To understand how a computer program thinks about truth, we must first understand how it handles the word "no." In the study of artificial intelligence, researchers often ask a model to judge a statement, such as "The sky is blue." If the model says "false," we know it believes the sky is not blue. But what if the model says "false" to the statement "The sky is blue" and also says "false" to the statement "The sky is not blue"? This creates a logical puzzle. The model is not simply saying the sky is a different color; it is rejecting both the claim and its opposite. In logic, this state is called a "gap," suggesting the model sees the statement as having no truth value at all, perhaps because the question itself is broken. This method of asking two separate questions—one about a claim and one about its negation—is called bilateral evaluation. It is designed to reveal whether an artificial mind can handle contradictions or if it simply gets confused by the structure of language.

The stakes of this question are high because of famous philosophical puzzles, like the "liar paradox," a sentence that says, "This sentence is false." If the sentence is true, then it must be false. If it is false, then it must be true. For decades, philosophers have debated whether such sentences are impossible to solve, or if they are both true and false at the same time. As artificial intelligence systems grow more capable, researchers have begun to use them to test these ancient ideas, hoping to see if machines naturally adopt one of these complex logical stances. However, a new study suggests that when we look closely at how these machines answer, we are often seeing a glitch in their ability to process the word "not," rather than a deep philosophical insight.

Maikel Leyva, a researcher from universities in Chile and Ecuador, set out to test six different language models to see if they truly reason in these complex ways. The study began with a simple but rigorous plan: before asking the models about difficult philosophical riddles, the researcher first asked them about facts that are undeniably true or false. He presented 150 clear statements, such as "The sum of 11 and 35 is 46" and "The capital of France is Paris," along with their negations. He then asked each model, in separate conversations, whether the original statement was true and whether the negated statement was true. If a model is working correctly, it should say "true" to one and "false" to the other. If it says "false" to both, it has created a gap. If it says "true" to both, it has created a "glut," or a contradiction.

The results of this initial test were startling. Out of the six models tested, four of them failed a basic check of their own logic. When faced with simple math problems, these models frequently denied both the correct sum and the incorrect sum. For instance, one model denied that 11 plus 35 equals 46, and also denied that 11 plus 35 does not equal 46. This happened in more than half of the math questions for some models. The researcher found that this failure was not due to a lack of knowledge; the models made these errors even when the math was correct. Instead, the models seemed to be struggling with the very act of negation. They were not discovering a logical gap in the nature of arithmetic; they were simply failing to process the word "not."

To make sense of these findings, the researcher established a strict rule, or "gate," for interpreting the results. A model's answers could only be considered a genuine logical gap if it performed well on the simple, known facts first. If a model made too many mistakes on the easy questions, its answers to the hard questions could not be trusted. When this rule was applied, the picture changed dramatically. Four of the models were disqualified because they failed the gate. A fifth model, while it did not make many logic errors, refused to answer or gave answers that could not be understood in over 30 percent of the cases, making its results impossible to measure. This left only one model, a smaller system named Gemini 3.1 Flash Lite, that was reliable enough to be studied further.

With this single, calibrated model, the researcher returned to the famous liar paradox. Before the calibration, an uncalibrated reading of the data from a large, open-source model had suggested that the machine treated the paradox as a logical gap, refusing to assign it a truth value. This looked like the machine was following the logic of a philosopher named Kripke, who argued that such sentences have no truth value. However, once the calibration was applied, that conclusion vanished. The only model that could be trusted did not treat the liar sentence as a gap. Instead, it answered that the sentence is not true and that it is false, effectively giving it a standard "false" verdict. It did not see the sentence as both true and false, nor did it see it as having no value at all. It simply rejected the statement.

The study concludes that the promise of using artificial intelligence to solve ancient logical puzzles is currently limited by the machines' own weaknesses. The "gaps" and "gluts" that researchers hoped to find in the behavior of these models were often just errors in how the models handle negation. The one model that passed the strict test did not offer a new philosophical solution to the liar paradox; it offered a standard, two-sided answer. The research does not prove that machines cannot think in complex ways, but it does prove that we cannot assume they do. To know what a machine is really thinking, we must first be certain that it understands the difference between a statement and its opposite. Until we can calibrate the models to handle the simple logic of "not," any deep conclusions we draw from their answers on complex riddles remain unproven.