Refutation of Chizzuk Emunah, Part II, Ch. LXVI: "Inconsistencies in Christian text regarding food" In Chizzuk Emunah, Part II, Ch. LXVI, 16th century Jewish theologian Yitzchak ben Avraham of Trakai wrote,


  Acts 10:11-15, "And Peter saw heaven opened; and a certain vessel descending unto him, wherein were all manner of four-footed beasts of the earth, and wild beasts, and creeping things, and fowls of the air. And there came a voice to him, saying, arise, Peter, kill and eat. But Peter said, Not so, Lord, for I have never eaten anything that is common or unclean. And the voice spake unto him again the second time, What God hath cleansed, that call not thou common." The same is stated ibid chapter 11:9. In Paul's first epistle to the Corinthians, chapter 10:25, the following doctrine is taught: "Whatsoever is sold in the shambles, that eat, asking no questions for conscience sake." Mark, (chapter 7:15) declares only such things unclean which come out of the mouth, but not those which go into the mouth. We have already animadverted on the inconsistency of such declarations when compared with the stringent injunction enforced in the very same book, to abstain most rigidly from blood and flesh of torn or strangled beasts. See what we have noticed before, when treating on Matthew 15 in the Second Part of this Work.


What is the refutation of his argument?
The supposed inconsistency here is clearly that some passages say it's OK to eat anything, but that in Acts 2:15, James judges that non-Jewish Christians should be "not troubled" with the Law except as follows:


  19 “Therefore it is my judgment that we do not trouble those who are
  turning to God from among the Gentiles, 20 but that we write to them
  that they abstain from things contaminated by idols and from
  fornication and from what is strangled and from blood. 21 “For Moses
  from ancient generations has in every city those who preach him, since
  he is read in the synagogues every Sabbath.”


So, in some passages, they say it's OK to eat these things, but in Acts, James says it's not.

The difference in context is, again, obvious.  All of the other passages are clear that eating a particular animal is not a sin.  Meat is meat.  So the question that must be answered is, why the restrictions in Acts 15?

There are several, but the most consistent with overall Scripture is that it's given so that the Gentiles won't be stumbling blocks for the Jewish Christians, as described in 1 Corinthians 8 (NASV):


  1 Now concerning things sacrificed to idols, we know that we all have
  knowledge. Knowledge makes arrogant, but love edifies. 2 If anyone
  supposes that he knows anything, he has not yet known as he ought to
  know; 3 but if anyone loves God, he is known by Him.
  
  4 Therefore concerning the eating of things sacrificed to idols, we
  know that there is no such thing as an idol in the world, and that
  there is no God but one. 5 For even if there are so-called gods whether
  in heaven or on earth, as indeed there are many gods and many
  lords, 6 yet for us there is but one God, the Father, from whom are all
  things and we exist for Him; and one Lord, Jesus Christ, by whom are
  all things, and we exist through Him.
  
  7 However not all men have this knowledge; but some, being accustomed
  to the idol until now, eat food as if it were sacrificed to an idol;
  and their conscience being weak is defiled. 8 But food will not commend
  us to God; we are neither the worse if we do not eat, nor the better
  if we do eat. 9 But take care that this liberty of yours does not
  somehow become a stumbling block to the weak. 10 For if someone sees
  you, who have knowledge, dining in an idol’s temple, will not his
  conscience, if he is weak, be strengthened to eat things sacrificed to
  idols? 11 For through your knowledge he who is weak is ruined, the
  brother for whose sake Christ died. 12 And so, by sinning against the
  brethren and wounding their conscience when it is weak, you sin
  against Christ. 13 Therefore, if food causes my brother to stumble, I
  will never eat meat again, so that I will not cause my brother to
  stumble.


Another possible answer is that the logic and decision recorded in Acts 15 isn't inspired, but is simply an accurate recording of James' own thoughts on the issue.  This would not negate the doctrine of inspiration of the Scriptures, as explained next.

This surrounding text makes this a very plausible suggestion, and it's not the only place in Scripture where someone's words and advice are recorded, and in the context it's clear that the advice isn't the direct commandment of God, but the opinion of the author.  See this previous question for an example.  

There are literally hundreds of places in Scripture where one person or another says something that is not the direct word of God.  It's just recording the historical events.  After all, some of Nebuchadnezzar's orders are recorded, but we don't take those as the Word of God.  We have to examine the context, and in this passage, the context could certainly be read that this is just James speaking his opinion and explaining why he holds the opinion.

As with the other question you posted in this vein, it seems odd that this would come from a Jewish perspective.  After all, the Old Testament has a similar "contradiction" on whether certain types of food are or are not allowed, a contradiction which is just as easily shown to be a non-issue.