There’s no need to to agree with theists or deists that gods’ non-existence can not be established. When they refer to god(s) to what (if anything) are they referring?
That is, the statement ‘the god X does not exist’ can be shown to be true. It’s up to claimants to specify just what concept of god they’re playing with. (Dealing with an irrationalist or a mystic requires different approaches not discussed here.)
Some concepts are simply inconsistent. For example is the concept of god X just like the concept of the round-square? “The” round-square does not exist because its (supposed) concept is incoherent.
In the Middle Ages an attempt was made to explicate “the” concept of God’s omnipresence by recourse to an analogy drawn from plane geometry. God is like . . . a circle whose circumference is nowhere and whose center is everywhere. Clever stuff.
But there can be no such circle. Among closed plane figures, the circle shares the property of always being finite. The analogy backfires — well if God’s omnipresence is like that; then, there can be no omnipresent God.
A different approach to showing conceptual limits of any concept of God also comes from the Middle Ages. “Can an omnipotent God create a stone too big for Him to lift?” To say either yes or no immediately implies that God is not omnipotent.
Language here is being misused. Absolute adjecitves are always relative to some context. A context free absolute adjective describes nothing. Stretching language past it limits is a commonplace in discourse about gods.
Obviously, most theists or deists won’t immediately offer up lucid concepts of god. Though the panto-divinity: all powerful, all knowing, all merciful, will often make His (Her, Its) appearance.
Can a “negative” be proved. Sure. Sometimes.
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