If nothing else, Paul's teaching seems to bring the question of clean and unclean foods to a head by accepting the proposition that Christians are cleansed from sin by means of the gospel of Jesus rather than for example by eating only clean foods (cf. Gal. 2:11ff, Rom. 14:1-12, 1 Tim. 4:3-5, also outside Paul in 1 John 1:7-2:2, John 13:10 & 15:3), an idea that Peter also apparently learned in Joppa and Caesarea when he went to eat erstwhile unclean animal food at Cornelius's house (Acts 10-11). God has "made clean" all the animals in the sheet (10:15, 11:9). There is no contextual indication rabbinic teachings were the crux for Peter or Luke (other than the purely Mosaic), while if no formerly ("Levitically") unclean animals were declared clean in the sheet, Peter would have hesitated to eat at (Roman) Cornelius's house (not to mention with Gentiles in Antioch) and the implied link between cleansing of formerly unclean animals and cleansing Gentiles from sin would have been lost on the Jerusalem apostles and elders (cf. 10:43-48, 11:14-18). As Jesus had taught, it is what goes out of a person that makes him unclean (e.g., the sins list of Mark 7:21-23), not what "goes into a person from outside" (v. 18) and enters his stomach (v. 19); in so saying, Jesus "declared all foods clean" (v. 19).
Galatians 2 doesn't say anything about eating clean or unclean animals, that's simply something that's inserted into the text. Living like a Jew is about keeping Jewish customs, not necessarily about keeping God's laws.
Romans 14 likewise isn't about clean and unclean animals either, but about disputable matters of opinion (14:1). Meat that had been sacrificed to idols was often later sold on the market, so if someone were at a community meal and did not know where the meat came from, they might be of the opinion that only vegetables should be be eaten (14:2). They were judging those who chose to eat everything at the meal and were in in turn being despised (14:3). God only commanded fasting on Yom Kippur, but as a matter of opinion fasting twice a week or to commemorate certain events was a common practice. Men who esteemed those days more than other days were judging those who didn't fast and were in turn being resented (14:5-6). Whether man esteems one day over another is an entirely different matter than whether God esteems one day over another. We aren't to keep the God's Sabbath and Festivals because we esteemed those days over others, but because God esteemed them and commanded us to keep them. So whether or not you choose to fast on other days is a disputable matter of human opinion, but whether or not you choose to fast on Yom Kippur is a matter of obedience to God. Romans 14 is about favoring each other in matters of opinion, not about whether we should obey God.
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Are those teaching obedience to God's commands really giving heed to seducing spirits and doctrines of devils (1 Timothy 4:1)? Is obeying God departing from the faith (4:1)? Would such teaching be considered lies and hypocrisy (4:2)? Are the same teaching others not to marry (4:3)? What things are stated to be consecrated by the word of God and declared to be food and to be received in thanksgiving (4:4-5)? Is Leviticus 11 no longer good doctrine (4:6)? Are God's commands old wives fables (4:7)? Are God's commands profane? (4:7) Are God's commands in Leviticus 11 not Godliness (4:7)? Is Leviticus 11 no longer Scripture and thus no longer instruction in righteousness or a basis for rebuking and correcting (2 Timothy 3:16)? If you think that 1 Timothy 4:3-5 is abolishing the teachings of Leviticus 11, then answering these questions becomes problematic.
2 Peter 3:14-18 Therefore, beloved, since you are waiting for these, be diligent to be found by him without spot or blemish, and at peace. 15 And count the patience of our Lord as salvation, just as our beloved brother
Paul also wrote to you according to the wisdom given him, 16 as he does in all his letters when he speaks in them of these matters.
There are some things in them that are hard to understand, which the ignorant and unstable twist to their own destruction, as they do the other Scriptures. 17 You therefore, beloved,
knowing this beforehand, take care that you are not carried away with the error of lawless people and lose your own stability. 18 But grow in the grace and knowledge of our Lord and Savior Jesus Christ. To him be the glory both now and to the day of eternity. Amen.
In other words, Paul letters plus being ignorant and unstable equals the error of breaking God's law.
1 Timothy 4:3-5 who forbid marriage and require abstinence from foods that God created
to be received with thanksgiving by those who believe and know the truth. 4 For everything created by God is good, and nothing is to be rejected if it is received with thanksgiving, 5 for
it is made holy by the word of God and prayer.
What does it mean to know the truth in verse 3? God's law, which includes dietary restrictions, is declared by Scripture to be the truth (Psalms 119:142). If all animals are clean and not suitable for eating, then Leviticus 11 is no longer the truth, but Paul said that all Scripture is true and instruction in righteousness (2 Timothy 3:16-17). Those who know the truth are those who know what animals God has given as food to eat, so verse 3 is talking about people are teaching people to abstain from eating meat that God has already said is good to eat according to the truth of His Word (Leviticus 11). So these false teachers were not telling people to abstain from eating pork, which is not defined as food in Scripture. Keep in mind that when this letter was written to Timothy, "Scripture" meant the OT. In verse 3, the greek word for "food" is "broma" which is used to refer to foods that have already declared to be clean.
In verse 5, what does it mean to be made holy by the word of God and prayer? For something to be holy, it means to be set apart, which is the opposite of common or profane. If all animals are clean, then the animals would not be holy or set apart because animals that are set apart for eating must be set apart or separate from animals that are not set apart for eating. By definition, the very fact that there are a group of animals that are holy means that there must be some other separate group of animals that are not holy. If all animals are made clean, then by definition they would all be common, unholy, or not set apart. If all animals are made clean and set apart, then what are they set apart from? It is an oxymoron to say that all animals are sanctified. So those who believe and know the truth, God's law, will understand the only creatures to be received with thanksgiving as food are creatures that have been set apart by the Word of God and prayer. This is why verse 4 uses the qualified "if" because the only animals to be received by prayer and thanksgiving are those listed as such in Leviticus 11. We are to be thankful for clean animals as food, not for unclean things. Why would we be thankful for eating animals God told us were unclean?
If we pull in even more context, then we run into more problems. Was Paul really saying that God's commands old wives fables, profane and ungodly? As absurd a that is, that's what must be asked if conclude that verse 4 is speaking against those teaching God's dietary commandments. In 1 Timothy 1:4, 2 Timothy 4:4, Titus 1:14, and 2 Peter 1:16, "fables" is contrasted with the truth, the law, and God's Word. The whole problem in the 1st century was false doctrines, traditions, and teaching of men that were nullifying the law of God (Mark 7:6-13). The commandments Leviticus 11 are not fables, common, unholy, ungodly and profane, and are not doctrines of devils, so 1 Timothy 4 is actually in favor of keeping God's dietary laws.
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I talked about how Peter's vision should be understood in my previous post. In that same respect in Mark 7, Jesus was also dismissing man-made laws while upholding God's laws. The best translation of Mark 7:19 that expresses Jesus' thought is in the ISV:
19 Because it doesn’t go into his heart but into his stomach, and then into the sewer, thereby expelling all foods.”