Wa `alaykum As-Salamu wa Rahmatullahi wa Barakatuh
In the Name of Allah, the All-Merciful, the Mercy-Giving
All praise is due to Allah. Peace and blessings be upon His Messenger Muhammad.
You should have allowed the dog in to carry out the inspection. Dogs in Islam can be trained to catch games, to perform works such as detection and inspection, and other vital services.
Allah tells us in the Qur’an:
They will ask you as to what is lawful to them. Say: “Lawful to you are all the good things of life.” And as for those hunting animals (including trained dogs: see the note below) which you train by imparting to them something of the knowledge that God has imparted to yourselves-eat of what they seize for you, but mention God’s name over it. and remain conscious of God: verily, God is swift in reckoning.
Note:
Lit., “such of the trained beasts of chase” (min al-jawarih mukallibin). The term mukallib signifies “trained like a [hunting) dog”, and is applied to every animal used for hunting -a hound, a falcon, a cheetah, etc.(Quran Ref: 5:4)
Therefore, you were wrong in not letting the dog do the work of inspection.
Even if the dog was not trained to do so, letting the dog entering the house in itself is not to be deemed to render your apartment filthy. How can assume this when we have reports from the Prophet’s companions that dogs would enter Prophet’s mosque through one door and exit through the other and yet the Prophet did not carry out any cleansing rituals.
In light of these, you have no right to sue the landlord for you were the one who was wrong in not allowing the dog in.
So I don’t think you will go anywhere with your unreasonable demands which you cannot even justify Islamically.
Almighty Allah knows best.
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