No P, No Q

Impeachment’s Missing Variable

On July 8, at the Senate impeachment court, Senator-judge Alan Peter Cayetano asked NBI witness John Mark Calilung a question that sounded more like a math class than a courtroom. In math, you can’t skip the “if” and jump straight to the “then.” Cayetano applied exactly that rule, asking, in Filipino, whether Vice President Sara Duterte’s statement was conditional, meaning it depended on something else happening first. If the person she named was not killed, Cayetano said, “the logical implication is that there was no threat,” as Inquirer.net quoted him on July 9, 2026.

Before I became a policy practitioner, I trained as an engineer, and engineers spend their careers thinking in “if this, then that.” A fire sprinkler turns on only if the temperature crosses a threshold. Electricity flows through a circuit only if the switch is turned on; leave it off, and no current passes. Neither system acts early or on suspicion. Each one waits for its condition. Acting before the condition is met isn’t normal or logical. It’s the kind of answer that fails an engineering exam.

This kind of thinking has a name in mathematics: the conditional statement, written as P → Q, read as “if P, then Q.” P is the part that has to happen first. Q is what follows, but only after P does. Readers more familiar with algebra than logic can think of it the same way: P behaves like the independent variable, Q like the dependent variable—Q’s applicability within the conditional depends entirely on P. If P has not happened, Q has no basis to be treated as real, active, or ongoing.

This idea is also much older than any lawyer in that courtroom. Around 300 BC, the Greek philosopher Philo of Megara worked out the rule that a conditional statement fails only when the “if” comes true and the “then” doesn’t. Historians of logic still call this the Philonian conditional, after him. Centuries later, modern logicians, including David Hilbert, adopted symbolic notation such as the implication arrow (→) to express conditionals with precision in lectures between 1917 and 1922.

Now apply it directly. During an online press briefing on November 23, 2024, reported that day by Rappler, Duterte said that if she were killed, she had arranged for someone to kill President Marcos, First Lady Liza Araneta-Marcos, and then-Speaker Martin Romualdez, adding “no joke, no joke,” as CNN also reported on November 24, 2024. Her death is P. The described act is Q. By that same rule, nothing says Q is true unless P has already happened. Q is the dependent variable here—it has no value of its own until P, the independent variable, happens first. P has not happened. Put simply, what the sentence describes doesn’t stand on its own. It only exists because of the condition attached to it.

Q needs P first. Ignore that, and you’re arguing with a sentence you manufactured, not the one she wrote. The rule works the same for everyone.

This also explains why the statement can be discussed and investigated as words without anyone having been harmed. A conditional statement is not silent or meaningless. It can be examined, quoted, and debated. What it cannot do, while its condition remains unmet, is assert that something is already underway.

Everyone is entitled to an opinion about what Duterte’s statement means, and that opinion deserves respect. Right or wrong is often a matter of perspective. But respect for an opinion is not exemption from having a basis. An interpretation needs something underneath it—a document, a transcript, a rule—or it isn’t an interpretation. It’s a guess wearing one. My basis here is formal logic, applied the same way Philo and Hilbert applied it to every conditional before this one. Readers are free to disagree with what I make of it. What the logical structure of the sentence is, however, is not a matter of opinion.

Mathematics will not decide how the impeachment court rules. What it settles, without opinion and without politics, is what can honestly be said about this sentence today, while its condition remains unmet. As of this writing, that condition has not occurred. Senator Cayetano did not need Philo or Hilbert to ask the right question in that courtroom. He needed only to read the sentence in the order it was written: condition first, consequence second. Mathematics cannot settle political disputes, but it can identify logical structure. Every “if, then” statement starts with the “if.” Until the “if” comes true, the “then” isn’t true either. In formal logic, the equation remains one variable short. No P. No Q.

Disclaimer: The views and opinions expressed in this article are those of the author and are intended to encourage public discussion on governance and national issues. They do not represent any official position of the institutions the author may be affiliated with.

About the author: Paul Y. Chua, PhD, holds doctoral degrees in Fiscal Management and Peace and Security, and a master’s degree in National Security Administration. He writes on Philippine governance, national security, transportation, and public policy. Full profile at pychua.com.