Assalamu alaikum
Jazakallah for this. I teach Muslim Thought and I have enjoyed reading these.
Perhaps what this post needs to do before going any further is to situate these terms in a discussion on their origins and at which point they became in vogue.
Q) What is the difference between kalam and falsifa?
A) Both disciplines are very similar but have subtle differences.
Let's analyse:
1) METHODOLOGY (usul) - they employ rational tools such as mantiq (syllogistic logic) and other forms.
It should be highlighted that mantiq in the syllogistic sense was incorporated only later as the early kalamic ulama (even the mutazila) rejected mantiq and adopted a non syllogistic logic/reasoning (stoic as well as inductive and analogical(qiyas) forms) and philological (linguistic) analysis.
But after the nascent phase of kalam mantiq was the hallmark of the later kalamic methodology - alongside stoic logic and other forms of reasoning.
It was regarded as essential thus Ghazali appropriated and naturalised mantiq (syllogistic logic) via his monograph such as qistat, miyar, etc.
Also there were mantiqs.
Kalam employed a reduced form of mantiq against the likes of al-farabi who proffered mantiq in the same way as his master Aristotle with all the problematic assumptions and within the metaphysical framework.
Ibn Taymiyya's critique of mantiq was a critique of mantiq in the sense of Aristotle and his framework rather than the reduced mantiq of kalam.
According to the kalamic scholars Ibn Taymiyya had falsely conflated and equvicated - he had missed the point. So his critique did not affect their use.
Also, Ibn Salah's and Nawawi's haram fatawa were on falsifa mantiq rather than kalamic mantiq according to Ibn Hajar al-makki. It seems they did not differentiate either. (Read hallaq and el Rouayheb for details)
So the philosophers called their syllogism as burhani (demonstrative binding proof) and pejoratively declare the kalamic syllogism as jadali (speculative non binding proof).
Nonsense if the syllogism is valid and sound it is demonstrative be it from kalam or falsifa.
So methodologically they are very similar but have technical differences in terms of scope, content and application.
Sypnosis - for kalamic mantiq is a tool deattached from Aristotle's metaphysics.
Falsifa mantiq is a tool embedded deeply within Aristotle's metaphysics.
Continue on furu in the future if time.....