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Jesus said, "I am the way, the truth, and the life", John 14:6. So talking "the truth", possibly, might be, talking about Jesus.

"speaking truthfully" may or may not mean the same as "speak the truth".

What does "aletheian" mean in Eph 4:25?

There are a number of verbal components/ideas in Ephesians 4:25.

So, notice that the manner of "setting aside everything that is false" would also include setting aside true statements that mislead the hearer. Likewise, notice the motivation: if we are one body, why would one member of the body hurt another body part? Omitting key truths for the sake of deception would be doing just that.

Finally, in the greater context, a few verses earlier, Paul invites them to "speak the truth in love" (“ἀληθεύοντες δὲ ἐν ἀγάπῃ” (Ἐφεσίους 4·15 THGNT-T)). Notice how this makes the same point in different words: we speak the truth. But the manner & motivation in our hearts is ⲁⲅⲁⲡⲏ love: that which seeks the best for the other person.