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

Ah, my friends, let us delve into the intricate realm of knowledge, where the rationalist posits that we arrive at understanding not through sensory experiences, which can at times deceive us, but through pure reason, a priori truths that reside within the very fabric of logic itself. Consider the nature of knowledge: is it not a wondrous gift that allows us to grasp the essence of existence? The rationalist would argue that this gift is bestowed upon us transcendentally, unshackling us from the ephemeral and often misleading sensations provided by our senses. For what is a priori knowledge, if not that which is intrinsic, arising from the depths of our rational faculties? It is a knowledge that exists independently of the physical world, a clarity of thought that shines like the sun at noon, illuminating the path of understanding. Herein lies the distinction between rationalism and empiricism—a tension that has rippled through the fabric of philosophical inquiry. Empiricists may claim that the sensory experience is paramount, yet I ask you: can we rely solely upon what our eyes perceive or our fingers touch? Perhaps there is a higher realm of comprehension, where logic reigns supreme, devoid of the clutter of sensory input. Might we then conclude that true knowledge emerges from the interplay of rational thought—the axioms, the principles by which we navigate this world? When we employ the logic of S5, stripping away unnecessary qualifers that clutter our understanding, we distill our propositions down to their essence, revealing the truths that must be true in all possible worlds.

In this light, let us ponder: if indeed something is necessarily true, what does that say about our reliance on experience? The laws of nature, immutable and constant, lend credence to the rationalist view. Our experience of the world may shift like sand under our feet, but the logical truths endure with a rigidity that belies their abstract nature. The axiom that states ‘if it is possible that a proposition necessarily exists, then it necessarily exists’ invites us to explore beyond mere contingencies of experience to realms where results are certain and immutable. A necessary truth cannot falter, for to do so would invite contradiction and chaos into the order of our universe. Thus, as we gather here in this philosophical pursuit, I implore you to consider not just the solidity of experience, but the radiant clarity of reason—an illumination that might reveal not just what is, but what must be, and indeed, what should guide our understanding of this magnificent tapestry we call reality.