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Unfortunately, the fields cannot be disabled. But, the error messages and validation should prevent the form to submit (which is the disabling part).
Another way for this is to set the fields to be required.
You can also use the conditional logic to show/hide fields if other inputs are empty. Here's an example form with those features enabled: http://form.jotformpro.com/form/52445378815968?. You'll see that the submit button is not visible unless you fill out the full name and email field.
But let me think through this and see if I can visualize how the disabling of input would work, either in the matrix field on in the plain form fields.
Thank you for your input.