek truth, not because only truths can qualify as Floridian information but because (in symbolic systems) only truth correctly characterizes the world as it is. Deriving a contradiction from one’s explicit views does not mean that one should accept a contradiction as being true. Rather, deriving a contradiction as a consequence of some prior commitments is a reason for denying at least one old commitment, because it is a consequence of one’s explicit theory (Michael Reference Michael2008, 54).

A theory of rational inference is a theory of the dynamics of commitment. Perhaps, a theory of semantic information need not be committed to either of the above epistemological accounts of Logic. We remain agnostic here about the need to hold to any particular view of rational inference, with one caveat. The simple story of the rational agent being under an obligation to infer any or all of the logical consequences of her explicit commitments is mistaken (Michael Reference Michael2008). A theory of inference should be built on a theory of semantic information—as a theory of what the agent’s commitments mean. It might be that we find merely formal rules for inference that abstract from the semantic information itself. Nevertheless, the justification of the inference rules cannot abstract from the meanings, that is, the information borne by the objects that these formal rules operate upon.

5.1. Information Processing without Truth

There has been much debate on whether information should be defined to include truth as briefly discussed in section 2. The Veridicality thesis is necessary for Floridi’s information-based epistemology (Reference Floridi2011, chap. 11, 12). Ever since Edmund Gettier showed that knowledge cannot be adequately equated with justified true belief (Reference Gettier1963), claims have been made such as that (a) knowledge requires nondefective justification, (b) ‘justification’ can be “weaker,” (c) knowledge requires a fourth condition, and (d) belief is not conceptually prior to knowledge (cf. Shope Reference Shope1983; Hetherington Reference Hetherington2012). Floridi has instead argued that the tripartite account of knowledge is not merely inadequate but demonstrably irreparable in principle: the Gettier problem is logically unsolvable—information flow can always be corrupted (Reference Floridi2011, chap. 9). Obviously, an argument of the form: ‘Weakly constrained semantic information is useful for cognitive science’ and ‘The Veridicality thesis imposes a strong constraint on information’, thus, ‘The Veridicality thesis is not useful for cognitive science’, is easily dismissible as a terminological disagreement. TSSI is meant to be a philosophical account of semantic information, and it may surely diverge from how cognitive scientists use ‘information’.

Nevertheless, our argument is different and cannot be similarly dismissed. It can be described as follows.

• P1. Information-based epistemology presupposes an account of human cognition.

• P2. Any complete account of human cognition requires a theory of weakly semantic information.

• P3. A theory of weakly semantic information cannot be based on the Veridicality thesis.

Thus, information-based epistemology cannot be based on the Veridicality thesis.

Its first premise seems trivially true insofar as we deal with knowledge of human cognizers. Its third premise is uncontentious (Floridi Reference Floridi2011, 114–17).

In the remaining section, we argue that the plausibility of the second premise rests on truth not being a property that our cognitive apparatus is sensitive to. Insofar as this apparatus manipulates information and produces representations (a process that is broadly called ‘information processing’), the information it processes can only be weakly constrained.

First, consider an imaginary organism that can roughly be identified with a Stoic sage, who never makes mistakes. According to Stoics, a sage only assents to kataleptic impressions. Kataleptic impressions arise from what is (i.e., from objective reality) and are stamped and impressed just as reality is with respect to those properties that are capable of being grasped (Bett Reference Bett2010, 63). Stoic epistemology involves a distinction between passively received impressions and actively made judgments on their basis. Crucially, what matters for our discussion is that the sage strictly avoids false judgment and so would never assent to false information and only have true beliefs. “Everything [the sage] believes, he knows to be true, and all of these truths are mutually supported and reinforced by everything else he believes” (63). If no impressions were kataleptic, this imaginary organism would have no beliefs, for it only processes veridical information.

