A philosopher of science studying the nature of scientific knowledge considers a logical sequence of hypotheses: Suppose a researcher forms hypotheses such that each new hypothesis is supported by two independent pieces of evidence, each of which increases the credibility of the hypothesis by 25%. If the initial credibility is 40%, what is the credibility after adding 4 such independent pieces of evidence?

["The Gradual Rise in Scientific Credibility: A Logical Approach to Hypothesis Strengthening", "In the pursuit of scientific truth, hypotheses rarely emerge fully formed. Instead, they evolve through careful testing and cumulative evidence. A philosopher of science examining how knowledge advances might consider a structured model where each hypothesis gains strength incrementally through independent verification. Recent research into the logical structure of hypothesis formation reveals an insightful pattern: when each new piece of evidence is independent and robust, it compounds credibility in measurable, multiplicative ways.", "Consider a scenario in which a researcher begins with a well-reasoned hypothesis formulated at 40% credibility—representing initial confidence based on theoretical plausibility and preliminary reasoning. According to a logical framework studied by philosophers of science, each successive piece of evidence should not be treated as mere addition but as a factor that multiplicatively increases confidence, provided the evidence is independent and credible.", "Each independent piece of evidence increases the hypothesis’s credibility by 25%—not cumulatively on a percentage scale, but geometrically: each new piece multiplies the current credibility by 1.25. Starting from an initial credibility of 0.40 (or 40%), the updates proceed as follows:", "- After 1st evidence:\n ( 0.40 \ imes 1.25 = 0.50 ) → 50%", "- After 2nd evidence:\n ( 0.50 \ imes 1.25 = 0.625 )", "- After 3rd evidence:\n ( 0.625 \ imes 1.25 = 0.78125 )", "- After 4th evidence:\n ( 0.78125 \ imes 1.25 = 0.9765625 )", "Thus, after four independent pieces of evidence, the hypothesis achieves approximately 97.66% credibility.", "This model reflects a deeper philosophical insight: scientific knowledge does not grow linearly but through compounding confidence under reliability and independence. Each piece of independent evidence acts as a logical gatekeeper, not just adding confidence, but reshaping the epistemic foundation. When evidence is independent, it avoids reinforcing biases and instead strengthens the hypothesis through verifiable corroboration.", "The philosopher of science thus recognizes that credibility is not static—it is dynamically calibrated by the logic of evidence integration. In this framework, starting from 40%, four independent 25%-boosting pieces elevate confidence to nearly 98%, demonstrating how cumulative, independent verification sharpens the pursuit of truth in science.", "Understanding this logical sequence deepens appreciation for both methodological rigor and the philosophical underpinnings of how we grow knowledge one credible step at a time."]