mirror of
https://github.com/open-thought/reasoning-gym.git
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201 lines
15 KiB
JSON
201 lines
15 KiB
JSON
{
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"metrics": {
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"dataset_name": "propositional_logic",
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"model": "google/gemini-2.0-flash-001",
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"size": 10,
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"seed": 42,
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"average_score": 0.059,
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"total_examples": 10,
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"timestamp": "2025-02-10T06:08:24.037697"
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},
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"results": [
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{
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"question": "Given:\n1. R\n2. Q\nWhat can we conclude?",
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"expected_answer": "(P \u2228 Q)",
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"model_answer": "Given only R and Q, we can conclude that **R and Q are true** (according to propositional logic).\n",
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"score": 0.01,
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"metadata": {
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"premises": [
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"R",
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"Q"
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],
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"variables": [
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"P",
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"Q",
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"R",
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"S"
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],
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"complexity": 3
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}
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},
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{
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"question": "Given:\n1. ((Q \u2192 P) \u2228 (Q \u2192 P))\n2. ((Q \u2194 Q) \u2192 (P \u2192 P))\n3. P\nWhat can we conclude?",
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"expected_answer": "(P \u2192 P)",
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"model_answer": "Let's break down each premise and see what we can derive.\n\n* **1. ((Q \u2192 P) \u2228 (Q \u2192 P))** This is logically equivalent to (Q \u2192 P). This is because any statement 'A or A' is simply 'A'. So we now have: **Q \u2192 P**\n\n* **2. ((Q \u2194 Q) \u2192 (P \u2192 P))** The statement (Q \u2194 Q) is a tautology (always true). The statement (P \u2192 P) is also a tautology. Therefore, (True \u2192 True) is True. This premise is always true and doesn't provide us with any new information that helps us directly derive something else. We can essentially ignore it.\n\n* **3. P** This is simply **P**.\n\nNow we have:\n\n* Q \u2192 P\n* P\n\nFrom these two statements, we can't directly conclude anything.\n\nRemember the Modus Ponens and Modus Tollens rules:\n\n* **Modus Ponens:** If you have P \u2192 Q and P, you can conclude Q.\n* **Modus Tollens:** If you have P \u2192 Q and \u00acQ, you can conclude \u00acP.\n\nIn our case, we have Q \u2192 P and P. This matches the *reverse* of Modus Ponens, which is *not* a valid inference rule. Knowing that the *consequent* (P) is true *doesn't* tell us anything definitively about the *antecedent* (Q). Q could be either true or false.\n\nTherefore, based on the given premises, we cannot conclude anything further about Q's truth value.\n",
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"score": 0.5,
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"metadata": {
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"premises": [
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"((Q \u2192 P) \u2228 (Q \u2192 P))",
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"((Q \u2194 Q) \u2192 (P \u2192 P))",
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"P"
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],
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"variables": [
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"P",
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"Q"
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],
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"complexity": 3
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}
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},
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{
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"question": "Given:\n1. ((Q \u2228 P) \u2227 \u00acP)\n2. P\n3. ((P \u2227 R) \u2227 \u00acR)\n4. ((Q \u2194 R) \u2192 \u00acQ)\nWhat can we conclude?",
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"expected_answer": "(Q \u2227 Q)",
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"model_answer": "Let's analyze each given statement:\n\n1. **((Q \u2228 P) \u2227 \u00acP)**: This says that either Q or P is true *and* P is false. Since P is false, Q must be true. So, we can conclude **Q**.\n\n2. **P**: This simply states that P is true.\n\n3. **((P \u2227 R) \u2227 \u00acR)**: This reads as \"Both P and R are true, *and* R is false.\" This is a contradiction. Since P and R are true, you cannot conclude that R is false.\n\n4. **((Q \u2194 R) \u2192 \u00acQ)**: This says \"If Q is equivalent to R (i.e., they have the same truth value), then Q is false.\"\n\nNow, let's see what we can derive, considering information from all the premises.\n\n* **From premise 1:** We *appear* to conclude Q. However premise 2 states that P is true. This contradicts the information provided in premise 1 that states P is false. Therefore, we have contradictory information. \n\n* **Premise 3 is a contradiction.**\n\nLet's consider the assumptions independently.