However, in the case of humans, their cognitive faculties cannot provide such impeccable perception and judgment. What this imaginary organism and humans have in common is that they receive and respond to environmental stimuli in a way that affects their doxastic states. Cognitive scientific research shows that the same human cognitive apparatus processes false and true information indiscriminately. Repeated exposure to assertions that are unsubstantiated by evidence increases the probability that people will believe those assertions (Gilbert Reference Gilbert1991). Cognitive load and time pressure cause adults to believe false information and use it in making consequential decisions about some relevant situation (Gilbert, Tafarodi, and Malone Reference Gilbert, Tafarodi and Malone1993; Chen and Blanchard-Fields Reference Chen and Blanchard-Fields2000). Moreover, it was shown that older adults, compared with young adults, are more likely to believe false information during dispositional attribution (Chen Reference Chen2002). Experimentally induced memory illusion in healthy people has shown that they are more likely to assert falsely that they recall a word when it was semantically related to words with which they actually had seen (Corlett et al. Reference Corlett, Simons, Pigott, Gardner, Murray, Krystal and Fletcher2009).

Fundamentally, human cognition—just as cognition in nonhuman animals—has developed to support the successful resolution of challenges related to survival and reproduction. This crucially depends on the recognition of opportunities and threats, on the one hand, and the generation of appropriate strategies for facing these challenges, on the other hand. Such recognition requires the agent’s attention to and perception of relevant environmental cues (Bodenhausen and Hugenberg Reference Bodenhausen, Hugenberg, Strack and Förster2009).

Yet, not all stimuli in one’s perceptual field receive equal attention: there is a threshold on the amount of information the receiver can perceive at any given moment. For example, some studies propose that stimuli signaling the potential need for immediate action attract attention, independently of explicit goals and beliefs. New objects, looming objects, and objects that move suddenly are all behaviorally significant for the receiver and thus capture attention in a bottom-up manner (Franconeri and Simons Reference Franconeri and Simons2003, 1008). Other studies have shown that evolutionary relevant threatening stimuli, namely, significant fear-relevant stimuli, such as spiders, snakes, and angry facial expressions, are similarly effective in capturing attention due to the survival benefits of quickly recognizing them (Öhman, Flykt, and Esteves Reference Öhman, Flykt and Esteves2001).

Our cognitive apparatus uses attention selection mechanisms to allow us to perceive and process functionally significant or goal-relevant inputs while ignoring distracting or irrelevant ones. Yet, in some cognitively impaired humans, such as those suffering from delirium disorder, these mechanisms do not function properly. This results in their inability to select significant stimuli, thereby leading to inattentiveness or distractibility (Waite et al. Reference Waite, Harwood, Morton and Connelly2009, 54). The upshot is that it is not the so-called veridicality of the stimuli that is a marker of information for our cognitive apparatus. For something to qualify as information it needs to be “a difference that makes a difference” (Bateson Reference Bateson1972, 428). That is, it needs to be some perceived difference that is functionally significant to the goal-oriented behavior of the receiver.

We note that Adams’s distinction between ‘information’ as veridical semantic content and nomic regularities of physical events should be extended. Data as physical patterns in the world only make a difference to perceivers: only those differences that are perceptually available to an agent can be informational. Yet, neither of Adams’s types of information are, strictly, the object of information processing in cognitive agents. (To reprise, it is not how we define information but what explanatory role ‘information’ plays in our theory.) The latter type cannot be the object of information processing, since our cognitive apparatus only allows us to discriminate some, but not all, nomic regularities. For example, while elephants are sensitive to infrasound, humans are not readily sensitive to nomic regularities transmitted via infrasound. (The former type cannot be the object of information processing, for the reason mentioned above.)

Instead, ‘information’ is somewhere between those two. Information that allows us to make sense of our cognitive processing is such that is perceptually available to us. Only nomic regularities that we can discriminate in the world are those that humans can cognitively process; that is, informational states are formed on the basis of discernible data. The fact that there are differences in the world is not going to be cognitively significant for us, if we cannot perceive or discern them. Physical data can become information when we have the ability to discern them using either our senses or some cognition-enhancing devices. Information is called that because of its impact on our cognitive states. Some of these states might turn out to be veridical, but that they are veridical does not affect the processing of information.

