Observation signatures: what the interface records

A complete valuation answers every variable question in our chosen language. In the card example, X means dot and Y means stripe. Where a state has a two-bit annotation, those bits give its fixed X,Y values: one for true, zero for false. A selected formula list will have its own ordered headers. A makes X true and Y false. That state remains A whether we observe many formulas or only one. Now choose the ordered list X, Y as our observations. We call the list Phi. Ordered means that each position has a fixed meaning: position one is X, and position two is Y. An index is just a position label. We can write formula one and formula two to name those entries. An indexed family can include repetitions, so it is more informative than an ordinary set of formula names. A state supplies the input. The formula list specifies what to evaluate there. Neither is the resulting list of answers. Evaluate the list at A. X is true, so the first entry is one. Y is false, so the second entry is zero. The signature of A for this list is one, zero. At B both entries are one. At C they are zero, one. At D they are zero, zero. The formula headers tell us how to read every vector. A signature is the complete truth vector of the selected tuple at one valuation. It may have fewer entries than the valuation, or more, because we choose the formulas independently of the variable list. The zero at A says that Y is false at A. It does not mean that a reply has not arrived. Every entry of this semantic profile already has a definite value. The observation interface answers a different question: which affirmative events are available as certificates? With X and Y selected, an X certificate establishes X, and a Y certificate establishes Y. Absence of a Y certificate does not itself establish not Y. The topology gives no waiting time or certificate-delivery rule that would license that conclusion. Knowing the complete semantic profile would tell us that Y is false at A. Having no Y certificate is a different situation. Keep the instructor's mathematical table separate from the evidence actually supplied to an observer. A viewer says, “The second bit is zero because no stripe certificate arrived.” Does that correctly explain the zero in A's complete signature? No. The bit is zero because Y is false at A. Missing delivery is not what the signature records, and our topology does not model delivery. Choose a different list: the single conjunction formula X and Y. There is just one coordinate. B has signature one, while A, C and D all have signature zero, because each makes that conjunction false. Those three states remain different valuations. This observation simply does not distinguish them. A zero signature is not an extra state and is not an unknown truth value. If the list is only X instead, A and B both give one, while C and D both give zero. The same carrier now has a different grouping of equal signatures. Selecting a test changes the presentation. It does not remove any possible valuation or assert that the selected formula is true. Learning a premise is a separate operation later in the course. We can also describe a signature by listing the positions where its entries are one. This set of indices is called its support. An index names a position, not merely a formula: two positions can carry the same formula and still be two separate members of the support. For the list X, Y, A's support is the singleton containing index one. B's support contains indices one and two. C's support contains index two. D's support is empty, because neither selected formula is true there. An empty support is still the profile of a perfectly definite state. D has not disappeared from the carrier. Order matters when reading the vector. If we list Y before X, A's vector becomes zero, one. The state did not change; the coordinate meanings did. Never compare bit strings after silently changing their headers. A selected observation may itself contain NOT. Consider the tuple X, not X. Four two-bit patterns can be written down. Which of them can actually occur at a valuation? Only one, zero and zero, one. A and B make X true and not X false. C and D make X false and not X true. Zero, zero is impossible because one of a statement and its negation must be true in classical logic. One, one is impossible because they cannot both be true. Written bit patterns need not all be realized by the selected formulas. A certificate for not X is affirmative evidence of that selected formula. “Positive observation” means affirming an available formula, not banning negation from its syntax. An indexed list may repeat a formula. For X, X, Y, A has signature one, one, zero. Its support contains indices one and two, even though those positions carry the same formula. The repeated X contributes the same truth region twice. The topology generated from these regions is therefore unchanged from the list X, Y: no new generating subset was added. But the presentation still remembers two separate coordinates. Later, a distance can count contributions coordinate by coordinate. Topology alone does not retain all of that presentation data. Use two observations. Observation one is Y. Observation two is the conjunction X and Y. At C, write the two-entry signature and its support. Then find D’s signature. Does this two-observation tuple distinguish A from D? At C, Y is true but X and Y is false. The signature is one, zero and its support contains only index one. At D both formulas are false, so the signature is zero, zero. A also gives zero, zero: Y is false and the conjunction is false. This tuple cannot distinguish A from D, despite their different X values. A valuation, a selected formula tuple and its complete signature are three different objects. Equal signatures identify the distinctions the chosen observations cannot make. Next we will use their truth regions to construct the observation space explicitly.