\n\n* **Assuming premise 1 is true in isolation:** Then, P is false and Q is true.\n\n* **Assuming premise 2 is true in isolation:** Then, P is true.\n\n* **Assuming premise 3 is true in isolation:** Then, P and R are true and R is false. This is itself a contradiction and therefore cannot be true in isolation.\n\n* **Assuming premise 4 is true in isolation:** It tells us the dependence between the truth values of Q and R, but doesn't give us the true/false result of either variable on their own.\n\nSince we have contradicting information in the premises, we must be able to reach a conclusion.\n\n**Analyzing for Contradiction:**\n\nSince premise 1 stated that Q is true only if \u00acP is True, yet premise 2 asserts that P is True, then we know that premise (1\u22272) \u22a2 False.\n\nPremise 3 also claims that ((P \u2227 R) \u2227 \u00acR) is true which again cannot be true since (R\u2227\u00acR) evaluates to False.\n\nPremise 4 will only create a contradiction if Q \u2194 R evaluates to True but \u00acQ evaluates to False. Since Q is true from Premise 1, this implies that R must also be True, yet we cannot determine whether this is indeed true or false. We can only determine this, if we remove the contradictory information given regarding Q and R across all the premises.\n\n**Conclusion with contradiction:**\n\nWith contradictory premises, we can conclude *anything*. This is a principle of classical logic called \"ex falso quodlibet\" (from falsehood, anything follows), often simplified as \"explosion\". Since the collection of premises contains a contradiction, we can use it derive any conclusion.\n\n**Example:**\n\n1. Q is true (from premise 1, under the assumption it's only true premise)\n2. P is true is also true (from premise 2 under the assumption it's the only true premise)\n3. P and R is true and R is false (from premise 3 with the same provision).\n4. (Q \u2194 R) \u2192 \u00acQ (from premise 4 and depends on the premise results from the above to determine whether Q is true, but we can only determine that when the contradictory information is omitted from the statement/statements/premises).\n\nBecause premises do not provide consistent rules to determine truth values of P, Q, R, and therefore contradiction, then anything follows.\n",
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"score": 0.01,
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"metadata": {
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"premises": [
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"((Q \u2228 P) \u2227 \u00acP)",
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"P",
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"((P \u2227 R) \u2227 \u00acR)",
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"((Q \u2194 R) \u2192 \u00acQ)"
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],
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"variables": [
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"P",
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"Q",
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"R"
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],
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"complexity": 3
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}
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},
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{
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"question": "Given:\n1. \u00acP\n2. \u00acP\n3. Q\nWhat can we conclude?",
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"expected_answer": "(Q \u2228 R)",
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"model_answer": "Given the premises:\n\n1. \u00acP\n2. \u00acP\n3. Q\n\nWe can conclude **Q**.\n\nThe repetition of \u00acP doesn't add any new information. This is a very simple case where 'Q' is stated directly in the premises. There's no need for any complex logical deductions.\n",
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"score": 0.01,
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"metadata": {
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"premises": [
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"\u00acP",
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"\u00acP",
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"Q"
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],
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"variables": [
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"P",
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"Q",
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"R"
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],
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"complexity": 3
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}
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},
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{
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"question": "Given:\n1. P\n2. ((P \u2228 P) \u2194 (Q \u2227 P))\n3. P\nWhat can we conclude?",
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"expected_answer": "(Q \u2194 Q)",
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"model_answer": "Let's analyze the given statements:\n\n1. P is true.\n2. ((P \u2228 P) \u2194 (Q \u2227 P)) - This statement tells us that the truth value of (P \u2228 P) is equivalent to the truth value of (Q \u2227 P).\n\nSince we know P is true from premise 1, we can simplify:\n\n* **P \u2228 P is true:** Because P is true, disjoining it with itself makes the disjunction true.\n* **Since (P \u2228 P) \u2194 (Q \u2227 P) is true and (P \u2228 P) is true, then (Q \u2227 P) must also be true.**\n* **(Q \u2227 P) is true, and P is true, therefore Q must be true.** For the conjunction (Q \u2227 P) to be true, both Q and P must be true.\n\nTherefore, we can conclude that **Q is true**.\n\nSo the conclusion includes P and Q.\n",
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"score": 0.01,
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"metadata": {
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"premises": [
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"P",
