Hullin, Chapter Two, Mishnah Seven
In our mishnah the sages debate whether an animal slaughtered on behalf of a non-Jew can be eaten by a Jew. The fear is that the animal was slaughtered as an idolatrous sacrifice and hence is prohibited.
1) If one slaughtered for a non-Jew, the slaughtering is valid.
a) Rabbi Eliezer declares it invalid.
b) Rabbi Eliezer said: even if one slaughtered a beast with the intention that a non-Jew should eat [only] its liver, the slaughtering is invalid, for the thoughts of a non-Jew are usually directed towards idolatry.
2) Rabbi Yose said: is there not a kal vehomer argument? For if in the case of consecrated animals, where a wrongful intention can render invalid, it is established that everything depends solely upon the intention of him who performs the service, how much more in the case of unconsecrated animals, where a wrongful intention cannot render invalid, is it not logical that everything should depend solely upon the intention of him who slaughters!
Section one: The first opinion in the mishnah holds that an animal slaughtered for a non-Jew is valid.
Rabbi Eliezer holds that even if the animal was slaughtered mostly for a Jews consumption and only a small part was meant for a non-Jew, the animal is still invalid because we can assume that the non-Jew intended to use it for idolatrous purposes. We see here that Rabbi Eliezer was very strict on relations between non-Jews and Jews because of the ever present fear of idolatry.
Section two: Rabbi Yose makes a kal vehomer (a fortiori) argument that this animal should be permitted. When it comes to sacrificial animals, an invalid intention can render the animal invalid, as we learned in Zevahim, chapter two. For instance, if the priest slaughters the animal in order to eat it after the time when it may be eaten or outside of the place where it may be eaten, the sacrifice is invalid. Nevertheless, when it comes to sacrifices we only take into consideration the intention of the slaughterer, and not the owner of the sacrifice. The same therefore should be true when it comes to non-sacrificial animals, whose laws are less strict. We should only care about the intention of the slaughterer, in this case a Jew, and not the non-Jew for whom he was slaughtering the animal.