Did Hashem violate Pharaoh's free will by "hardening his heart"?
The image above, from the 17th-century Venice Haggadah, illustrates the text which says that Hashem will send evil angels to punish the Egyptians. The evil angels are depicted like demons, the breath of each emitting one of the 10 plagues. One of the figures crouches by the shore of the Nile while breathing on it and causing the water to turn to blood, another emits lice from his mouth, and a third - locusts.
The Jewish concept of free will finds is hinted at in the creation account of Genesis, wherein Adam is formed from the dust of the earth. The verb "formed" is vayyitzer (וַיִּ֩יצֶר֩; note the doubling dot in the yod), which is said to be the point at which humankind was imbued with the two inclinations - the good inclination, or yetzer hatov (יצר הטוב), and the evil inclination, or yetzer hara (יצר הרע). It is taught in Bereshit Rabbah and Radak (Rabbi David Kimchi) that the doubled yod in vayyitzer (וַיִּ֩יצֶר֩) represents the two yetzers. Thus, from Adam's first breath, he had the ability to choose between the good inclination within him and the evil inclination (also within him; he had free will.
In Jewish hashqafah, free will is inviolable and the idea of absolute fatalism is farcical. Given the centrality of the free will concept to Judaism, the question arises in light of the English translations of Shemot (Exodus) re: how Hashem could "harden Pharaoh's heart" without violating his free will. The English translations are troublesome with regard to the "hardening" of Pharaoh's heart, as there are three unrelated roots all rendered with the same English word.
The first of these - חזק - is a root meaning "to strengthen" and is understood in its usage in Shemot to mean strengthening one's resolve.
Rabbi Marcus Jastrow defined the root thus:
חזק to be or grow strong, to hold fast.
— Qal - חָזַק 1 he was strong; 2 he held fast.
— Pi. - חִזֵּק 1 he made strong, strengened, reinforced; 2 he encouraged.
— Pu. - חֻזַּק was strengened, was reinforced.
— Hith. - הִתְחַזֵּק 1 he strengened himself, took courage; 2 he put for strength, exerted himself.
— Hiph. - הֶחֱזִיק 1 he made strong, strengthened; 2 he seized, took hold of; 3 he clung to; NH 4 he kept, held; PBH 5 it contained, held.
— Hoph. - הָחֳזַק, הֻחְזַק PBH 1 was seized, was held; NH 2 was kept; PBH 3 was regarded; PBH 4 was sure, was certain; NH 5 was maintained.
It occurs at Shemot 4:21 (piel impf 1cs), 9:35 (qal wci 3ms), 10:20 (piel wci 3ms), and 10:27 (piel wci 3ms).
The second verb - קשה - is only encountered in the plague pericope at Shemot 7:3 (hiph impf 1cs). The meaning here is more in line with the rendering of "harden, stiffen." It is the same root used in the description of Israel as a "stiff-necked people" (עם־קשה־ערף) in Shemot 3:29, in which context it carries the adjectival meaning of stubborn-hearted, as it also does here in verb form.
Shemot Rabbah suggests that Pharaoh himself was the active party in the hardening of his own heart. This seems to find support in Rashi, who opines that Hashem passively "allowed his heart to harden."
Ibn Ezra and Ramban both read the passage differently than the Sages cited above, i.e. understanding Hashem to be the active party.
Sforno's explanation for the hardening of Pharaoh's heart attempts to harmonize Ezekiel 33:11 with the passage to find a rationale for it.
With most of the plagues, the verb used is כבד - to be heavy, stubborn - and it is used of Pharaoh's own decisions regarding the state of his heart. This aligns with what would be his own (Egyptian) soteriology. In the Egyptian worldview, at death a person's heart (specifically the ka, i.e. "life spirit," within it) is weighed against a feather. If his or her sins weigh less than the feather, admission into the afterlife is granted, but if they are heavier admission is denied. Pharaoh's decisions are adding weight to his heart. This verb כבד is used at 8:11 (hiph inf abs), 8:28 (hiph wci 3ms), 9:34 (hiph wci 3ms), and 10:1 (hiph pf 1cs).