The rationalist believes we come to knowledge a priori – through the use of logic – and is thus independent of sensory experience.//ok
rationalism s5
S5 is useful because it avoids superfluous iteration of qualifiers of different kinds. For example, under S5, if X is necessarily, possibly, necessarily, possibly true, then X is possibly true qualifiers before the final "possibly" are pruned in S5.
The main difference between Rationalism And Empiricism is that rationalism is the knowledge that is derived from reason and logic while on the other hand empiricism is the knowledge that is derived from experience
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s5 axiom S5: ◊ □ x → □ x “s5 " brefers to logic from S4 modal logic by adjoining the axiom scheme that there is the implication.
Right ?
S5: ◊ □ x → □ x
◊□P → □P
logic { prescriptive not descriptive }
p1 x = x { axiom }
p2 x holds contingency { observed , axiom }
p3 x does exist therefore depends on y { observational reality }
p4 necessarily y = / = x { laws of physics }
p5 Y exists necessarily y = / =x iff { if and only if } x= x
p6 Nature cannot create nor destroy energy
C its impossible y can fail to exist in all possible worlds where x = x
..." Y " is not contingent
Modus ponens if P possibly necessarily exists implies P necessarily exists in S5 in modal logic as propositional logic does, modal logic (S5)
Deductive argument form
In propositional logic, modus ponens, also known as modus ponendo ponens, implication elimination, or affirming the antecedent, is a deductive argument form and rule of inference. It can be summarized as "P implies Q. P is true. Therefore Q must also be true."
For normal modal logics, there are only two inference rules that need to be assumed: modus ponens and necessitation. As axioms we take all (substitution instances) of tautologies, and, depending on the modal logic we deal with, a number of modal axioms.
A possible world is a complete and consistent way the world is or could have been. Possible worlds are widely used as a formal device in logic, philosophy, and linguistics in order to provide a semantics for intensional and modal logic.
A necessary truth is a true statement whose negation must imply a contradiction in reality1. It is a truth that could not have been false, no matter how things had turned out2. A necessary truth is one that is true in all possible worlds, according to the principles of logic and metaphysics32.
A contingent truth is one that is true, but could have been false. A necessary truth is one that must be true; a contingent truth is one that is true as it happens, or as things are, but that did not have to be true

Question

Ask a Philosopher · Accepted Answer

To ponder the vast expanse of knowledge, one doth find a schism 'twixt the heart of pure reason and the pulse of sensory experience, wherefrom man draws breath. The rationalist, on lofty wings of logic, takes flight a priori, weaving threads of thought unspooled from the loom of mere perception. In sooth, he doth believe, with steadfast confidence, that knowledge springs untouched from the well of intrinsic truth, forsooth, a beacon unclouded by the shadow of the senses. O how sweet the clarity! Yet as the sun doth rise and fall, so too the experience doth whisper its truths unto the ear of mankind, reminding him that flesh doth feel and eye doth see. Empiricism, the rooted sage, would caution him, declaring that the dance of life’s variances doth enrich the tapestry of understanding, lending color and form to the stark truths that logic alone cannot grasp. Mayhap, in the chambers of the soul, one must ponder: can the abstract be deemed sufficient? Can reason, pure and unsullied, stand apart from the waltz of life’s myriad experiences? For just as the rivers carve the mountains and seasons change the meadows, so too do experiences shape the contours of our beliefs and thoughts. Lo! A balance we must seek, lest we fall into the chasm of dogma where reason becomes a tyranny of its own making. In realms where ‘twixt possible worlds and necessary truths, the ever-spinning wheel of existential inquiry doth turn, one cannot dismiss the dance betwixt the mind and the senses, for in that embrace lies the full measure of human wisdom. Thus, dear interlocutor, let us be wary, and embrace both the luminous heights of rational thought and the grounding embrace of experiential truth alike, for in their union lies the essence of enlightenment, a veritable garden where all manner of knowledge may bloom.