The claim that information processed by human cognition need not be true is premised on information requiring a consumer that uses it. Pace Floridi (as well as Dretske and Adams), we argue that information does not exist in the world independently of a consumer. Nevertheless, the underlying perceived data still constrain the consumer’s formation of information. On Floridi’s view, for example, concentric rings (as structured data) in the trunk of a tree that are used to calculate the tree’s age (the data are meaningful) qualify as information also in the absence of any perceiver. But we argue that it is only potential information that rises to the level of information for a specific consumer using it for some goal (e.g., analyzing the growing conditions of the tree in past years).

Cognition is broader than knowledge, yet it is not clear that Floridian information has an explanatory advantage over information without truth in information-based epistemology. Suppose, for argument’s sake, that human cognition includes some knowledge processing mechanism. According to Floridi, any information processed by this mechanism must be true. Assume that two cognitively identical agents in two different possible worlds come to believe p ∨ q on the basis of perceptual stimuli, and, by stipulation, p is true in both worlds, yet q is only true in one of them. According to TSSI, only the agent holding the true information q processes information and might qualify as knowing p ∨ q, while the other cannot process the information p ∨ q. (We remain agnostic about whether the second agent can come to know p ∨ q.) Although the information processing in both agents is the same, according to TSSI, only one of them qualifies as such, and the other qualifies as data processing. We cannot use Floridian information to make sense of the difference between our two agents, since veridicality is already presupposed.

5.2. BCP Is Not Paradoxical but a Feature of Logic

While the Veridicality thesis is necessary for Floridi’s epistemology, he explicitly defends it on logical grounds pertaining to IRP and BCP. However, given that information processed by human cognition seems to accord with IRP and IRP entails BCP, we argue that BCP is not paradoxical but a feature of the relation between information and Logic.

Information earns its keep in theories of cognition by being that which explains the dynamics of cognition. Whatever the criteria for information are, what makes some x informational, has to do with how a cognitive agent either processes x or can process x. Even if some new semantic content p leads to, or is itself, a contradiction, it can be classified as information on our approach. Even though a self-contradictory statement is “one which no ideal receiver would accept,” it is a false statement “which happens to say much [and] is thereby highly informative” (Bar-Hillel and Carnap Reference Bar-Hillel and Carnap1952, 12). The expansion of an agent’s belief set by p (to use the terminology of Belief Revision models) results in an inconsistent belief set, which the agent strives to avoid as much as possible. So, a contradiction is informational for the cognitive agent without being maximally informative. For it simultaneously entails all possible answers to a given question without the agent being able to pick the “right” answer.

There seem to be three interrelated reasons for not classifying BCP as a paradox. The first, as is discussed in section 4.1, is the conflation of the notions ‘information’ and ‘informativeness’. If they are distinguished, BCP should no longer be seen as being paradoxical. The second reason is the conflation of Logic as a theory of consequence (or commitment) and Logic as a theory of inference (to which we alluded in sec. 4.2). The third reason is that BCP being a paradox undermines the very notion of validity in entailment semantics.

If we treated Logic as a theory of inference, the counterintuitive result of IRP would indeed be paradoxical.Footnote ¹³ But this view is mistaken. As a theory of commitment, Logic provides an agent a reason to revise back. Suppose the agent believes p, and also that p → q. The agent is clearly, at least implicitly, committed to q. But if she thought that q is absurd, that would give her a reason to revise back and question the veracity of p or p → q. In Belief Revision models of knowledge, if new information about the world contradicts at least one of the agent’s old beliefs, she has to give up one or more of her beliefs in order to accept this new belief and maintain a consistent set of beliefs. In that sense, identifying a contradiction as a consequence of the cognitive agent’s theory can be both informational and informative for the agent. But there is more than just Logic that plays a role in the agent’s belief revision.

Floridi seems to take the role of Logic in belief revision to be a theory of inference in rejecting the deductive closure principle. An infon—a potential instance of semantic information—σ could be false. Thus, if an agent commits herself to accepting σ as epistemically preferable, she “is not necessarily committing herself to accepting all the deductive consequences of σ as epistemically preferable” (Floridi Reference Floridi2011, 119). On TSSI, the target situation S “functions as a benchmark to test the Boolean truth-value and the degree of discrepancy of each σ” (118).

