Despite the many designs of devices operating with the DNA strand displacement, surprisingly none is explicitly devoted to the implementation of logical deductions. The present article introduces a new model of biosensor device that uses nucleic acid strands to encode simple rules such as "IF DNA_strand(1) is present THEN disease(A)" or "IF DNA_strand(1) AND DNA_strand(2) are present THEN disease(B)". Taking advantage of the strand displacement operation, our model makes these simple rules interact with input signals (either DNA or any type of RNA) to generate an output signal (in the form of nucleotide strands). This output signal represents a diagnosis, which either can be measured using FRET techniques, cascaded as the input of another logical deduction with different rules, or even be a drug that is administered in response to a set of symptoms. The encoding introduces an implicit error cancellation mechanism, which increases the system scalability enabling longer inference cascades with a bounded and controllable signal-noise relation. It also allows the same rule to be used in forward inference or backward inference, providing the option of validly outputting negated propositions (e.g. "diagnosis A excluded"). The models presented in this paper can be used to implement smart logical DNA devices that perform genetic diagnosis in vitro.