According to Molinism, God's omniscience can be broken down into four distinct parts: Natural Knowledge, Free Knowledge, Creative Command, and Middle Knowledge.
Natural Knowledge is virtually uncontested by any orthodox theologian. It is God's knowledge of literally all truths, including that which could be true, even if it isn't. Interestingly, these truths exist, in a sense, separately from God, as in they are independent of His will. God, for example, did not choose to exist, and nor could He even choose not to.
Free knowledge, by contrast, is entirely subject to God's ultimate will. It is His knowledge of what He actually created, as opposed to what He could have. This is linked to Creative Command, which is the moment where God decides to act (or not) on His knowledge of what could be.
Middle Knowledge is the "in between" part of Natural and Free Knowledge. It is His knowledge of what would happen in a given set of circumstances, up to and including the free choices that would be made by a free creature.
Middle Knowledge, if I understand Molinism correctly, is a cornerstone of Molinism. If Middle Knowledge is false, so is Molinism. If Middle Knowledge is true, Molinism would seem to be also. Pending further study, I withhold my judgement on Molinism itself. However, Middle Knowledge, under the definitions provided thus far, seems to be what I have always believed in, and I believe a stone solid case can be made for it.
To begin with, Middle Knowledge is simply a logical extension of God's omniscience. There is not now, nor will there ever be, nor has there ever been, anything God does not know. Therefore, even if you don't believe God created, or even can create a free creature (!!!), you must logically believe God knows what would happen at every given moment of that free creature's existence in any given scenario.
If you do not believe God created free creatures, then the logical extension of that is you do not believe you or I are free creatures. Therefore, this dispute is both illogical and inevitable. Illogical because neither of us are free to draw any conclusion, but inevitable because neither of us are free to cease the discussion. This would seem to be incompatible with the nature of God, for He is a logical being.
It would also seem to be in conflict with Scripture, which consistently treats human beings (and angels) as if we are free beings. The Bible often describes us doing things that God neither commanded us, nor did it enter His heart (Jeremiah 7:31), resisting His will by not obeying His commands (Luke 7:30), and even Him delaying the fulfillment of His will because we are not willing (Matthew 23:37; Luke 13:34). Furthermore, Scripture clearly tells us that though the wicked die, this does not please God (Ezekiel 18:32; 33:11), and though not all will be saved (Matthew 7:13-14), He is certainly willing that they are (1 Timothy 2:1-4). Thus, it is beyond dispute that God created us with such freedom that, for reasons I am content to let be His alone, He allows us to even resist His will in order to bring about something He desires more.
Furthermore, God shows His Middle Knowledge by specifically revealing it to us. The most obvious example would be Matthew 11:23, where Jesus says "And you, Capernaum, who are exalted to heaven, will be brought down to Hades; for if the mighty works which were done in you had been done in Sodom, it would have remained until this day."
In His condemnation of certain cities, Jesus points to other cities which were destroyed. The destruction of Sodom and Gomorrah is one of the earliest large-scale judgments recorded in the Bible, first seen as early as Genesis 19. There is no doubt any devout Jew would have known both how terrible Sodom was, and how spectacular their destruction was. Yet, Jesus says if He had done His mighty works in Sodom, "it would have remained until this day", which is only possible if they repented. So, Jesus says if He had provided a given circumstance (doing His mighty works in Sodom), they would have repented. In other words, Jesus is showing that He knows what the free creatures in Sodom would have done if He had put them in a given set of circumstances. That is free knowledge, indisputably displayed in a single verse.
But then why did God not perform mighty works in Sodom to bring about their repentance, since His word says so clearly that He wants all men to be saved and come to the knowledge of the truth? The "simple" answer is that, much like us, God has two wills, a lesser and a greater. Jesus had a lesser will not to be crucified, yet He had a greater will to obey His Father, and so He said "...nevertheless, not as I will, but as You will." (Matthew 26:39).
Now, in Jesus dwells the fullness of Godhead bodily (Colossians 2:9). It's not like when He became incarnate, He gained a separate will from God. Thus, we can reasonably conclude that the fullness of the Godhead, Father, Son, and Holy Spirit, was unwilling for Jesus to be unjustly seized, put through a bogus trial, and slain for sins He never committed. And one can hardly blame Him! Even knowing I deserve such punishment, I don't especially want to go through it. Jesus never deserved it, and the fact that He isn't some freaky masochist who enjoyed every second He hung on that cross means it was a real sacrifice. So what was the greater will of God? The sacrifice itself. God didn't will for Jesus to suffer, but through that suffering, He did will to save all who would come to Him in faith.
So we can logically say that God willed for Sodom to repent, yet even knowing how to achieve that, He had a greater will. Now, it seems unwise to speculate on the fullness of it, as we do not possess God's Middle Knowledge. All we can say for sure is that God, knowing every possible scenario, chose this one. But His word does mention Sodom a lot later on, most notably in 2 Peter 2:6, where He tells us the destruction of Sodom and Gomorrah made them "...an example to those who afterward would live ungodly;".
Ultimately, not even God can have His cake and eat it too. And I'm not denying His omnipotence there, His word literally says "...He cannot deny Himself." (2 Timothy 2:13). Thus, faced with a set of contradictory options, even God must choose. If He loves a sinner, yet also loves justice, and in order to bring about justice, He must withdraw His love for the sinner, there are only a limited set of options available to Him.
But for us, as Christians, this is fantastic news! Knowing every possible outcome of our lives, God chose one of the ones (if, indeed, there are so many) in which we are saved. He is still sovereign over our salvation, and He chose to save us. To me, this is far more glorious than any view in which God is sovereign over all things, yet abuses that to force us to sin, then force us to repent. For one, it shows great wisdom on His part. How much harder is it to control a free choice? For God, no extra effort is required. He's God, and though I fought Him, He still won. But also, it shows real mercy. I fought Him, I deserve to be condemned for that, yet He instead includes me in His victory, dividing the spoils with me. I chose to live as a child of Hell, yet He went to the cross, collecting my wages (death), and instead calling me to repent and become a child of God? I still do not know if I fully understand Molinism, and until I do, I will not accept that label, but knowing what Middle Knowledge is, I wholeheartedly affirm it, and give glory to my Creator for it.