Consider an example to emphasise the perplexity of degrees of discrepancy in this context. Suppose that S is precisely described by ‘There are 9 people in the library’. Suppose further that σ ₁ stands for ‘There are some people in the library’ and σ ₂ stands for ‘There are 9 or 10 people in the library’. Both σ ₁ and σ ₂ are true (and, thus, qualify as Floridian information), but they deviate more or less significantly from S (Floridi Reference Floridi2011, 119). Taking Logic as a theory of commitments serves us well in explaining why σ ₃: ‘There are less than 11 people in the library’ has less informational content than σ ₂ (σ ₂ logically entails σ ₃). However, the degree of discrepancy from some target situation as a measure of informational content is not well defined in general when there is no logical entailment relation.Footnote ¹⁴ ‘There are more than 8 people in the library’ is true, yet it entails neither σ ₂ nor σ ₃ (and it is only logically entailed by σ ₂). (In general, for any two true instances of semantic content σ _x and σ _y indicating one less person in the library, the loss of information in σ _x is equal to that in σ _y relative to S [Wolf Reference Wolf2011, 251].)

As regards the third reason, if BCP were a paradox, the very notion of validity in entailment semantics would be undermined. The key motivating idea in IRP is that the consequences of an informational state determine its information. The principle of Explosion guarantees, then, that a contradiction is not just informational but maximally informational: it dominates anything that logically follows from it. But this only shows that the informational content of the consequences of a contradiction is contained within that of the contradiction. This is plausibly one way of spelling out the idea that Logic is nonampliative, as Kant put it: we cannot get more information in a conclusion of a valid argument than there is in the premises. It is a notable consequence of TSSI that there are many arguments in which the premises have less information than the conclusion. Any valid argument with false premises and a true conclusion would suffice as an example of that. The special case of an argument with contradiction as premise and true statement as conclusion is another clear example.

The converse of this principle also provides us with an account of validity: if p contains all the information in q, it is a sufficient condition for p entailing q. “In any valid deduction, the information carried by the conclusion must be already contained in the information carried by the (conjunction of) the premises” (Floridi Reference Floridi2010a, 55). (Jaakko Hintikka called it the Scandal of Deduction [Reference Hintikka1970].) The notion of validity rests on this feature. This is the reason, for example, that one can prove that a conclusion c logically follows from the set of premises, S, by adding ¬c to S and showing that the resulting set is inconsistent.

While validity is one criterion for rational inference, another important criterion is soundness. According to Logic, although the principle of “Explosion is valid, it will never be sound in the sense of having both true premises and a true conclusion” (Allo Reference Allo2010, 27). Inference as a reliable source of knowing the world requires more than just validity: soundness for epistemic progress has to be guaranteed. An agent equipped with Logic in search of truth would know that even though p ∧ ¬p ⊨ q, she cannot soundly infer q, since either p or ¬p is necessarily false. Defining information as requiring truth is of no help here. On TSSI, either p or ¬p is Floridian information but not both. The contradiction informs the agent that one of the premises is false, and she, therefore, has to reject q.

A contradiction simply marks the end of a “logical discourse.” Once it is reached, from one perspective, there are, strictly, no more “logical moves” to be made, and, from another perspective, the agent is free to make any “logical move” whatsoever. Rational inquiry does not stop at this point. But the idea that Logic is the sole engine of rational inquiry is for that reason inadequate.

The same principles apply, mutatis mutandis, in the case of a self-contradictory statement, too. Denying an explicit self-contradiction is a cognitive judgment one would expect of any rational agent. An agent should not accept a self-contradictory statement into her theory, because it informs her that at least one of the conjuncts in the statement is necessarily false. It is, thus, useful to know that a statement is contradictory. However, a contradiction may also be implicit, and a rational agent might fail to identify it. Russell’s paradox shows that naive set theory leads to a contradiction (Shapiro Reference Shapiro2005, 145–48), yet this contradiction is not self-evident, and so an agent may well accept it into her set of beliefs.