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"((P \u2228 P) \u2194 (Q \u2227 P))",
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"P"
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],
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"variables": [
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"P",
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"Q"
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],
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"complexity": 3
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}
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},
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{
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"question": "Given:\n1. (Q \u2194 R)\n2. \u00acR\nWhat can we conclude?",
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"expected_answer": "(Q \u2192 R)",
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"model_answer": "From the given premises:\n\n1. Q \u2194 R (Q is equivalent to R)\n2. \u00acR (R is false)\n\nSince Q is equivalent to R, this means that if R is false, then Q must also be false.\n\nTherefore, we can conclude:\n\n\u00acQ (Q is false)\n",
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"score": 0.01,
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"metadata": {
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"premises": [
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"(Q \u2194 R)",
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"\u00acR"
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],
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"variables": [
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"P",
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"Q",
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"R"
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],
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"complexity": 3
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}
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},
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{
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"question": "Given:\n1. R\n2. ((Q \u2192 Q) \u2228 (S \u2227 Q))\n3. Q\nWhat can we conclude?",
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"expected_answer": "(S \u2228 Q)",
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"model_answer": "Let's analyze the given statements:\n\n* **1. R** - This is a simple assertion: R is true.\n\n* **2. ((Q \u2192 Q) \u2228 (S \u2227 Q))** - Let's break this down:\n * `(Q \u2192 Q)`: This is a tautology. Any proposition implying itself is always true.\n * `(S \u2227 Q)`: This is true if and only if both S and Q are true.\n * `\u2228`: The entire statement is true if either `(Q \u2192 Q)` is true, or `(S \u2227 Q)` is true. Since `(Q \u2192 Q)` is always true, this entire statement is always true, regardless of the truth value of S.\n\n* **3. Q** - This asserts that Q is true.\n\nNow, what can we conclude?\n\n* **We know R is true.** (Directly from statement 1).\n* **We know Q is true.** (Directly from statement 3).\n\nFrom the second statement, `((Q \u2192 Q) \u2228 (S \u2227 Q))`, we can note the following:\n\n* Because `Q \u2192 Q` is always true (a tautology), `((Q \u2192 Q) \u2228 (S \u2227 Q))` is always true, regardless of the truth value of S. This statement doesn't help us conclude anything new about S, but it does provide some information about the relationship between Q and itself, and Q and S.\n* Because we also know that Q is true, and that `((Q \u2192 Q) \u2228 (S \u2227 Q))` is true, the `(S \u2227 Q)` portion could be true or false. If it's true, then S must also be true. If it's false, then S must be false. However, because `((Q \u2192 Q) \u2228 (S \u2227 Q))` is true, and Q is true, we cannot know for sure if S must be true or false. \n\nTherefore, the key conclusions we can draw are:\n\n* **R is true.**\n* **Q is true.**\n\nWe **cannot** definitively conclude anything about the truth value of S.\n",
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"score": 0.01,
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"metadata": {
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"premises": [
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"R",
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"((Q \u2192 Q) \u2228 (S \u2227 Q))",
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"Q"
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],
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"variables": [
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"P",
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"Q",
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"R",
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"S"
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],
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"complexity": 3
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}
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},
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{
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"question": "Given:\n1. (Q \u2227 P)\n2. (\u00acP \u2227 (P \u2192 P))\n3. \u00ac(Q \u2194 Q)\nWhat can we conclude?",
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"expected_answer": "(P \u2227 P)",
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"model_answer": "Let's analyze each statement:\n\n1. **(Q \u2227 P):** This statement tells us that both Q and P are true.\n\n2. **(\u00acP \u2227 (P \u2192 P)):**\n * `\u00acP`: This means P is false.\n * `(P \u2192 P)`: This is always true (a tautology). Any proposition implies itself.\n\n3. **\u00ac(Q \u2194 Q):**\n * `(Q \u2194 Q)`: This means \"Q if and only if Q\". This is always true (a tautology).\n * `\u00ac(Q \u2194 Q)`: This means \"not (Q if and only if Q)\". Since `(Q \u2194 Q)` is always true, its negation is always false.\n\nNow let's look for contradictions:\n\n* Statement 1 says P is true.\n* Statement 2 says P is false.\n\nTherefore, **we have a contradiction**. The set of premises is **inconsistent**. We can conclude that the statements cannot all be true simultaneously; there's no truth assignment that will satisfy all three statements.\n\nTherefore, we can conclude that the premises are inconsistent.\n\n**Final Answer: The set of premises is inconsistent.**\n",
