My inductive argument against both the reliability of early church canonical statements and the authenticity of the Gospel of Matthew based on their statements.

(1) According to Ireneaus and Papias, Matthew the apostle authored a gospel, and he is not spoken of writing more than one. As an inductive basis: we can affirm only one Matthean Gospel narrative was written.
(2) This gospel was written "in the language in the Hebrews" according to the earliest sources, which is Ireneaus of Lyons and Papias.
(3) The gospel of Matthew has no linguistic ties or evidence of it being translated from  hebrew/aramaic or any semitic language to greek. It is synoptic to the gospel of mark, which is greek. Inductively, it has no source in semitic language.
(4)  either the true gospel of Matthew was lost, or the earliest sources on the cannon are false, and another possibility that can be true in both instances: the gospel of matthew is a late writing.

Conclusion 1: By induction, it is less likely for the later writers to be wrong, rather than Matthew's gospel being lost. If the Apostle Matthew wrote a gospel, one would expect it would likely not have been lost. As it is a very important apostolic writing, and would have circulated quickly, or atleast in alot Christian communities and churches.
Conclusion 2: The earliest church fathers are likely wrong about the cannon, or, of necessity, the Gospel of Matthew is not the sayings spoken about by Ireneaus and Papias, because it has no ties to a semitic language, and it not described accurately by Papias (He describes sayings instead of a Gospel).

Ireneaus of Lyons, Against Heresies Book 3, Ch.1: "Matthew also issued a written Gospel **among the Hebrews** in their own dialect.."

Eusebius, Historia Ecclesiastica Book 3, Ch.39 quoting Papias: "But concerning Matthew he writes as follows: So then Matthew wrote the oracles in the Hebrew language, and every one interpreted them as he was able...."

Note that i am using induction because it is necessary in historical studies. I hate induction, to make absolute logical claims.

Eusebius, Historia Ecclesiastica Book 3, Ch.39 quoting Papias: "There are extant five books of Papias, which bear the title Expositions of Oracles of the Lord... Matthew also issued a written Gospel among the Hebrews in their own dialect,..."


Ireneaus of Lyons, Against Heresies Book 3, Ch.1: "Matthew also issued a written Gospel among the Hebrews in their own dialect.."

Eusebius, Historia Ecclesiastica Book 6, Ch.25:  "I have learned by tradition that the first was written by Matthew, who was once a publican, but afterwards an apostle of Jesus Christ, and it was prepared for the converts from Judaism, and published in the Hebrew language."

Eusebius, Historia Ecclesiastica Book 3, Ch.24: "For Matthew, who had at first preached to the Hebrews, when he was about to go to other peoples, committed his Gospel to writing in his native tongue, and thus compensated those whom he was obliged to leave for the loss of his presence."

There is even more agreement to that fact among the fathers (ex: Jerome and Epiphanius of Salamis).