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"score": 0.01,
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"metadata": {
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"premises": [
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"(Q \u2227 P)",
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"(\u00acP \u2227 (P \u2192 P))",
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"\u00ac(Q \u2194 Q)"
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],
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"variables": [
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"P",
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"Q"
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],
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"complexity": 3
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}
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},
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{
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"question": "Given:\n1. (R \u2228 Q)\n2. (R \u2227 Q)\n3. R\nWhat can we conclude?",
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"expected_answer": "(P \u2194 P)",
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"model_answer": "From the given statements:\n\n* **1. (R \u2228 Q)** - This tells us that R or Q (or both) is true.\n* **2. (R \u2227 Q)** - This tells us that both R and Q are true.\n* **3. R** - This tells us that R is true.\n\nWhile statement 3. R is already confirmed in statement 2., so, knowing that both R and Q are true from statement 2, we can definitively conclude:\n\n* **Q** is true.\n\nHere's a breakdown:\n\nStatement 2 tells us R \u2227 Q. For a conjunction (\u2227) to be true, both parts must be true therefore R and Q must be true.\n\nSo, the most direct and strongest conclusion we can make is:\n\n* **Q is true**\n",
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"score": 0.01,
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"metadata": {
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"premises": [
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"(R \u2228 Q)",
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"(R \u2227 Q)",
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"R"
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],
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"variables": [
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"P",
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"Q",
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"R"
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],
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"complexity": 3
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}
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},
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{
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"question": "Given:\n1. ((P \u2228 Q) \u2228 (Q \u2192 Q))\n2. \u00ac(Q \u2192 Q)\n3. ((P \u2227 Q) \u2227 (Q \u2227 P))\n4. (Q \u2194 Q)\nWhat can we conclude?",
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"expected_answer": "(P \u2192 P)",
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"model_answer": "Let's analyze each statement:\n\n1. **((P \u2228 Q) \u2228 (Q \u2192 Q))**: We know that `Q \u2192 Q` is always true (a tautology). Therefore, `(Q \u2192 Q)` is equivalent to `T` (True). The statement becomes `((P \u2228 Q) \u2228 T)`. Since anything ORed with True is always True, this whole statement simplifies to **T (True)**. It tells us nothing about the values of P or Q.\n\n2. **\u00ac(Q \u2192 Q)**: We know that `Q \u2192 Q` is always true. Therefore, `\u00ac(Q \u2192 Q)` is equivalent to `\u00acT`, which is **F (False)**. This is a contradiction given that the argument is valid.\n\n3. **((P \u2227 Q) \u2227 (Q \u2227 P))**: This statement is equivalent to `(P \u2227 Q)`. This is because `P \u2227 Q` is logically equivalent to `Q \u2227 P`. So, we can say that `P` is True **AND** `Q` is True.\n\n4. **(Q \u2194 Q)**: This is a biconditional, meaning \"Q if and only if Q\". This statement is always true (a tautology) regardless of the value of Q. So this is equivalent to **T (True)**, and again, it tells us nothing about the value of Q.\n\n**Conclusion**:\n\nThe argument presents contradictory statements:\n\nStatement 1: Always True.\nStatement 2: Always False.\nStatement 3: P is true AND Q is true.\nStatement 4: Always true.\n\nStatement 2 contradicts both statement 1 and statement 4.\n\n**Therefore, we can conclude that the set of premises is inconsistent due to the contradiction in premise 2 \"\u00ac(Q \u2192 Q)\".**\n\nWithout premise 2, we could conclude that P is true AND Q is true from premise 3. The logical equivalence `(P \u2227 Q) \u2194 (Q \u2194 Q)` can be derived by simply checking truth tables.\n",
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"score": 0.01,
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"metadata": {
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"premises": [
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"((P \u2228 Q) \u2228 (Q \u2192 Q))",
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"\u00ac(Q \u2192 Q)",
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"((P \u2227 Q) \u2227 (Q \u2227 P))",
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"(Q \u2194 Q)"
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],
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"variables": [
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"P",
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"Q"
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],
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"complexity": 3
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}
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}
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]
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}
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